{
  "name": "Quantum Information and Quantum Computation Open Problem Zoo",
  "shortName": "QIQCOP Zoo",
  "siteUrl": "https://qiqc-op.com/",
  "repositoryUrl": "https://github.com/Naixu-Guo/quantum-open-problems",
  "generated": "2026-09-08",
  "updated": "2026-09-08",
  "counts": {
    "distinctQuestions": {
      "total": 104,
      "unsolved": 95,
      "solved": 9
    },
    "total": 105,
    "unsolved": 95,
    "solved": 10,
    "fields": 6,
    "topics": 58,
    "references": 452,
    "equations": 356
  },
  "problems": [
    {
      "id": "op_36ac6718d2c37628",
      "ulid": "01M20QZS0HSJKXWFBD1YS9VVTS",
      "aliases": [
        "op_36ac6718d2c37628",
        "01M20QZS0HSJKXWFBD1YS9VVTS",
        "op-36ac6718d2c37628"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T14:50:37.457Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M20H9K0GJGBPXAKZBJ2YEBSH",
          "01M20J3M572VGZM83E2GJ0AG2D"
        ]
      },
      "title": "Polynomial-time quantum algorithm for approximate Shortest Vector Problem",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability"
      ],
      "statement": "Does the polynomial-factor approximate Shortest Vector Problem admit a polynomial-time quantum algorithm?\n\nLet $B\\in\\mathbb Q^{n\\times n}$ be a nonsingular lattice basis whose entries have polynomially bounded bit length, and let\n\n \\begin{equation}\n \\mathcal L(B)=\\{Bz:z\\in\\mathbb Z^n\\}\\subset\\mathbb R^n .\n\\tag{1}\n\\end{equation} \nDefine the length of a shortest nonzero lattice vector by\n\n \\begin{equation}\n \\lambda_1(\\mathcal L)=\\min_{v\\in\\mathcal L\\setminus\\{0\\}}|v|_2 .\n\\tag{2}\n\\end{equation} \nFor an approximation factor $\\gamma=\\gamma(n)\\geq 1$, the $\\gamma$-approximate Shortest Vector Problem asks for a nonzero vector $v\\in\\mathcal L(B)$ satisfying\n\n \\begin{equation}\n |v|_2\\leq\\gamma(n)\\lambda_1(\\mathcal L(B)).\n\\tag{3}\n\\end{equation} \nThe open question is whether, for polynomial approximation factors such as $\\gamma(n)=n^c$ for a fixed constant $c>0$, there exists a bounded-error quantum algorithm that outputs a vector satisfying (3) in time polynomial in the bit length of $B$.",
      "url": "https://qiqc-op.com/problem/op_36ac6718d2c37628/",
      "json": "https://qiqc-op.com/api/problems/op_36ac6718d2c37628.json",
      "tex": "https://qiqc-op.com/problem/op_36ac6718d2c37628/op_36ac6718d2c37628.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T14:54:06.000Z",
      "updatedAt": "2026-09-08T16:06:44.000Z",
      "sha256": "129b1d830f02b8dd9917cff60415f79efde7645c8a58ed44c8d6bd3027ad428d"
    },
    {
      "id": "op_a381e2ccd80cec9c",
      "ulid": "01M208FN774T1EFE5GK2DMR654",
      "aliases": [
        "op_a381e2ccd80cec9c",
        "01M208FN774T1EFE5GK2DMR654",
        "op-a381e2ccd80cec9c"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:19:40.647Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-measures",
          "additivity-and-regularization",
          "quantum-relative-entropy"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME780JCWZMCJMARNSAXH3"
        ]
      },
      "title": "Additivity of the relative entropy of entanglement",
      "status": "Solved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement measures",
        "Additivity and regularization",
        "Quantum relative entropy"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement measures",
        "Additivity and regularization",
        "Quantum relative entropy"
      ],
      "statement": "Does the relative entropy of entanglement of every bipartite state equal its regularization, or is regularization genuinely necessary? For a finite-dimensional bipartite system $A{:}B$, write $\\operatorname{Sep}(A{:}B)$ for the set of separable states and $D(\\rho\\Vert\\sigma)=\\operatorname{Tr}[\\rho(\\log_2\\rho-\\log_2\\sigma)]$ for the Umegaki relative entropy, defined when $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$, with the trace evaluated on $\\operatorname{supp}\\rho$ and $0\\log_2 0:=0$. Define the relative entropy of entanglement and its regularization by\n\n \\begin{equation}\n E_R(\\rho)\n :=\\min_{\\sigma\\in\\operatorname{Sep}(A{:}B)}D(\\rho\\Vert\\sigma),\n \\qquad\n E_R^\\infty(\\rho)\n :=\\lim_{n\\to\\infty}\\frac1nE_R\\bigl(\\rho^{\\otimes n}\\bigr).\n\\tag{1}\n\\end{equation} \nThe limit in Eq. (1) exists and equals $\\inf_{n\\geq1}\\frac1nE_R(\\rho^{\\otimes n})$ by Fekete’s lemma: the product of minimizing separable states is separable and $D$ is additive on tensor products, which gives the subadditivity $E_R(\\rho\\otimes\\sigma)\\leq E_R(\\rho)+E_R(\\sigma)$, and subadditivity implies convergence of the normalized terms to their infimum, not that each of them is nonincreasing. The archived question is whether single copies already suffice, that is, whether\n\n \\begin{equation}\n E_R^\\infty(\\rho)=E_R(\\rho)\n \\quad\\text{for every finite-dimensional bipartite state }\\rho.\n\\tag{2}\n\\end{equation} \nSince $E_R^\\infty(\\rho)\\leq E_R(\\rho)$ always holds, Eq. (2) can only fail strictly, through a single state $\\rho$ with\n\n \\begin{equation}\n E_R^\\infty(\\rho)<E_R(\\rho).\n\\tag{3}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_a381e2ccd80cec9c/",
      "json": "https://qiqc-op.com/api/problems/op_a381e2ccd80cec9c.json",
      "tex": "https://qiqc-op.com/problem/op_a381e2ccd80cec9c/op_a381e2ccd80cec9c.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:23:02.000Z",
      "updatedAt": "2026-09-08T15:38:43.000Z",
      "sha256": "255bd83ef26d5ed84c69fb2442f08a3b4e3268c8f2730f2513752d30426b8044"
    },
    {
      "id": "op_ac5fa4581b04f4c2",
      "ulid": "01M208AZBKSZTSQ6KS8YV38RFK",
      "aliases": [
        "op_ac5fa4581b04f4c2",
        "01M208AZBKSZTSQ6KS8YV38RFK",
        "op-ac5fa4581b04f4c2"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:17:07.187Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-measures",
          "additivity-and-regularization"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1Q787QRSHGZH7NDSMFG88GH"
        ]
      },
      "title": "Additivity of the entanglement of purification",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement measures",
        "Additivity and regularization"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement measures",
        "Additivity and regularization"
      ],
      "statement": "Is the entanglement of purification additive on tensor products? For a bipartite density operator $\\rho_{AB}$ on finite-dimensional Hilbert spaces $A$ and $B$, define its entanglement of purification by\n\n \\begin{equation}\n E_P(A{:}B)_\\rho\n :=\\min_{\\psi}\\ S(AA')_\\psi,\n\\tag{1}\n\\end{equation} \nwhere the minimum ranges over all pure states $\\psi$ on $AA'BB'$ with purifying systems $A'$, $B'$ of arbitrary finite dimension such that $\\operatorname{Tr}_{A'B'}\\psi=\\rho_{AB}$, and $S$ denotes the von Neumann entropy. Taking the product of optimal purifications gives $E_P(\\rho\\otimes\\sigma)\\leq E_P(\\rho)+E_P(\\sigma)$, so by Fekete’s lemma for subadditive sequences the normalized values $\\frac1nE_P(\\rho^{\\otimes n})$ converge to their infimum rather than decrease monotonically, and the regularization\n\n \\begin{equation}\n E_P^\\infty(\\rho)\n :=\\lim_{n\\to\\infty}\\frac1nE_P\\bigl(\\rho^{\\otimes n}\\bigr)\n =\\inf_{n\\geq1}\\frac1nE_P\\bigl(\\rho^{\\otimes n}\\bigr)\n\\tag{2}\n\\end{equation} \nis well defined. The question is whether regularization is unnecessary: does\n\n \\begin{equation}\n E_P(\\rho\\otimes\\sigma)=E_P(\\rho)+E_P(\\sigma)\n \\quad\\text{for all finite-dimensional bipartite }\\rho\\text{ and }\\sigma,\n\\tag{3}\n\\end{equation} \nhold? Applied inductively to the pairs $(\\rho,\\rho^{\\otimes(n-1)})$, Eq. (3) would give $E_P(\\rho^{\\otimes n})=nE_P(\\rho)$ and hence make the regularization of Eq. (2) collapse to $E_P^\\infty(\\rho)=E_P(\\rho)$ for every $\\rho$; whether such collapse on single-state tensor powers would already force Eq. (3) for arbitrary pairs is not known. Since Eq. (3) can only fail through strict subadditivity for some pair, the simplest potential witness is\n\n \\begin{equation}\n E_P\\bigl(\\rho^{\\otimes2}\\bigr)<2E_P(\\rho),\n\\tag{4}\n\\end{equation} \na violating pair with $\\sigma=\\rho$; no reduction from a violation with distinct $\\rho\\neq\\sigma$ to such a same-state violation is known. The sharpest openly posed subcase concerns two-qubit classical states $\\rho=\\sum_{i,j\\in\\{0,1\\}}p_{ij}|ij\\rangle\\langle ij|$ at the von Neumann order: for this family the Rényi generalizations of Eq. (1) are settled non-additive for every order $\\alpha\\in[0,1)$ and additive for every $\\alpha\\in[2,\\infty]$, while the interval $\\alpha\\in[1,2)$, which includes the von Neumann case $\\alpha=1$ of Eqs. (3)–(4), remains open.",
      "url": "https://qiqc-op.com/problem/op_ac5fa4581b04f4c2/",
      "json": "https://qiqc-op.com/api/problems/op_ac5fa4581b04f4c2.json",
      "tex": "https://qiqc-op.com/problem/op_ac5fa4581b04f4c2/op_ac5fa4581b04f4c2.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:19:32.000Z",
      "updatedAt": "2026-09-08T15:38:43.000Z",
      "sha256": "15ef09074d7e5b0ff985941542f20ef281300ce44ad70a7ba758d2faee9d83bc"
    },
    {
      "id": "op_e724615844c52297",
      "ulid": "01M207QTTXDG3NHDWTRA9R0203",
      "aliases": [
        "op_e724615844c52297",
        "01M207QTTXDG3NHDWTRA9R0203",
        "op-e724615844c52297"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:06:39.965Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "derived",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "additivity-and-regularization",
          "matrix-and-entropy-inequalities",
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME78010TTEQK6NFPRCGZT",
          "01M202HP8CMK4SGZZE0FHGAXC1"
        ]
      },
      "title": "Smallest output dimension violating minimum output entropy additivity",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Additivity and regularization",
        "Matrix and entropy inequalities",
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Communication",
        "Additivity and regularization",
        "Matrix and entropy inequalities",
        "Quantum channel structure"
      ],
      "statement": "What is the smallest output dimension in which the minimum output von Neumann entropy of quantum channels fails to be additive? For a density operator $\\sigma$ on a finite-dimensional Hilbert space write $H(\\sigma)=-\\operatorname{Tr}(\\sigma\\log_2\\sigma)$ for its von Neumann entropy, and for a quantum channel $\\Phi$ define its minimum output entropy by\n\n \\begin{equation}\n S_{\\min}(\\Phi):=\\min_{\\rho\\in\\mathcal D(A)}H\\bigl(\\Phi(\\rho)\\bigr),\n\\tag{1}\n\\end{equation} \nwhere $\\mathcal D(A)$ is the set of density operators on the input space $A$ of $\\Phi$. A product input is always available in $\\Phi\\otimes\\Psi$, so Eq. (1) gives $S_{\\min}(\\Phi\\otimes\\Psi)\\leq S_{\\min}(\\Phi)+S_{\\min}(\\Psi)$ for any two channels; the phenomenon at issue is the strict inequality\n\n \\begin{equation}\n S_{\\min}(\\Phi\\otimes\\Psi)<S_{\\min}(\\Phi)+S_{\\min}(\\Psi),\n\\tag{2}\n\\end{equation} \nand a pair $(\\Phi,\\Psi)$ of channels satisfying Eq. (2) is called a violation. Define the output-dimension threshold\n\n \\begin{equation}\n d_{\\min}:=\\min\\bigl\\{d\\in\\mathbb N:\\ \\text{some violation }(\\Phi,\\Psi)\n \\text{ has both output spaces of dimension at most }d\\bigr\\},\n\\tag{3}\n\\end{equation} \nwith input and environment dimensions arbitrary. Determine the exact value of $d_{\\min}$ in Eq. (3). The companion constructive target, also open at every output dimension, is a practically computable, explicitly presented pair of finite-dimensional channels with a certified instance of Eq. (2): deterministic asymptotic algorithms now output violating pairs once their size parameter is sufficiently large, but none supplies a practically computable finite-dimensional instance.",
      "url": "https://qiqc-op.com/problem/op_e724615844c52297/",
      "json": "https://qiqc-op.com/api/problems/op_e724615844c52297.json",
      "tex": "https://qiqc-op.com/problem/op_e724615844c52297/op_e724615844c52297.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:16:57.000Z",
      "updatedAt": "2026-09-08T15:38:43.000Z",
      "sha256": "19bfa5fa477ea12207f336644f01f66200d08cfe0f2aa164066d300015e3bf1a"
    },
    {
      "id": "op_61cc0e261b3889c6",
      "ulid": "01M208JBB0XT10DQ8KTKMYPTPH",
      "aliases": [
        "op_61cc0e261b3889c6",
        "01M208JBB0XT10DQ8KTKMYPTPH",
        "op-61cc0e261b3889c6"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:21:08.832Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": "2001",
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability",
          "quantum-separability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME780S5JZKCQN8X0RR8TG",
          "01M1HME78078BW0JG3X252BVX5"
        ]
      },
      "title": "QMA(2) versus QMA",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability",
        "Quantum separability"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability",
        "Quantum separability"
      ],
      "statement": "Is every promise problem verifiable by a quantum Merlin-Arthur protocol with two unentangled witnesses also verifiable by a protocol with a single arbitrary witness, that is, is $\\mathsf{QMA}(2) = \\mathsf{QMA}$ in the standard, unrelativized setting? A promise problem $L = (L_{yes}, L_{no})$ is in $\\mathsf{QMA}(2)$ if some uniform polynomial-time quantum verifier, given an input $x$ of length $n$ and two witnesses of $\\operatorname{poly}(n)$ qubits each that are promised to be unentangled across the two registers, accepts some product state $\\sigma_1 \\otimes \\sigma_2$ with probability at least $2/3$ for every $x \\in L_{yes}$ and every product state $\\tau_1 \\otimes \\tau_2$ with probability at most $1/3$ for every $x \\in L_{no}$; $\\mathsf{QMA}$ is the same model with a single, unrestricted witness. Discarding one witness gives $\\mathsf{QMA} \\subseteq \\mathsf{QMA}(2)$, and nondeterministically guessing classical descriptions of the two witnesses and simulating the verifier gives $\\mathsf{QMA}(2) \\subseteq \\mathsf{NEXP}$, so\n\n \\begin{equation}\n \\mathsf{QMA} \\subseteq \\mathsf{QMA}(2) \\subseteq \\mathsf{NEXP},\n\\tag{1}\n\\end{equation} \nwhile the SWAP-test reduction of Harrow and Montanaro gives $\\mathsf{QMA}(k) = \\mathsf{QMA}(2)$ for every constant $k \\ge 2$. The open question is whether the first inclusion in Eq. (1) is an equality: can every protocol whose soundness is required only against separable pairs of witnesses be simulated by a single-witness protocol with polynomial overhead? No improvement of either containment in Eq. (1) is known in the unrelativized setting, and relativized evidence such as an oracle separation does not settle this question.",
      "url": "https://qiqc-op.com/problem/op_61cc0e261b3889c6/",
      "json": "https://qiqc-op.com/api/problems/op_61cc0e261b3889c6.json",
      "tex": "https://qiqc-op.com/problem/op_61cc0e261b3889c6/op_61cc0e261b3889c6.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:22:31.000Z",
      "updatedAt": "2026-09-08T15:36:03.000Z",
      "sha256": "702a86b775822c4f05fd18836b39ac00b859adc93558fac4a3fa00ee0a78109d"
    },
    {
      "id": "op_37f7b003ec78a028",
      "ulid": "01M208CJ9MKVF4X0B83M9V7QZM",
      "aliases": [
        "op_37f7b003ec78a028",
        "01M208CJ9MKVF4X0B83M9V7QZM",
        "op-37f7b003ec78a028"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:17:59.348Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": "2016",
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "quantum-magic",
          "quantum-circuit-complexity",
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1Q787QRD6APNHX659G4CTEF"
        ]
      },
      "title": "Asymptotic growth of the stabilizer rank of T-state tensor powers",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Quantum magic",
        "Quantum circuit complexity",
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Quantum magic",
        "Quantum circuit complexity",
        "Computational complexity and computability"
      ],
      "statement": "Does the exact stabilizer rank of the tensor powers of the single-qubit magic state $|T\\rangle$ grow polynomially in the number of copies, or is it not polynomially bounded? The state is\n\n \\begin{equation}\n |T\\rangle := 2^{-1/2}\\bigl(|0\\rangle + e^{i\\pi/4}|1\\rangle\\bigr),\n\\tag{1}\n\\end{equation} \nand, for an $n$-qubit pure state $|\\psi\\rangle$, its exact stabilizer rank is\n\n \\begin{equation}\n \\chi(\\psi) := \\min\\Bigl\\{r : |\\psi\\rangle = \\sum_{j=1}^{r} c_j\\,|\\varphi_j\\rangle\n \\text{ for some } c_j \\in \\mathbb{C} \\text{ and } n\\text{-qubit stabilizer states }\n |\\varphi_j\\rangle\\Bigr\\},\n\\tag{2}\n\\end{equation} \nwhere an $n$-qubit stabilizer state is the common $+1$ eigenstate of $n$ independent commuting Hermitian Pauli operators. Which of the two alternatives holds for $\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr)$: does there exist a constant $k$ with $\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr) \\le n^{k}$ for all sufficiently large $n$, or is $\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr)$ not bounded by any polynomial in $n$? The best current bounds are\n\n \\begin{equation}\n \\Omega\\!\\left(\\frac{n^{2}}{\\operatorname{polylog} n}\\right) \\le\n \\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr) \\le O\\bigl(2^{\\alpha n}\\bigr),\n \\qquad \\alpha = \\tfrac{1}{4}\\log_{2} 3 \\le 0.3963,\n\\tag{3}\n\\end{equation} \nwhich leave both alternatives open. A sharper sub-question is whether the exponential rate \\(\\gamma = \\lim_{n\\to\\infty} \\frac{1}{n}\\log_{2}\n\\chi\\bigl(|T\\rangle^{\\otimes n}\\bigr)\\), which exists by sub-multiplicativity of the rank under tensor product and Fekete’s lemma, is strictly positive, since Eq. (3) only pins $\\gamma$ to the interval $[0, 0.3963]$. The small-copy frontier of the quantity in Eq. (2) on copies of the state in Eq. (1) is also open: the exact values $\\chi\\bigl(|T\\rangle^{\\otimes 2}\\bigr) = 2$ and $\\chi\\bigl(|T\\rangle^{\\otimes 3}\\bigr) = 3$ are proved, while at four through seven copies only upper bounds are known, $\\chi\\bigl(|T\\rangle^{\\otimes 4}\\bigr) \\le 4$, $\\chi\\bigl(|T\\rangle^{\\otimes 5}\\bigr) \\le 6$, $\\chi\\bigl(|T\\rangle^{\\otimes 6}\\bigr) \\le 6$, and $\\chi\\bigl(|T\\rangle^{\\otimes 7}\\bigr) \\le 12$, the four-copy bound having since been certified as the exact value \\(\\chi\\bigl(|T\\rangle^{\\otimes\n4}\\bigr) = 4\\) by an exhaustive computer search; determine the exact values at $t = 5$, $6$, and $7$.",
      "url": "https://qiqc-op.com/problem/op_37f7b003ec78a028/",
      "json": "https://qiqc-op.com/api/problems/op_37f7b003ec78a028.json",
      "tex": "https://qiqc-op.com/problem/op_37f7b003ec78a028/op_37f7b003ec78a028.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:20:59.000Z",
      "updatedAt": "2026-09-08T15:36:03.000Z",
      "sha256": "80cf4390aecb53eb618b735716dc03baf272fe17fb25b3127fd031e219903c93"
    },
    {
      "id": "op_291a943fdec5bd1d",
      "ulid": "01M20868F4GEPX6FBJZF8T0GRJ",
      "aliases": [
        "op_291a943fdec5bd1d",
        "01M20868F4GEPX6FBJZF8T0GRJ",
        "op-291a943fdec5bd1d"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:14:32.676Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "derived",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-communication"
        ],
        "topicIds": [
          "entanglement-measures",
          "quantum-capacity",
          "private-capacity"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME7809M71BG24CSMYKA8A",
          "01M1HME780NTMHKFB95TSXKKFW",
          "01M20868D5Z2B0W2KKC7QGRGRR"
        ]
      },
      "title": "Squashed entanglement of the qubit depolarizing channel",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum Communication"
      ],
      "topics": [
        "Entanglement measures",
        "Quantum capacity",
        "Private capacity"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum Communication",
        "Entanglement measures",
        "Quantum capacity",
        "Private capacity"
      ],
      "statement": "What is the squashed entanglement $E_{\\mathrm{sq}}(\\Lambda_p)$ of the qubit depolarizing channel, defined for a mixing parameter $p\\in[0,1]$ by\n\n \\begin{equation}\n \\Lambda_p(\\rho):=(1-p)\\,\\rho+p\\,\\mathrm{Tr}[\\rho]\\,\\frac{I}{2}\n =\\bigl(1-\\tfrac{3p}{4}\\bigr)\\rho\n +\\tfrac{p}{4}\\bigl(X\\rho X+Y\\rho Y+Z\\rho Z\\bigr),\n\\tag{1}\n\\end{equation} \nwhere $\\rho$ ranges over qubit states, $I$ is the qubit identity, and $X,Y,Z$ are the Pauli matrices, so that the total Pauli error probability, read off from the second expression in Eq. (1), is $\\tfrac{3p}{4}$? Let $U:A'\\to BE$ be an isometric extension of $\\Lambda_p$, let $\\varphi_{RA'}$ range over pure bipartite input states, and let $S:E\\to E'$ range over completely positive trace-preserving maps of arbitrary finite output dimension. For \\(\\omega_{RBE'}:=(\\mathrm{id}_{RB}\\otimes S)\\bigl[(\\mathrm{id}_R\\otimes\nU)\\,\\varphi_{RA'}\\,(\\mathrm{id}_R\\otimes U^\\dagger)\\bigr]\\) the squashed entanglement of the channel is\n\n \\begin{equation}\n E_{\\mathrm{sq}}(\\Lambda_p)\n :=\\max_{\\varphi_{RA'}}\\ \\frac{1}{2}\\inf_{S}I(R;B|E')_\\omega,\n\\tag{2}\n\\end{equation} \nwhere $I(R;B|E')_\\omega$ is the conditional mutual information and all logarithms are base 2 [TGW14]. The quantity in Eq. (2) is additive over tensor products of channels and upper-bounds the two-way-assisted quantum and private capacities, $Q_2(\\Lambda_p)\\le E_{\\mathrm{sq}}(\\Lambda_p)$ and $P_2(\\Lambda_p)\\le E_{\\mathrm{sq}}(\\Lambda_p)$ [TGW14]; this is its operational role, since an exact value would be a single-letter upper bound on secret and quantum transmission over depolarizing noise. Determine $E_{\\mathrm{sq}}(\\Lambda_p)$ as an explicit function of $p$ on $[0,1]$.",
      "url": "https://qiqc-op.com/problem/op_291a943fdec5bd1d/",
      "json": "https://qiqc-op.com/api/problems/op_291a943fdec5bd1d.json",
      "tex": "https://qiqc-op.com/problem/op_291a943fdec5bd1d/op_291a943fdec5bd1d.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:25:04.000Z",
      "updatedAt": "2026-09-08T15:31:47.000Z",
      "sha256": "bc4d1ae69362ab8cd4ce71914e822131e8a64895d0c7068680cb0302a2ff2820"
    },
    {
      "id": "op_d2813fe3fcdf09ad",
      "ulid": "01M20868D5Z2B0W2KKC7QGRGRR",
      "aliases": [
        "op_d2813fe3fcdf09ad",
        "01M20868D5Z2B0W2KKC7QGRGRR",
        "op-d2813fe3fcdf09ad"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T10:14:32.613Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-cryptography"
        ],
        "topicIds": [
          "private-capacity",
          "additivity-and-regularization"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME7809M71BG24CSMYKA8A",
          "01M1Q787QR8FTR00QF4PMHHWPE",
          "01M20868F4GEPX6FBJZF8T0GRJ"
        ]
      },
      "title": "Private capacity of the qubit depolarizing channel",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum Cryptography"
      ],
      "topics": [
        "Private capacity",
        "Additivity and regularization"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Cryptography",
        "Private capacity",
        "Additivity and regularization"
      ],
      "statement": "What is the private classical capacity $P(\\Lambda_p)$ of the qubit depolarizing channel, defined for a mixing parameter $p\\in[0,1]$ by\n\n \\begin{equation}\n \\Lambda_p(\\rho):=(1-p)\\,\\rho+p\\,\\mathrm{Tr}[\\rho]\\,\\frac{I}{2}\n =\\bigl(1-\\tfrac{3p}{4}\\bigr)\\rho\n +\\tfrac{p}{4}\\bigl(X\\rho X+Y\\rho Y+Z\\rho Z\\bigr),\n\\tag{1}\n\\end{equation} \nwhere $\\rho$ ranges over qubit states, $I$ is the qubit identity, and $X,Y,Z$ are the Pauli matrices? The second expression in Eq. (1) exhibits the Pauli error probabilities \\(\\bigl(p_I,p_X,p_Y,p_Z\\bigr)=\\bigl(1-\\tfrac{3p}{4},\\tfrac{p}{4},\\tfrac{p}{4},\n\\tfrac{p}{4}\\bigr)\\), with total Pauli error probability $\\tfrac{3p}{4}$. Fix a Stinespring isometry $V:A\\to BE$ of $\\Lambda_p$ and a finite ensemble $\\{q_x,\\rho_x\\}_x$ of qubit states, and set $\\omega^{XBE}:=\\sum_x q_x\\,|x\\rangle\\!\\langle x|^X\\otimes V\\rho_xV^\\dagger$. The one-shot private information and the private classical capacity of the channel are\n\n \\begin{equation}\n P^{(1)}(\\Lambda_p):=\\max_{\\{q_x,\\rho_x\\}}\n \\bigl[I(X;B)_\\omega-I(X;E)_\\omega\\bigr],\n \\qquad\n P(\\Lambda_p):=\\sup_{n\\ge1}\\frac{1}{n}\\,\n P^{(1)}\\bigl(\\Lambda_p^{\\otimes n}\\bigr),\n\\tag{2}\n\\end{equation} \nwhere $I(\\cdot\\,;\\cdot)_\\omega$ is the quantum mutual information and all logarithms are base 2 [Dev05], [CWY04]. Determine the value of $P(\\Lambda_p)$ in Eq. (2) as an explicit function of $p$ on $[0,1]$. For context, every channel satisfies $P\\ge\\mathcal Q$, where $\\mathcal Q$ denotes the quantum capacity, the supremum of achievable coherent-information rates: for any input state $\\rho=\\sum_x q_x\\,\\psi_x$ decomposed into an ensemble of pure states, each $V\\psi_xV^\\dagger$ is pure on $BE$, so its $B$ and $E$ marginals have equal entropy, the conditional-entropy terms cancel in $I(X;B)_\\omega-I(X;E)_\\omega$, and the private information of the ensemble equals the coherent information $S(\\Lambda_p(\\rho))-S(\\Lambda_p^{c}(\\rho))$ at $\\rho$, where $S(\\cdot)$ is the von Neumann entropy and $\\Lambda_p^{c}(\\sigma):=\\mathrm{Tr}_B[V\\sigma V^\\dagger]$ is the complementary channel; the identity holds verbatim for every tensor power, so the private-information maximum dominates the coherent-information maximum for the channel and all its powers, giving $P\\ge\\mathcal Q$ [Dev05]. Whether $P(\\Lambda_p)$ and $\\mathcal Q(\\Lambda_p)$ differ anywhere on $0<p<\\tfrac{1}{3}$, i.e. whether an ensemble with a nontrivial classical index register strictly beats every pure-state-ensemble rate on this channel, is part of what is open.",
      "url": "https://qiqc-op.com/problem/op_d2813fe3fcdf09ad/",
      "json": "https://qiqc-op.com/api/problems/op_d2813fe3fcdf09ad.json",
      "tex": "https://qiqc-op.com/problem/op_d2813fe3fcdf09ad/op_d2813fe3fcdf09ad.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T10:24:52.000Z",
      "updatedAt": "2026-09-08T15:31:47.000Z",
      "sha256": "a46cf94047fa7c68a0df54366c4dc4f027e15b808d3e97ddb69bfb067264f6ef"
    },
    {
      "id": "op_3596f8c955c66d94",
      "ulid": "01M20H9K0GJGBPXAKZBJ2YEBSH",
      "aliases": [
        "op_3596f8c955c66d94",
        "01M20H9K0GJGBPXAKZBJ2YEBSH",
        "op-3596f8c955c66d94"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T12:53:38.960Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M20J3M572VGZM83E2GJ0AG2D",
          "01M20QZS0HSJKXWFBD1YS9VVTS"
        ]
      },
      "title": "Polynomial-time quantum algorithm for Learning With Errors",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability"
      ],
      "statement": "Does the Learning With Errors problem in its standard worst-case-hard parameter regime admit a polynomial-time quantum algorithm?\n\nLet $n$ be the dimension, let $q=q(n)\\geq 2$ be an integer modulus, and let $\\alpha=\\alpha(n)\\in(0,1)$ be a noise rate. A search-LWE instance is generated by choosing a secret $s\\in\\mathbb Z_q^n$ uniformly at random and providing polynomially many independent samples\n\n \\begin{equation}\n (a_i,b_i)\\in\\mathbb Z_q^n\\times\\mathbb Z_q,\n \\qquad\n b_i=\\langle a_i,s\\rangle+e_i\\pmod q,\n\\tag{1}\n\\end{equation} \nwhere each $a_i$ is uniform in $\\mathbb Z_q^n$, and each error $e_i$ is sampled independently from a discrete Gaussian of width $\\alpha q$. For $x\\in\\mathbb Z$,\n\n \\begin{equation}\n \\Pr[e_i=x]=\\frac{\\exp\\left(-\\pi x^2/(\\alpha q)^2\\right)}\n {\\sum_{z\\in\\mathbb Z}\\exp\\left(-\\pi z^2/(\\alpha q)^2\\right)} .\n\\tag{2}\n\\end{equation} \nThe task is to recover the secret $s$ from samples satisfying (1). The open question is whether, for standard parameter families for which LWE has worst-case lattice-hardness guarantees, there exists a bounded-error quantum algorithm whose running time is polynomial in $n$ and $\\log q$.",
      "url": "https://qiqc-op.com/problem/op_3596f8c955c66d94/",
      "json": "https://qiqc-op.com/api/problems/op_3596f8c955c66d94.json",
      "tex": "https://qiqc-op.com/problem/op_3596f8c955c66d94/op_3596f8c955c66d94.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:57:29.000Z",
      "updatedAt": "2026-09-08T15:21:28.000Z",
      "sha256": "2c89e62bea1936b8b5f5668b16bd4a7942b166871e3cc578dbfa5df52422b56d"
    },
    {
      "id": "op_95fccb9df34a08b1",
      "ulid": "01M20F505SW8RJ6Z0WK8B31TEH",
      "aliases": [
        "op_95fccb9df34a08b1",
        "01M20F505SW8RJ6Z0WK8B31TEH",
        "op-95fccb9df34a08b1"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T12:16:11.449Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "quantum-circuit-complexity"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Parity is not in QAC0",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Quantum circuit complexity"
      ],
      "tags": [
        "Quantum algorithm",
        "Quantum circuit complexity"
      ],
      "statement": "Can polynomial-size constant-depth $\\mathsf{QAC}^0$ circuits compute the parity function, or equivalently, is $\\mathrm{PARITY}\\notin\\mathsf{QAC}^0$?\n\nFor $x=(x_1,\\ldots,x_n)\\in\\{0,1\\}^n$, define\n\n \\begin{equation}\n \\mathrm{PARITY}_n(x)=x_1\\oplus x_2\\oplus\\cdots\\oplus x_n.\n\\tag{1}\n\\end{equation} \nA $\\mathsf{QAC}^0$ circuit family consists of polynomial-size quantum circuits of constant depth built from arbitrary single-qubit gates and generalized Toffoli gates, with polynomially many ancilla qubits initialized to a fixed computational-basis state.\n\nThe conjecture is that there do not exist a constant $d$, a polynomial $p$, and a family of depth-at-most-$d$ $\\mathsf{QAC}^0$ circuits $C_n$ of size and ancilla count at most $p(n)$ such that, for every $n$ and every $x\\in\\{0,1\\}^n$, measurement of a designated output qubit gives\n\n \\begin{equation}\n \\Pr\\bigl[\\,C_n(x)\\text{ outputs }\\mathrm{PARITY}_n(x)\\,\\bigr]=1.\n\\tag{2}\n\\end{equation} \nEquivalently, the conjecture asserts that no polynomial-size constant-depth $\\mathsf{QAC}^0$ family satisfies (2) for the function defined in (1).",
      "url": "https://qiqc-op.com/problem/op_95fccb9df34a08b1/",
      "json": "https://qiqc-op.com/api/problems/op_95fccb9df34a08b1.json",
      "tex": "https://qiqc-op.com/problem/op_95fccb9df34a08b1/op_95fccb9df34a08b1.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:20:35.000Z",
      "updatedAt": "2026-09-08T15:21:28.000Z",
      "sha256": "93a6db75380b5c57e763d92db98ac38d9ab9f281fef2f1d09c72bb3eafd8c23a"
    },
    {
      "id": "op_90a05e57e44c086e",
      "ulid": "01M20J3M572VGZM83E2GJ0AG2D",
      "aliases": [
        "op_90a05e57e44c086e",
        "01M20J3M572VGZM83E2GJ0AG2D",
        "op-90a05e57e44c086e"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T13:07:52.103Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M20H9K0GJGBPXAKZBJ2YEBSH",
          "01M20QZS0HSJKXWFBD1YS9VVTS"
        ]
      },
      "title": "Polynomial-time quantum algorithm for the Dihedral Hidden Subgroup Problem",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability"
      ],
      "statement": "Does the Dihedral Hidden Subgroup Problem admit a quantum algorithm whose running time is polynomial in the input length?\n\nFor a positive integer $N$, let the dihedral group be\n\n \\begin{equation}\n D_N=\\langle r,s\\mid r^N=1,\\ s^2=1,\\ srs=r^{-1}\\rangle ,\n\\tag{1}\n\\end{equation} \nso that $D_N$ has order $2N$. Consider an oracle $f:D_N\\to X$ promised to hide a subgroup generated by an unknown reflection. Equivalently, for an unknown $a\\in\\mathbb Z_N$, let\n\n \\begin{equation}\n H_a=\\{1,sr^a\\},\n \\qquad\n f(x)=f(y)\\iff xH_a=yH_a .\n\\tag{2}\n\\end{equation} \nGiven quantum oracle access to $f$, the task is to recover $a$, and hence $H_a$ in (2). The open question is whether this can be done with bounded error using a number of elementary quantum operations polynomial in $\\log N$, where $D_N$ is defined by (1).",
      "url": "https://qiqc-op.com/problem/op_90a05e57e44c086e/",
      "json": "https://qiqc-op.com/api/problems/op_90a05e57e44c086e.json",
      "tex": "https://qiqc-op.com/problem/op_90a05e57e44c086e/op_90a05e57e44c086e.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T13:12:36.000Z",
      "updatedAt": "2026-09-08T13:12:36.000Z",
      "sha256": "16795972e5790df0b4480c6a7f8559d0f09cc4adb727154fa7c8c9f2040df03e"
    },
    {
      "id": "op_25e23d6e92ebce9d",
      "ulid": "01M202HP8CMK4SGZZE0FHGAXC1",
      "aliases": [
        "op_25e23d6e92ebce9d",
        "01M202HP8CMK4SGZZE0FHGAXC1",
        "op-25e23d6e92ebce9d"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T08:35:55.788Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "classical-capacity",
          "additivity-and-regularization",
          "entanglement-measures"
        ],
        "keywords": [
          "entanglement of formation",
          "Holevo capacity",
          "classical capacity",
          "minimum output entropy",
          "additivity",
          "explicit counterexample"
        ],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME78010TTEQK6NFPRCGZT",
          "01M1HME780XSRZ7K9HQJSZ176R"
        ]
      },
      "title": "Closed-form nonadditivity of the Holevo capacity and entanglement of formation",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Classical capacity",
        "Additivity and regularization",
        "Entanglement measures"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Resource Theory",
        "Classical capacity",
        "Additivity and regularization",
        "Entanglement measures"
      ],
      "statement": "Construct a simple closed-form or practically computable counterexample to additivity of the Holevo capacity or the entanglement of formation, together with a rigorous certificate of a strict violation.\n\nFor a density operator $\\rho$, write $S(\\rho)=-\\operatorname{Tr}\\rho\\log_2\\rho$. For a bipartite state $\\rho_{AB}$, the entanglement of formation is\n\n \\begin{equation}\n E_F(\\rho_{AB})=\\inf_{\\{p_x,\\psi_x\\}}\\ \\sum_x p_x\\, S\\big(\\operatorname{Tr}_B\\lvert\\psi_x\\rangle\\!\\langle\\psi_x\\rvert\\big),\n\\tag{1}\n\\end{equation} \nwhere the infimum runs over pure-state ensembles with $\\sum_x p_x\\lvert\\psi_x\\rangle\\!\\langle\\psi_x\\rvert=\\rho_{AB}$; and for a channel $\\mathcal N$, the Holevo capacity is\n\n \\begin{equation}\n \\chi(\\mathcal N)=\\sup_{\\{p_x,\\rho_x\\}}\\ \\Big[ S\\Big(\\sum_x p_x\\,\\mathcal N(\\rho_x)\\Big)-\\sum_x p_x\\,S\\big(\\mathcal N(\\rho_x)\\big)\\Big],\n\\tag{2}\n\\end{equation} \nwhere the supremum runs over finite input ensembles. The target is at least one of the inequalities\n\n \\begin{equation}\n E_F(\\rho\\otimes\\sigma)<E_F(\\rho)+E_F(\\sigma)\n \\qquad\\text{and}\\qquad\n \\chi(\\mathcal N_1\\otimes\\mathcal N_2)>\\chi(\\mathcal N_1)+\\chi(\\mathcal N_2),\n\\tag{3}\n\\end{equation} \nA solution must specify the finite states or channel operators completely, give their dimensions, and certify the corresponding inequality in (3) for the quantities defined in (1) and (2). For example, a certificate may combine rigorous one-copy bounds with an explicit two-copy input or ensemble. A simple closed-form construction must include a proof; a computational construction must include the actual instance and reproducible error bounds. An asymptotic algorithm or an existence proof alone does not supply such an instance. No universal upper bound on the witness dimensions is imposed.\n\nThe two formulations share the universal additivity equivalences discussed in Progress. Any use of a reduction to produce a witness must also specify the resulting data and certificate; the equivalence alone does not guarantee practical size. Counterexamples for minimum output Rényi entropy at orders $p\\ne1$ do not answer this question.",
      "url": "https://qiqc-op.com/problem/op_25e23d6e92ebce9d/",
      "json": "https://qiqc-op.com/api/problems/op_25e23d6e92ebce9d.json",
      "tex": "https://qiqc-op.com/problem/op_25e23d6e92ebce9d/op_25e23d6e92ebce9d.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T08:42:45.000Z",
      "updatedAt": "2026-09-08T13:06:10.000Z",
      "sha256": "d40a1a9457220043cc2062fcc7e8027b98ea92110c3beea88ad838a69f542240"
    },
    {
      "id": "op_69520395226dc45a",
      "ulid": "01M1HME780FM4P69SQZX74G5NE",
      "aliases": [
        "op_69520395226dc45a",
        "01M1HME780FM4P69SQZX74G5NE",
        "op-69520395226dc45a",
        "v2-the-ppt-squared-conjecture",
        "open-problem-v2-problem-32"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 3,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-separability",
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "The PPT-squared conjecture",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum Communication"
      ],
      "topics": [
        "Quantum separability",
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum Communication",
        "Quantum separability",
        "Quantum channel structure"
      ],
      "statement": "Must the composition of any two compatible PPT completely positive maps be entanglement breaking? Let $\\Phi:M_{d_1}(\\mathbb C)\\to M_{d_2}(\\mathbb C)$ and $\\Psi:M_{d_2}(\\mathbb C)\\to M_{d_3}(\\mathbb C)$ be completely positive. The Choi operator of $\\Phi$ is\n\n \\begin{equation}\n J(\\Phi)\n :=\\sum_{i,j=1}^{d_1}\\lvert i\\rangle\\!\\langle j\\rvert\n \\otimes\\Phi(\\lvert i\\rangle\\!\\langle j\\rvert),\n\\tag{1}\n\\end{equation} \nand $J(\\Psi)$ is defined analogously. In the convention of Eq. (1), a map $\\Phi$ is PPT when\n\n \\begin{equation}\n (T\\otimes\\operatorname{id})\\bigl(J(\\Phi)\\bigr)\\succeq0,\n\\tag{2}\n\\end{equation} \nand it is entanglement breaking when $J(\\Phi)$ is separable; the same definitions apply to $\\Psi$. Under condition Eq. (2) for both maps, is $J(\\Psi\\circ\\Phi)$ necessarily separable? Equivalently, does postselection on any joint measurement outcome on the middle systems of two PPT bipartite states always leave a separable state on the two outer systems?",
      "url": "https://qiqc-op.com/problem/op_69520395226dc45a/",
      "json": "https://qiqc-op.com/api/problems/op_69520395226dc45a.json",
      "tex": "https://qiqc-op.com/problem/op_69520395226dc45a/op_69520395226dc45a.tex",
      "created": "2026-09-01",
      "updated": "2026-09-08",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-08T12:46:16.000Z",
      "sha256": "d6e8c19a03625d2a4d96408db91f29eb39d3db29d60f3582f5b9c11a46d060ea"
    },
    {
      "id": "op_1576e5f52f7ff3b2",
      "ulid": "01M20E31B526DAX81WYGMDD01T",
      "aliases": [
        "op_1576e5f52f7ff3b2",
        "01M20E31B526DAX81WYGMDD01T",
        "op-1576e5f52f7ff3b2"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T11:57:38.533Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability",
          "hamiltonian-complexity",
          "quantum-max-cut"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Is bipartite Quantum Max-Cut in BPP?",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability",
        "Hamiltonian complexity",
        "Quantum Max-Cut"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability",
        "Hamiltonian complexity",
        "Quantum Max-Cut"
      ],
      "statement": "Is the following bipartite Quantum Max-Cut promise problem in $\\mathrm{BPP}$? Given a bipartite graph $G=(V,E)$ with $|V|=n$, polynomially bounded nonnegative rational weights $w_{ij}$, and rational thresholds $a<b$ with $b-a\\geq1/\\operatorname{poly}(n)$, define\n\n \\begin{equation}\n H_G:=\\sum_{\\{i,j\\}\\in E}w_{ij}(X_iX_j+Y_iY_j+Z_iZ_j),\n \\qquad E_0:=\\lambda_{\\min}(H_G).\n\\tag{1}\n\\end{equation} \nHere $X_i,Y_i,Z_i$ are Pauli operators on qubit $i$. For the energy in Eq. (1), distinguish $E_0\\leq a$ from $E_0\\geq b$, promised one holds, using a randomized classical algorithm polynomial in the input length and correct with probability at least $2/3$.",
      "url": "https://qiqc-op.com/problem/op_1576e5f52f7ff3b2/",
      "json": "https://qiqc-op.com/api/problems/op_1576e5f52f7ff3b2.json",
      "tex": "https://qiqc-op.com/problem/op_1576e5f52f7ff3b2/op_1576e5f52f7ff3b2.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:16:40.000Z",
      "updatedAt": "2026-09-08T12:16:40.000Z",
      "sha256": "21a45e49e084c1bad608b73e18e9385577f314e97a903ec4b51f49e9e3252894"
    },
    {
      "id": "op_378472b2da3c533d",
      "ulid": "01M20BHBAFHFJEYHE131WS8EJX",
      "aliases": [
        "op_378472b2da3c533d",
        "01M20BHBAFHFJEYHE131WS8EJX",
        "op-378472b2da3c533d"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T11:13:01.775Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability",
          "quantum-supremacy",
          "iqp-sampling"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Average-case approximation hardness of random Ising partition functions",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability",
        "Quantum supremacy",
        "IQP sampling"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability",
        "Quantum supremacy",
        "IQP sampling"
      ],
      "statement": "Is it $\\#\\mathrm{P}$-hard to approximate $|Z_R|^2$ to relative multiplicative error $a+o(1)$ on a $b$ fraction of random Ising instances? Here $a>0$ and $0<b\\leq1$ are constant parameters independent of the number of vertices $n$, specifying the relative error and instance fraction.\n\nTake the complete graph on $n$ vertices. Choose each edge weight $w_{ij}$ and each vertex weight $v_k$ independently and uniformly from $\\{0,\\ldots,7\\}$. With $\\omega=e^{i\\pi/8}$, define\n\n \\begin{equation}\n Z_R=\\sum_{z\\in\\{-1,1\\}^n}\n \\omega^{\\sum_{i<j}w_{ij}z_i z_j+\\sum_{k=1}^n v_k z_k}.\n\\tag{1}\n\\end{equation} \nThe target is the squared modulus of Eq. (1). An estimate $\\widetilde Q_R$ is required to satisfy\n\n \\begin{equation}\n \\bigl|\\widetilde Q_R-|Z_R|^2\\bigr|\n \\leq (a+o(1))|Z_R|^2.\n\\tag{2}\n\\end{equation} \nThe $b$ fraction in the question is measured over the random vertex and edge weights for which Eq. (2) holds; $o(1)$ tends to zero as $n\\to\\infty$.",
      "url": "https://qiqc-op.com/problem/op_378472b2da3c533d/",
      "json": "https://qiqc-op.com/api/problems/op_378472b2da3c533d.json",
      "tex": "https://qiqc-op.com/problem/op_378472b2da3c533d/op_378472b2da3c533d.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:16:40.000Z",
      "updatedAt": "2026-09-08T12:16:40.000Z",
      "sha256": "ef4ea9f0dda791c3d38099e8e4f7e4e3f6ca33814392291ea869cdf1ffecbaa1"
    },
    {
      "id": "op_406a00f7c5a9398c",
      "ulid": "01M20BHBBG223ZDPK3F526EHFH",
      "aliases": [
        "op_406a00f7c5a9398c",
        "01M20BHBBG223ZDPK3F526EHFH",
        "op-406a00f7c5a9398c"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T11:13:01.808Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability",
          "quantum-supremacy",
          "iqp-sampling"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Average-case approximation hardness of squared normalized gaps of random cubic polynomials",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability",
        "Quantum supremacy",
        "IQP sampling"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability",
        "Quantum supremacy",
        "IQP sampling"
      ],
      "statement": "Is it $\\#\\mathrm{P}$-hard to approximate $\\operatorname{ngap}(f)^2$ to relative multiplicative error $a+o(1)$ on a $b$ fraction of uniformly random degree-3 polynomials over $\\mathbb{F}_2$? Here $a>0$ and $0<b\\leq1$ are constant parameters independent of the number of variables $n$, specifying the relative error and polynomial fraction.\n\nWrite the random polynomial in multilinear form as\n\n \\begin{equation}\n f(x)=\\sum_{i<j<k}\\alpha_{ijk}x_i x_j x_k\n +\\sum_{i<j}\\beta_{ij}x_i x_j\n +\\sum_i\\gamma_i x_i \\pmod{2},\n \\qquad x\\in\\{0,1\\}^n.\n\\tag{1}\n\\end{equation} \nIn Eq. (1), all coefficients are independent uniform bits. The degree-3 ensemble allows terms of degrees one through three, without conditioning on a nonzero cubic term. A constant term is omitted because it changes only the sign of the gap. Define\n\n \\begin{equation}\n \\operatorname{gap}(f)\n =|\\{x\\in\\{0,1\\}^n:f(x)=0\\}|-|\\{x\\in\\{0,1\\}^n:f(x)=1\\}|,\n \\qquad\n \\operatorname{ngap}(f)=2^{-n}\\operatorname{gap}(f).\n\\tag{2}\n\\end{equation} \nThe target is the square of the normalized gap in Eq. (2). An estimate $\\widetilde Q_f$ is required to satisfy\n\n \\begin{equation}\n \\bigl|\\widetilde Q_f-\\operatorname{ngap}(f)^2\\bigr|\n \\leq(a+o(1))\\operatorname{ngap}(f)^2.\n\\tag{3}\n\\end{equation} \nThe $b$ fraction in the question is measured over the coefficient choices for which Eq. (3) holds; $o(1)$ tends to zero as $n\\to\\infty$.",
      "url": "https://qiqc-op.com/problem/op_406a00f7c5a9398c/",
      "json": "https://qiqc-op.com/api/problems/op_406a00f7c5a9398c.json",
      "tex": "https://qiqc-op.com/problem/op_406a00f7c5a9398c/op_406a00f7c5a9398c.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:16:40.000Z",
      "updatedAt": "2026-09-08T12:16:40.000Z",
      "sha256": "ce78328c76ddcbd9adbf2d88f37a255d5a8db8f190fd2f617f894e27317c9519"
    },
    {
      "id": "op_94fc1874fe23df16",
      "ulid": "01M20CXWDYD1RXVWDKFMFM675K",
      "aliases": [
        "op_94fc1874fe23df16",
        "01M20CXWDYD1RXVWDKFMFM675K",
        "op-94fc1874fe23df16"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T11:37:21.086Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability",
          "quantum-supremacy",
          "random-circuit-sampling"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Average-case approximation hardness of random-circuit output probabilities",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability",
        "Quantum supremacy",
        "Random circuit sampling"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability",
        "Quantum supremacy",
        "Random circuit sampling"
      ],
      "statement": "Does there exist a fixed family of $n$-qubit circuit layouts with $m=\\operatorname{poly}(n)$ one- and two-qubit gates for which the following task is $\\#\\mathrm{P}$-hard? Choose every gate independently from Haar measure, obtaining $C$. Given $C$ and $\\epsilon,\\delta\\in(0,1)$, estimate the all-zero output probability with\n\n \\begin{equation}\n p_0(C)=|\\langle0^n|C|0^n\\rangle|^2,\\qquad\n \\Pr_C[|\\widetilde p(C)-p_0(C)|\\leq\\epsilon 2^{-n}]\\geq1-\\delta.\n\\tag{1}\n\\end{equation} \nThe guarantee in Eq. (1) must hold for arbitrary $\\epsilon,\\delta$, with running time measured in $n,1/\\epsilon,1/\\delta$.",
      "url": "https://qiqc-op.com/problem/op_94fc1874fe23df16/",
      "json": "https://qiqc-op.com/api/problems/op_94fc1874fe23df16.json",
      "tex": "https://qiqc-op.com/problem/op_94fc1874fe23df16/op_94fc1874fe23df16.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:16:40.000Z",
      "updatedAt": "2026-09-08T12:16:40.000Z",
      "sha256": "f430b0c16d942c31269bdfcb841593d712a21426d4ecfaac988ccfa825d4e5cd"
    },
    {
      "id": "op_af68b033a88ef934",
      "ulid": "01M20DFNK3W29RR12PZ50N40AH",
      "aliases": [
        "op_af68b033a88ef934",
        "01M20DFNK3W29RR12PZ50N40AH",
        "op-af68b033a88ef934"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T11:47:03.907Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability",
          "quantum-supremacy",
          "boson-sampling"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Permanent-of-Gaussians Conjecture",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability",
        "Quantum supremacy",
        "Boson sampling"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability",
        "Quantum supremacy",
        "Boson sampling"
      ],
      "statement": "Is the following estimation task $\\#\\mathrm{P}$-hard under randomized polynomial-time Turing reductions? Given $X\\in\\mathbb{C}^{n\\times n}$ with independent entries $X_{ij}=(a_{ij}+ib_{ij})/\\sqrt{2}$, where all $a_{ij},b_{ij}\\sim\\mathcal{N}(0,1)$ independently, and $\\epsilon,\\delta\\in(0,1)$, output $z(X)\\in\\mathbb{C}$ satisfying\n\n \\begin{equation}\n \\operatorname{Per}(X):=\\sum_{\\sigma\\in S_n}\\prod_{j=1}^{n}X_{j,\\sigma(j)},\n \\qquad\n \\Pr_X\\!\\left[|z(X)-\\operatorname{Per}(X)|\n \\leq\\epsilon|\\operatorname{Per}(X)|\\right]\\geq1-\\delta.\n\\tag{1}\n\\end{equation} \nHere $S_n$ is the set of permutations of $\\{1,\\ldots,n\\}$. The guarantee in Eq. (1) is required for arbitrary $\\epsilon,\\delta$, with running time measured in $n,1/\\epsilon,1/\\delta$.",
      "url": "https://qiqc-op.com/problem/op_af68b033a88ef934/",
      "json": "https://qiqc-op.com/api/problems/op_af68b033a88ef934.json",
      "tex": "https://qiqc-op.com/problem/op_af68b033a88ef934/op_af68b033a88ef934.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:16:40.000Z",
      "updatedAt": "2026-09-08T12:16:40.000Z",
      "sha256": "c43a02ac9fc42737fdc7d9c5f6cb1c4298004e167b7385af3d0beb899033bccc"
    },
    {
      "id": "op_0c17d9fef967858d",
      "ulid": "01M20EG8DVMFZX9TWEXN602APF",
      "aliases": [
        "op_0c17d9fef967858d",
        "01M20EG8DVMFZX9TWEXN602APF",
        "op-0c17d9fef967858d"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T12:04:51.771Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Quantum query complexity of Triangle Finding",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability"
      ],
      "statement": "What is the bounded-error quantum query complexity of finding a triangle in an $n$-vertex graph given oracle access to its adjacency matrix?\n\nLet $G=(V,E)$ be a simple undirected graph with $V=[n]$, and let $A\\in\\{0,1\\}^{n\\times n}$ be its adjacency matrix. The input is accessed through a quantum oracle acting as\n\n \\begin{equation}\n O_A\\lvert u,v,z\\rangle\n =\\lvert u,v,z\\oplus A_{uv}\\rangle .\n\\tag{1}\n\\end{equation} \nThe task is to output three distinct vertices $u,v,w\\in[n]$ satisfying\n\n \\begin{equation}\n A_{uv}=A_{vw}=A_{wu}=1,\n\\tag{2}\n\\end{equation} \nif such vertices exist, and otherwise report that the graph is triangle-free. Let $Q_{\\triangle}(n)$ denote the minimum number of queries to the oracle in (1) required by a bounded-error quantum algorithm for this task.\n\nThe best known general bounds are\n\n \\begin{equation}\n Q_{\\triangle}(n)=O\\left(n^{5/4}\\right)\n \\qquad\\text{and}\\qquad\n Q_{\\triangle}(n)=\\Omega(n).\n\\tag{3}\n\\end{equation} \nThe open question is to determine the asymptotic behavior of $Q_{\\triangle}(n)$, equivalently to improve either the upper or lower bound in (3) until the bounds match.",
      "url": "https://qiqc-op.com/problem/op_0c17d9fef967858d/",
      "json": "https://qiqc-op.com/api/problems/op_0c17d9fef967858d.json",
      "tex": "https://qiqc-op.com/problem/op_0c17d9fef967858d/op_0c17d9fef967858d.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T12:10:01.000Z",
      "updatedAt": "2026-09-08T12:10:01.000Z",
      "sha256": "96e71b2d8ec67d066f9f8d5ca424e557c9eff3fc16f37ef30a968a54e558b947"
    },
    {
      "id": "op_7249a79e1534f6ab",
      "ulid": "01M20CKB78AVP4GX3MKGMJ1VAE",
      "aliases": [
        "op_7249a79e1534f6ab",
        "01M20CKB78AVP4GX3MKGMJ1VAE",
        "op-7249a79e1534f6ab"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-08T11:31:35.784Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Polynomial-time quantum algorithm for Graph Isomorphism",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Computational complexity and computability"
      ],
      "statement": "Does the Graph Isomorphism problem admit a polynomial-time quantum algorithm?\n\nLet $G=(V_G,E_G)$ and $H=(V_H,E_H)$ be finite simple graphs with $|V_G|=|V_H|=n$. The graphs are isomorphic if there exists a bijection $\\pi:V_G\\to V_H$ satisfying\n\n \\begin{equation}\n \\{u,v\\}\\in E_G\n \\Longleftrightarrow\n \\{\\pi(u),\\pi(v)\\}\\in E_H\n \\qquad\n \\forall u,v\\in V_G .\n\\tag{1}\n\\end{equation} \nThe open question is whether there is a bounded-error quantum algorithm running in time polynomial in $n$ that, given $G$ and $H$, decides whether a bijection satisfying (1) exists.",
      "url": "https://qiqc-op.com/problem/op_7249a79e1534f6ab/",
      "json": "https://qiqc-op.com/api/problems/op_7249a79e1534f6ab.json",
      "tex": "https://qiqc-op.com/problem/op_7249a79e1534f6ab/op_7249a79e1534f6ab.tex",
      "created": "2026-09-08",
      "updated": "2026-09-08",
      "createdAt": "2026-09-08T11:55:53.000Z",
      "updatedAt": "2026-09-08T11:55:53.000Z",
      "sha256": "8328f35846632bb98226b0e1ce5e6354f41d1053a590622f8a05c17044636cd3"
    },
    {
      "id": "op_fd75613c5bab4164",
      "ulid": "01M1Q787QRG3HGYA0Y8F8JBPTE",
      "aliases": [
        "op_fd75613c5bab4164",
        "01M1Q787QRG3HGYA0Y8F8JBPTE",
        "op-fd75613c5bab4164"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "channel-degradability",
          "private-capacity"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1HME780WGX30SANKVAEX4S2"
        ]
      },
      "title": "Strict inclusion of degradable channels in the less-noisy class",
      "status": "Solved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Channel degradability",
        "Private capacity"
      ],
      "tags": [
        "Quantum Communication",
        "Channel degradability",
        "Private capacity"
      ],
      "statement": "Do there exist finite-dimensional quantum channels that are less noisy in Watanabe’s regularized sense but are not degradable? Let $V_{A\\to BE}$ be a Stinespring isometry defining a channel and a complementary channel by\n\n \\begin{equation}\n \\mathcal N_{A\\to B}(X)\n :=\\operatorname{Tr}_E[VXV^\\dagger],\n \\qquad\n \\mathcal N^c_{A\\to E}(X)\n :=\\operatorname{Tr}_B[VXV^\\dagger].\n\\tag{1}\n\\end{equation} \nThe channel in Eq. (1) is degradable if there is a completely positive trace-preserving map $\\mathcal D_{B\\to E}$ such that\n\n \\begin{equation}\n \\mathcal N^c=\\mathcal D\\circ\\mathcal N.\n\\tag{2}\n\\end{equation} \nWriting $P$ for the unassisted private classical capacity, Watanabe calls $\\mathcal N$ less noisy when\n\n \\begin{equation}\n P(\\mathcal N^c)=0.\n\\tag{3}\n\\end{equation} \nEquivalently, Eq. (3) requires that, for every $n\\ge1$ and every classical–quantum ensemble on $UA^{n}$, the receiver and environment outputs obey\n\n \\begin{equation}\n I(U:B^n)_{(\\operatorname{id}_U\\otimes\\mathcal N^{\\otimes n})(\\omega)}\n \\ge\n I(U:E^n)_{(\\operatorname{id}_U\\otimes(\\mathcal N^c)^{\\otimes n})(\\omega)}.\n\\tag{4}\n\\end{equation} \nThus the question asks whether the inclusion implied by Eqs. (2)–(4) is strict.",
      "url": "https://qiqc-op.com/problem/op_fd75613c5bab4164/",
      "json": "https://qiqc-op.com/api/problems/op_fd75613c5bab4164.json",
      "tex": "https://qiqc-op.com/problem/op_fd75613c5bab4164/op_fd75613c5bab4164.tex",
      "created": "2026-09-02",
      "updated": "2026-09-06",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-06T11:06:52.000Z",
      "sha256": "6333b4ff4a95b17e81336a67ad535738207e15d152e1d8ff17dfd03d3f54c869"
    },
    {
      "id": "op_12fc55f67580588e",
      "ulid": "01M1HME780WGX30SANKVAEX4S2",
      "aliases": [
        "op_12fc55f67580588e",
        "01M1HME780WGX30SANKVAEX4S2",
        "op-12fc55f67580588e",
        "v2-regularized-less-noisy-channels-beyond-degradability",
        "open-problem-v2-problem-52"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "channel-degradability",
          "private-capacity"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": [
          "01M1Q787QRG3HGYA0Y8F8JBPTE"
        ],
        "equivalentToProblemId": "01M1Q787QRG3HGYA0Y8F8JBPTE"
      },
      "title": "Regularized less-noisy channels beyond degradability",
      "status": "Solved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Channel degradability",
        "Private capacity"
      ],
      "tags": [
        "Quantum Communication",
        "Channel degradability",
        "Private capacity"
      ],
      "statement": "Does there exist a finite-dimensional regularized less-noisy quantum channel that is not degradable? Let $\\mathcal T:\\mathcal L(\\mathcal H_A)\\to\\mathcal L(\\mathcal H_B)$ have a Stinespring isometry $V:\\mathcal H_A\\to\\mathcal H_B\\otimes\\mathcal H_E$, defining the channel and one complementary channel by\n\n \\begin{equation}\n \\mathcal T(\\rho)=\\operatorname{Tr}_E(V\\rho V^\\dagger),\n \\qquad\n \\mathcal T^c(\\rho)=\\operatorname{Tr}_B(V\\rho V^\\dagger).\n\\tag{1}\n\\end{equation} \nEquation (1) fixes the complementary-channel notation; changing the Stinespring representation only changes $\\mathcal T^c$ by an output isometry.\n\nThe channel $\\mathcal T$ is degradable if there exists a channel $\\mathcal D:\\mathcal L(\\mathcal H_B)\\to\\mathcal L(\\mathcal H_E)$ satisfying\n\n \\begin{equation}\n \\mathcal T^c=\\mathcal D\\circ\\mathcal T.\n\\tag{2}\n\\end{equation} \nThus Eq. (2) requires exact, rather than approximate, degradability.\n\nFor a channel $\\mathcal N:A\\to B$ with complement $\\mathcal N^c:A\\to E$, let \\(\\omega_{XBE}:=\\sum_x p_x\\lvert x\\rangle\\!\\langle x\\rvert_X\\otimes\nV_{\\mathcal N}\\rho_A^xV_{\\mathcal N}^\\dagger\\) for a finite cq ensemble $\\{p_x,\\rho_A^x\\}$. Define its one-shot private information and private capacity by\n\n \\begin{equation}\n \\begin{aligned}\n \\mathcal C_p^{(1)}(\\mathcal N)\n &:=\\max_{\\{p_x,\\rho_A^x\\}}\n \\bigl[I(X;B)_\\omega-I(X;E)_\\omega\\bigr],\\\\\n \\mathcal C_p(\\mathcal N)\n &:=\\sup_{n\\ge1}\\frac1n\n \\mathcal C_p^{(1)}(\\mathcal N^{\\otimes n}).\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nThe regularization in Eq. (3) distinguishes the question from its single-letter counterpart. In particular, define\n\n \\begin{equation}\n \\mathfrak D\n :=\\{\\mathcal T:\\mathcal T\\text{ satisfies\n Eq.~(2)}\\}.\n\\tag{4}\n\\end{equation} \n \\begin{equation}\n \\mathfrak L_1\n :=\\{\\mathcal T:\\mathcal C_p^{(1)}(\\mathcal T^c)=0\\}.\n\\tag{5}\n\\end{equation} \n \\begin{equation}\n \\mathfrak L_\\infty\n :=\\{\\mathcal T:\\mathcal C_p(\\mathcal T^c)=0\\}.\n\\tag{6}\n\\end{equation} \n \\begin{equation}\n \\mathfrak D\\subseteq\\mathfrak L_\\infty\\subseteq\\mathfrak L_1.\n\\tag{7}\n\\end{equation} \nA channel in $\\mathfrak L_\\infty$, as defined in Eq. (6), is regularized less noisy. The problem is whether the first inclusion in Eq. (7) is strict.",
      "url": "https://qiqc-op.com/problem/op_12fc55f67580588e/",
      "json": "https://qiqc-op.com/api/problems/op_12fc55f67580588e.json",
      "tex": "https://qiqc-op.com/problem/op_12fc55f67580588e/op_12fc55f67580588e.tex",
      "created": "2026-09-02",
      "updated": "2026-09-06",
      "createdAt": "2026-09-02T00:20:51.000Z",
      "updatedAt": "2026-09-06T11:06:52.000Z",
      "sha256": "357873b588a51480aa6f16e6cd1bdcb29733e69be72b68f3fffe3279758ed5a4"
    },
    {
      "id": "op_06e9f0c7b3b62f3b",
      "ulid": "01M1Q787QRCCSDNVA159Y6S261",
      "aliases": [
        "op_06e9f0c7b3b62f3b",
        "01M1Q787QRCCSDNVA159Y6S261",
        "op-06e9f0c7b3b62f3b"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "channel-simulation",
          "superchannels-and-quantum-combs",
          "quantum-error-mitigation"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Multi-slot overhead of virtual channel conjugation",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Channel simulation",
        "Superchannels and quantum combs",
        "Quantum error mitigation"
      ],
      "tags": [
        "Quantum algorithm",
        "Quantum Resource Theory",
        "Channel simulation",
        "Superchannels and quantum combs",
        "Quantum error mitigation"
      ],
      "statement": "What is the optimal quasiprobability overhead of implementing the complex conjugate of an unknown quantum channel from $n$ queries? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be an unknown channel with $d_A:=\\dim A$ and $d_B:=\\dim B$, and fix orthonormal bases of $A$ and $B$. The complex conjugate of $\\mathcal N$ is the channel\n\n \\begin{equation}\n \\mathcal N^{*}(X):=\\overline{\\mathcal N(\\overline X)},\n\\tag{1}\n\\end{equation} \nwhere the bar is entrywise complex conjugation in the fixed bases, so the Choi operator of $\\mathcal N^{*}$ is the entrywise conjugate of that of $\\mathcal N$. An $n$-slot quantum comb is a physically realizable circuit with $n$ open slots, each receiving one use of the unknown channel, whose overall action is again a channel from $A$ to $B$; write $\\mathrm{Comb}_n$ for the set of such combs. An $n$-slot virtual comb is a real linear combination $\\widetilde{\\mathcal C}=\\sum_i c_i\\mathcal C_i$ with $\\mathcal C_i\\in\\mathrm{Comb}_n$, and its base norm\n\n \\begin{equation}\n \\|\\widetilde{\\mathcal C}\\|_{\\mathrm{base}}\n :=\\min\\Bigl\\{\\sum_i|c_i|:\n \\widetilde{\\mathcal C}=\\sum_ic_i\\mathcal C_i,\\\n c_i\\in\\mathbb R,\\ \\mathcal C_i\\in\\mathrm{Comb}_n\\Bigr\\}\n\\tag{2}\n\\end{equation} \nis the sampling overhead: estimating an expectation value of the output of $\\widetilde{\\mathcal C}$ to additive error $\\varepsilon$ by Monte Carlo sampling of the $\\mathcal C_i$ costs $O(\\|\\widetilde{\\mathcal C}\\|_{\\mathrm{base}}^{2}\\varepsilon^{-2})$ runs. Define the optimal $n$-query overhead of universal conjugation by\n\n \\begin{equation}\n g_n(d_A,d_B)\n :=\\inf\\Bigl\\{\\|\\widetilde{\\mathcal C}\\|_{\\mathrm{base}}:\n \\widetilde{\\mathcal C}\\text{ is an $n$-slot virtual comb with }\n \\widetilde{\\mathcal C}(\\mathcal N^{\\otimes n})=\\mathcal N^{*}\n \\text{ for every channel }\\mathcal N\\Bigr\\}.\n\\tag{3}\n\\end{equation} \nSince the extra slots may be discarded, $g_n\\leq g_1$. Determine $g_n(d_A,d_B)$ in Eq. (3) for $n\\geq2$: is $g_n(d_A,d_B)<g_1(d_A,d_B)$ for some $n$, and what is $\\inf_{n}g_n(d_A,d_B)$?",
      "url": "https://qiqc-op.com/problem/op_06e9f0c7b3b62f3b/",
      "json": "https://qiqc-op.com/api/problems/op_06e9f0c7b3b62f3b.json",
      "tex": "https://qiqc-op.com/problem/op_06e9f0c7b3b62f3b/op_06e9f0c7b3b62f3b.tex",
      "created": "2026-09-04",
      "updated": "2026-09-04",
      "createdAt": "2026-09-04T01:19:11.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "dc3b608dbdba5c2fe5b774bf0d591fd2d7fc9ea96242f7925515e593e082d03b"
    },
    {
      "id": "op_6006e574028a48fd",
      "ulid": "01M1Q787QR7SBS31M6PYNF20RT",
      "aliases": [
        "op_6006e574028a48fd",
        "01M1Q787QR7SBS31M6PYNF20RT",
        "op-6006e574028a48fd"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "channel-simulation",
          "quantum-communication-complexity",
          "quantum-relative-entropy",
          "one-shot-and-finite-blocklength-bounds"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Bidirectional classical-communication cost of bipartite channel simulation",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Channel simulation",
        "Quantum communication complexity",
        "Quantum relative entropy",
        "One-shot and finite-blocklength bounds"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Resource Theory",
        "Channel simulation",
        "Quantum communication complexity",
        "Quantum relative entropy",
        "One-shot and finite-blocklength bounds"
      ],
      "statement": "What is the asymptotic classical-communication cost of simulating a bipartite quantum channel with bidirectional classical communication and non-signalling assistance? Let $\\mathcal N:\\mathcal L(A_0\\otimes B_0)\\to\\mathcal L(A_1\\otimes B_1)$ be a quantum channel on finite-dimensional systems, with Alice holding $A_0,A_1$ and Bob holding $B_0,B_1$. A bidirectional simulation protocol consists of a bipartite channel shared in advance that can signal in neither direction (a non-signalling correlation, which includes every shared entangled state), one classical message with $m_\\to$ values from Alice to Bob, one classical message with $m_\\leftarrow$ values from Bob to Alice, and local operations; it uses $m:=m_\\to m_\\leftarrow$ classical values in total. For $\\varepsilon\\in[0,1)$ define the one-shot cost\n\n \\begin{equation}\n S^{(1)}_{\\leftrightarrow,\\varepsilon}(\\mathcal N)\n :=\\log_2\\min\\Bigl\\{m\\in\\mathbb N:\n \\tfrac12\\|\\Upsilon-\\mathcal N\\|_\\diamond\\leq\\varepsilon\n \\text{ for some protocol }\\Upsilon\\text{ using }m\\text{ values}\\Bigr\\},\n\\tag{1}\n\\end{equation} \nwhere $\\|\\cdot\\|_\\diamond$ is the diamond norm. The asymptotic exact and vanishing-error costs are\n\n \\begin{equation}\n S_{\\leftrightarrow,0}(\\mathcal N)\n :=\\lim_{n\\to\\infty}\\frac1n\n S^{(1)}_{\\leftrightarrow,0}(\\mathcal N^{\\otimes n}),\n \\qquad\n S_{\\leftrightarrow}(\\mathcal N)\n :=\\lim_{\\varepsilon\\downarrow0}\\limsup_{n\\to\\infty}\n \\frac1n S^{(1)}_{\\leftrightarrow,\\varepsilon}(\\mathcal N^{\\otimes n}).\n\\tag{2}\n\\end{equation} \nLet $\\mathrm{NS}$ denote the set of bipartite channels from $A_0B_0$ to $A_1B_1$ that are non-signalling in both directions, and define the max-relative entropy of bidirectional communication\n\n \\begin{equation}\n \\mathfrak D^{\\leftrightarrow}_{\\max}(\\mathcal N)\n :=\\min_{\\mathcal E\\in\\mathrm{NS}}D_{\\max}(\\mathcal N\\|\\mathcal E),\n \\qquad\n D_{\\max}(\\mathcal N\\|\\mathcal E)\n :=\\log_2\\min\\{\\lambda\\geq0:J_{\\mathcal N}\\leq\\lambda J_{\\mathcal E}\\},\n\\tag{3}\n\\end{equation} \nwhere $J$ denotes the Choi operator. Determine $S_{\\leftrightarrow,0}(\\mathcal N)$ and $S_{\\leftrightarrow}(\\mathcal N)$ in Eq. (2) for a general bipartite channel. In particular, is either cost equal to the regularization \\(\\lim_{n\\to\\infty}\\frac1n\n\\mathfrak D^{\\leftrightarrow}_{\\max}(\\mathcal N^{\\otimes n})\\) of Eq. (3), and does a single-letter formula exist?",
      "url": "https://qiqc-op.com/problem/op_6006e574028a48fd/",
      "json": "https://qiqc-op.com/api/problems/op_6006e574028a48fd.json",
      "tex": "https://qiqc-op.com/problem/op_6006e574028a48fd/op_6006e574028a48fd.tex",
      "created": "2026-09-04",
      "updated": "2026-09-04",
      "createdAt": "2026-09-04T01:19:11.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "078f5429fc7aaec4897401b6eb32b37c3d35c47d3fbaa9ae6ca3765f461ed4da"
    },
    {
      "id": "op_75b91a20dd384110",
      "ulid": "01M1Q787QRPDH1Y9ADAGSB1AGN",
      "aliases": [
        "op_75b91a20dd384110",
        "01M1Q787QRPDH1Y9ADAGSB1AGN",
        "op-75b91a20dd384110"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-distillation",
          "ppt-preserving-operations",
          "additivity-and-regularization",
          "one-shot-and-finite-blocklength-bounds"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Closed-form exact PPT distillable entanglement",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement distillation",
        "PPT-preserving operations",
        "Additivity and regularization",
        "One-shot and finite-blocklength bounds"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement distillation",
        "PPT-preserving operations",
        "Additivity and regularization",
        "One-shot and finite-blocklength bounds"
      ],
      "statement": "What computable expression, if any, equals the regularized exact PPT distillable entanglement of a bipartite state? Let $\\rho_{AB}$ be a state on $\\mathbb C^{d_A}\\otimes\\mathbb C^{d_B}$ with support projector $P:=\\Pi_{\\operatorname{supp}(\\rho)}$, and let $\\Gamma$ denote partial transposition on $B$. Exact (zero-error) distillation under PPT-preserving operations converts $\\rho^{\\otimes n}$ into a maximally entangled state of Schmidt rank $M_n$ with unit fidelity. The largest one-shot rate is governed by the semidefinite program\n\n \\begin{equation}\n W_0(P):=\\min\\bigl\\{\\|E^{\\Gamma}\\|_\\infty:\\ P\\leq E\\leq\\mathbb 1\\bigr\\},\n \\qquad\n E^{(1)}_{0,\\mathrm{PPT}}(\\rho):=-\\log_2W_0(P),\n\\tag{1}\n\\end{equation} \nwhich depends on $\\rho$ only through its support. The regularized exact PPT distillable entanglement is\n\n \\begin{equation}\n E^{\\infty}_{0,\\mathrm{PPT}}(\\rho)\n :=\\lim_{n\\to\\infty}\\frac1n E^{(1)}_{0,\\mathrm{PPT}}(\\rho^{\\otimes n})\n =\\lim_{n\\to\\infty}-\\frac1n\\log_2W_0(P^{\\otimes n}).\n\\tag{2}\n\\end{equation} \nIs there a single-letter, efficiently computable formula, for example a semidefinite program in $P$ alone, that equals Eq. (2) for every bipartite state?",
      "url": "https://qiqc-op.com/problem/op_75b91a20dd384110/",
      "json": "https://qiqc-op.com/api/problems/op_75b91a20dd384110.json",
      "tex": "https://qiqc-op.com/problem/op_75b91a20dd384110/op_75b91a20dd384110.tex",
      "created": "2026-09-04",
      "updated": "2026-09-04",
      "createdAt": "2026-09-04T01:19:11.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "61fad6b1ed4e1f295b3a1f1530a0bd7af5f030dd581eed8a052535b32125b0f4"
    },
    {
      "id": "op_a64dc63d6ae49127",
      "ulid": "01M1Q787QRD6APNHX659G4CTEF",
      "aliases": [
        "op_a64dc63d6ae49127",
        "01M1Q787QRD6APNHX659G4CTEF",
        "op-a64dc63d6ae49127"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-algorithm"
        ],
        "topicIds": [
          "quantum-magic",
          "resource-conversion",
          "quantum-state-preparation"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Universal purification with classically simulable operations",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum algorithm"
      ],
      "topics": [
        "Quantum magic",
        "Resource conversion",
        "Quantum state preparation"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum algorithm",
        "Quantum magic",
        "Resource conversion",
        "Quantum state preparation"
      ],
      "statement": "Can classically simulable operations purify an unknown depolarized pure state from any number of copies? Fix a dimension $d$ that is either $2$ or odd. For an unknown pure state $\\psi=|\\psi\\rangle\\langle\\psi|$ on $\\mathbb C^d$ and a noise parameter $0<\\delta<1$, consider the depolarized copy\n\n \\begin{equation}\n \\mathcal D_\\delta(\\psi):=(1-\\delta)\\psi+\\delta\\,\\frac{\\mathbb 1_d}{d},\n \\qquad\n \\operatorname{Tr}\\bigl[\\psi\\,\\mathcal D_\\delta(\\psi)\\bigr]\n =1-\\frac{d-1}{d}\\,\\delta.\n\\tag{1}\n\\end{equation} \nLet $\\mathcal A_2$ be the set of completely stabilizer-preserving maps on qubits and, for odd $d$, let $\\mathcal A_d$ be the set of completely positive-Wigner-preserving maps on qudits; both classes are efficiently classically simulable, and trace-nonincreasing (probabilistic) members are allowed. For $n\\geq2$ copies and a success probability $0<s\\leq1$, the optimal Haar-averaged purification fidelity is\n\n \\begin{equation}\n F^{\\mathcal A_d}_{\\delta}(n,s)\n :=\\sup\\Bigl\\{\\frac1s\\int d\\psi\\,\n \\operatorname{Tr}\\bigl[\\psi\\,\n \\mathcal E\\bigl(\\mathcal D_\\delta(\\psi)^{\\otimes n}\\bigr)\\bigr]\n :\\ \\mathcal E\\in\\mathcal A_d,\\\n \\int d\\psi\\,\\operatorname{Tr}\\,\n \\mathcal E\\bigl(\\mathcal D_\\delta(\\psi)^{\\otimes n}\\bigr)=s\n \\Bigr\\},\n\\tag{2}\n\\end{equation} \nwhere $\\mathcal E$ maps the $n$ copies to one $d$-dimensional system and $d\\psi$ is the Haar measure on pure states. Is\n\n \\begin{equation}\n F^{\\mathcal A_d}_{\\delta}(n,s)=1-\\frac{d-1}{d}\\,\\delta\n\\tag{3}\n\\end{equation} \nfor every $n\\geq2$, every $0<\\delta<1$, and every $0<s\\leq1$, so that no classically simulable protocol, deterministic or postselected, improves on the single-copy fidelity in Eq. (1)?",
      "url": "https://qiqc-op.com/problem/op_a64dc63d6ae49127/",
      "json": "https://qiqc-op.com/api/problems/op_a64dc63d6ae49127.json",
      "tex": "https://qiqc-op.com/problem/op_a64dc63d6ae49127/op_a64dc63d6ae49127.tex",
      "created": "2026-09-04",
      "updated": "2026-09-04",
      "createdAt": "2026-09-04T01:19:11.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "6da974780ed72da58a3378a6e8228bc9d7f6557afd3c136fe575e3f282efdaf4"
    },
    {
      "id": "op_02bb8f8228649ac3",
      "ulid": "01M1Q787QR3RWGKBRKK8CQSZF6",
      "aliases": [
        "op_02bb8f8228649ac3",
        "01M1Q787QR3RWGKBRKK8CQSZF6",
        "op-02bb8f8228649ac3"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "channel-discrimination",
          "quantum-hypothesis-testing",
          "superchannels-and-quantum-combs"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Advantage of fully general superchannel-discrimination strategies",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Channel discrimination",
        "Quantum hypothesis testing",
        "Superchannels and quantum combs"
      ],
      "tags": [
        "Quantum metrology",
        "Channel discrimination",
        "Quantum hypothesis testing",
        "Superchannels and quantum combs"
      ],
      "statement": "Can a fully general adaptive strategy attain a larger Stein exponent than every nested-adaptive strategy for discriminating two quantum superchannels? Fix finite-dimensional physical realizations\n\n \\begin{equation}\n \\Theta_i(\\mathcal N)\n =\\mathcal D_i\\circ(\\mathcal N\\otimes\\operatorname{id}_{S_i})\n \\circ\\mathcal E_i,\n \\qquad i\\in\\{1,2\\}.\n\\tag{1}\n\\end{equation} \nThe systems $S_i$ in Eq. (1) are internal memories. A nested strategy recursively places complete uses of $\\Theta_i$ inside one another. A fully general strategy may interleave the preprocessing and postprocessing components of different uses in any causally valid order, provided each $\\mathcal E_i$ precedes its matched $\\mathcal D_i$ and the tester cannot access the internal memory $S_i$. Define the vanishing-type-I-error Stein exponent for a strategy class $\\mathsf S$ by\n\n \\begin{equation}\n \\zeta_{\\mathsf S}(\\Theta_1\\|\\Theta_2)\n :=\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\n -\\frac1n\\log_2\n \\inf_{\\substack{P\\in\\mathsf S_n:\\alpha_n(P)\\leq\\varepsilon}}\n \\beta_n(P).\n\\tag{2}\n\\end{equation} \nThe definition in Eq. (2) uses the type-I and type-II errors $\\alpha_n(P)$ and $\\beta_n(P)$ of protocol $P$. Does there exist a pair of realizations for which\n\n \\begin{equation}\n \\zeta_{\\mathrm{fg}}(\\Theta_1\\|\\Theta_2)\n >\\zeta_{\\mathrm{nest}}(\\Theta_1\\|\\Theta_2),\n\\tag{3}\n\\end{equation} \nwhere $\\mathrm{fg}$ denotes fully general strategies? If not, prove equality in Eq. (3) with $>$ replaced by $=$ for all superchannel pairs.",
      "url": "https://qiqc-op.com/problem/op_02bb8f8228649ac3/",
      "json": "https://qiqc-op.com/api/problems/op_02bb8f8228649ac3.json",
      "tex": "https://qiqc-op.com/problem/op_02bb8f8228649ac3/op_02bb8f8228649ac3.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "c22990104178f1d81384b56196e6d0139c3eb176e106271a92dac11acf10a7f8"
    },
    {
      "id": "op_1482756b02794495",
      "ulid": "01M1Q787QRTZXCRVQWGE6DXEKN",
      "aliases": [
        "op_1482756b02794495",
        "01M1Q787QRTZXCRVQWGE6DXEKN",
        "op-1482756b02794495"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology",
          "quantum-communication"
        ],
        "topicIds": [
          "channel-discrimination",
          "quantum-relative-entropy",
          "superchannels-and-quantum-combs"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Amortization collapse for superchannel divergences",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology",
        "Quantum Communication"
      ],
      "topics": [
        "Channel discrimination",
        "Quantum relative entropy",
        "Superchannels and quantum combs"
      ],
      "tags": [
        "Quantum metrology",
        "Quantum Communication",
        "Channel discrimination",
        "Quantum relative entropy",
        "Superchannels and quantum combs"
      ],
      "statement": "Does amortization collapse for the max-relative entropy and geometric Rényi divergence of arbitrary finite-dimensional quantum superchannels? For compatible states, let\n\n \\begin{equation}\n \\begin{aligned}\n D_{\\max}(\\rho\\|\\sigma)\n &:=\\inf\\{\\lambda:\\rho\\leq2^\\lambda\\sigma\\},\\\\\n \\widehat D_\\alpha(\\rho\\|\\sigma)\n &:=\\frac1{\\alpha-1}\\log_2\\operatorname{Tr}\\!\\left[\n \\sigma\\bigl(\\sigma^{-1/2}\\rho\\sigma^{-1/2}\\bigr)^\\alpha\n \\right],\\qquad 1<\\alpha\\leq2,\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nwith the standard support conventions. For either divergence $\\mathbf D\\in\\{D_{\\max},\\widehat D_\\alpha\\}$, define its channel extension and channel-amortized extension by\n\n \\begin{equation}\n \\begin{aligned}\n \\mathbf D_{\\rm ch}(\\mathcal N\\|\\mathcal M)\n &:=\\sup_{\\rho_{RA}}\n \\mathbf D(\\mathcal N(\\rho)\\|\\mathcal M(\\rho)),\\\\\n \\mathbf D_{\\rm ch}^{A}(\\mathcal N\\|\\mathcal M)\n &:=\\sup_{\\rho_{RA},\\sigma_{RA}}\n \\{\\mathbf D(\\mathcal N(\\rho)\\|\\mathcal M(\\sigma))\n -\\mathbf D(\\rho\\|\\sigma)\\},\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nIn Eq. (2), identity maps on $R$ are implicit, and the optimizations allow an arbitrary reference of sufficient finite dimension. For superchannels $\\Theta_1,\\Theta_2$, set\n\n \\begin{equation}\n \\begin{aligned}\n \\mathbf D_{\\rm sc}(\\Theta_1\\|\\Theta_2)\n &:=\\sup_{\\mathcal N}\n \\mathbf D_{\\rm ch}(\\Theta_1(\\mathcal N)\\|\\Theta_2(\\mathcal N)),\\\\\n \\mathbf D_{\\rm sc}^{A}(\\Theta_1\\|\\Theta_2)\n &:=\\sup_{\\mathcal N,\\mathcal M}\n \\{\\mathbf D_{\\rm ch}^{A}\n (\\Theta_1(\\mathcal N)\\|\\Theta_2(\\mathcal M))\n -\\mathbf D_{\\rm ch}^{A}(\\mathcal N\\|\\mathcal M)\\}.\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nIs\n\n \\begin{equation}\n \\mathbf D_{\\rm sc}^{A}(\\Theta_1\\|\\Theta_2)\n =\\mathbf D_{\\rm sc}(\\Theta_1\\|\\Theta_2)\n\\tag{4}\n\\end{equation} \nfor both choices of $\\mathbf D$ in Eq. (1) and all superchannel pairs?",
      "url": "https://qiqc-op.com/problem/op_1482756b02794495/",
      "json": "https://qiqc-op.com/api/problems/op_1482756b02794495.json",
      "tex": "https://qiqc-op.com/problem/op_1482756b02794495/op_1482756b02794495.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "349ad8b59198273ad5fbc5049932141c0a2d50e3cdaac6900167efff6c6fee37"
    },
    {
      "id": "op_2c9c3abb0983004c",
      "ulid": "01M1Q787QR4C97BS9QTECX5TNG",
      "aliases": [
        "op_2c9c3abb0983004c",
        "01M1Q787QR4C97BS9QTECX5TNG",
        "op-2c9c3abb0983004c"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm",
          "quantum-communication"
        ],
        "topicIds": [
          "decoding-algorithms",
          "classical-capacity",
          "quantum-state-discrimination"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Uniformly efficient HSW pretty-good decoding",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm",
        "Quantum Communication"
      ],
      "topics": [
        "Decoding algorithms",
        "Classical capacity",
        "Quantum state discrimination"
      ],
      "tags": [
        "Quantum algorithm",
        "Quantum Communication",
        "Decoding algorithms",
        "Classical capacity",
        "Quantum state discrimination"
      ],
      "statement": "Can Holevo–Schumacher–Westmoreland codebooks operating at every rate below their ensemble Holevo information be chosen so that their square-root, or pretty-good, measurements have uniform quantum implementations whose cost is polynomial in the blocklength and the logarithm of the codebook size? Fix an efficiently implementable finite-dimensional memoryless channel $\\mathcal N:A\\to B$ and an efficiently preparable ensemble $\\{p(x),\\phi_x^A\\}_{x\\in\\mathcal X}$. For\n\n \\begin{equation}\n \\omega_{XB}:=\\sum_{x\\in\\mathcal X}p(x)|x\\rangle\\!\\langle x|_X\n \\otimes\\mathcal N(\\phi_x),\n \\qquad\n \\chi(\\mathcal N,p):=I(X;B)_\\omega,\n\\tag{1}\n\\end{equation} \nfix a rate $0<R<\\chi(\\mathcal N,p)$ and choose a fixed typicality parameter $\\delta>0$ sufficiently small as a function of the gap $\\chi(\\mathcal N,p)-R$. A length-$n$ codebook consists of $M_n$ words $x^n(m)$, with $n^{-1}\\log_2M_n\\to R$, and corresponding output states\n\n \\begin{equation}\n \\rho_m^{B^n}:=\\bigotimes_{i=1}^{n}\\mathcal N(\\phi_{x_i(m)}),\n \\qquad m\\in\\{1,\\ldots,M_n\\}.\n\\tag{2}\n\\end{equation} \nLet $\\Pi_{\\bar\\rho,\\delta}^{(n)}$ be the typical projector of $\\bar\\rho^B:=\\sum_x p(x)\\mathcal N(\\phi_x)$, and let $\\Pi_{m,\\delta}^{(n)}$ be the conditionally typical projector for the codeword $x^n(m)$ and the product state in Eq. (2). The detection operators used in the standard HSW decoder are\n\n \\begin{equation}\n \\Gamma_{m,n}\n :=\\Pi_{\\bar\\rho,\\delta}^{(n)}\n \\Pi_{m,\\delta}^{(n)}\n \\Pi_{\\bar\\rho,\\delta}^{(n)},\n \\qquad\n G_n:=\\sum_{m=1}^{M_n}\\Gamma_{m,n}.\n\\tag{3}\n\\end{equation} \nUsing the Moore–Penrose inverse on $\\operatorname{supp}G_n$, their square-root measurement, including its failure outcome, is\n\n \\begin{equation}\n \\Lambda_{m,n}:=G_n^{-1/2}\\Gamma_{m,n}G_n^{-1/2},\n \\qquad\n \\Lambda_{0,n}:=I-\\sum_{m=1}^{M_n}\\Lambda_{m,n}.\n\\tag{4}\n\\end{equation} \nThe codebook must be succinct rather than an explicit list of exponentially many words. Require uniform polynomial-size coherent circuits that (i) compute $x_i(m)$, (ii) prepare a purification $|\\psi_m\\rangle_{B^nR_n}$ of $\\rho_m^{B^n}$ coherently in $m$, and (iii) give a controlled block encoding of the operators in Eq. (3). Denote these circuits by $C_n$, $O_n$, and $U_{\\Gamma,n}$, so that\n\n \\begin{equation}\n \\begin{aligned}\n C_n|m,i,0\\rangle&=|m,i,x_i(m)\\rangle,\\\\\n O_n|m,0\\rangle&=|m\\rangle|\\psi_m\\rangle,\n \\qquad \\operatorname{Tr}_{R_n}|\\psi_m\\rangle\\!\\langle\\psi_m|=\\rho_m,\\\\\n (\\langle0|_Z\\otimes I)U_{\\Gamma,n}(|0\\rangle_Z\\otimes I)\n &=\\sum_{m=1}^{M_n}|m\\rangle\\!\\langle m|\\otimes\\Gamma_{m,n}.\n \\end{aligned}\n\\tag{5}\n\\end{equation} \nThe promise in Eq. (5) includes efficient inverse circuits and a classical algorithm that outputs their gates in time polynomial in $n$ and $\\log M_n$.\n\nThe target is a uniform decoder circuit producing a POVM $\\{\\widetilde\\Lambda_{j,n}\\}_{j=0}^{M_n}$ whose output distribution, averaged over transmitted messages, approximates that of Eq. (4). For every $0<\\varepsilon<1/2$, require\n\n \\begin{equation}\n \\Delta_n\n :=\\frac{1}{2M_n}\\sum_{m=1}^{M_n}\\sum_{j=0}^{M_n}\n \\left|\n \\operatorname{Tr}\\!\\left[\n (\\widetilde\\Lambda_{j,n}-\\Lambda_{j,n})\\rho_m^{B^n}\n \\right]\n \\right|\n \\leq\\varepsilon,\n\\tag{6}\n\\end{equation} \nwith gate and oracle complexity $\\operatorname{poly}(n,\\log M_n,\\log(1/\\varepsilon))$, rather than polynomial in $M_n$ or in the inverse of an exponentially small singular value. Taking, for example, an inverse-polynomial sequence $\\varepsilon=\\varepsilon_n\\to0$, the implemented decoder must retain vanishing average message error. The problem is to construct such codebooks and decoders for every instance in Eq. (1), or to prove that this uniform target is impossible under the access model in Eq. (5).",
      "url": "https://qiqc-op.com/problem/op_2c9c3abb0983004c/",
      "json": "https://qiqc-op.com/api/problems/op_2c9c3abb0983004c.json",
      "tex": "https://qiqc-op.com/problem/op_2c9c3abb0983004c/op_2c9c3abb0983004c.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "eac466a7693c6d2c7192ebc43c275dc6c30a8719fe3ec3609324e0f874d75c28"
    },
    {
      "id": "op_3cd14aef409b226b",
      "ulid": "01M1Q787QRE9WF0NXMX32BQDCQ",
      "aliases": [
        "op_3cd14aef409b226b",
        "01M1Q787QRE9WF0NXMX32BQDCQ",
        "op-3cd14aef409b226b"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "bosonic-channels",
          "matrix-and-entropy-inequalities"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Multimode constrained output entropy of a pure-loss channel",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Bosonic channels",
        "Matrix and entropy inequalities"
      ],
      "tags": [
        "Quantum Communication",
        "Bosonic channels",
        "Matrix and entropy inequalities"
      ],
      "statement": "Is Guha’s multimode Strong Conjecture 2 true for every correlated input state? Fix $n\\geq1$, $K\\geq0$, and $0<\\eta<1$. Let the modes $A_1,\\ldots,A_n$ be in the vacuum and let $\\rho_{B^n}$ be an arbitrary joint $n$-mode state. Apply identical beam splitters whose output annihilation operators satisfy\n\n \\begin{equation}\n \\hat c_i=\\sqrt{\\eta}\\,\\hat a_i\n +\\sqrt{1-\\eta}\\,\\hat b_i,\n \\qquad i\\in\\{1,\\ldots,n\\},\n\\tag{1}\n\\end{equation} \nand denote the resulting joint state of $C_1,\\ldots,C_n$ by $\\rho_{C^n}$. In the convention of Eq. (1), $\\eta$ multiplies the vacuum $A$ port, so the attenuator from the information-bearing $B$ port to $C$ has transmissivity $1-\\eta$. Define the thermal entropy function by\n\n \\begin{equation}\n g(x):=(x+1)\\log_2(x+1)-x\\log_2x,\n \\qquad x\\geq0,\n\\tag{2}\n\\end{equation} \nwith $0\\log_2 0:=0$. Using the function in Eq. (2), impose the sole input-entropy constraint $S(\\rho_{B^n})=ng(K)$. Does the state produced in Eq. (1) always obey\n\n \\begin{equation}\n S(\\rho_{C^n})\\geq ng((1-\\eta)K)?\n\\tag{3}\n\\end{equation} \nEquality in Eq. (3) is attained when $\\rho_{B^n}$ is a tensor product of $n$ thermal states of mean photon number $K$; the conjecture asserts that arbitrary correlations cannot lower the output entropy further.",
      "url": "https://qiqc-op.com/problem/op_3cd14aef409b226b/",
      "json": "https://qiqc-op.com/api/problems/op_3cd14aef409b226b.json",
      "tex": "https://qiqc-op.com/problem/op_3cd14aef409b226b/op_3cd14aef409b226b.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "e472b68bfe2150139aa8a330d98ff92480b41c7b312f285438cb0baf30ab7653"
    },
    {
      "id": "op_3ea0de34a1fe6e0b",
      "ulid": "01M1Q787QR8FTR00QF4PMHHWPE",
      "aliases": [
        "op_3ea0de34a1fe6e0b",
        "01M1Q787QR8FTR00QF4PMHHWPE",
        "op-3ea0de34a1fe6e0b"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-cryptography"
        ],
        "topicIds": [
          "quantum-capacity",
          "private-capacity",
          "additivity-and-regularization"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Universal finite truncation of quantum and private capacities",
      "status": "Solved",
      "fields": [
        "Quantum Communication",
        "Quantum Cryptography"
      ],
      "topics": [
        "Quantum capacity",
        "Private capacity",
        "Additivity and regularization"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Cryptography",
        "Quantum capacity",
        "Private capacity",
        "Additivity and regularization"
      ],
      "statement": "Do there exist channel-independent finite integers $m_Q$ and $m_P$ that determine, respectively, the quantum capacity and the private classical capacity of every finite-dimensional quantum channel? Let $V_{A\\to BE}$ be a Stinespring isometry for a channel and its complement, and define coherent information by\n\n \\begin{equation}\n \\mathcal N_{A\\to B}(\\rho):=\\operatorname{Tr}_{E}[V\\rho V^\\dagger],\n \\qquad\n \\mathcal N^c_{A\\to E}(\\rho):=\\operatorname{Tr}_{B}[V\\rho V^\\dagger],\n \\qquad\n I_{\\rm c}(\\rho,\\mathcal N):=\n S(\\mathcal N(\\rho))-S(\\mathcal N^c(\\rho)).\n\\tag{1}\n\\end{equation} \nFor each $m\\geq1$, use Eq. (1) to set\n\n \\begin{equation}\n Q^{(m)}(\\mathcal N)\n :=\\frac1m\\max_{\\rho_{A^m}}\n I_{\\rm c}(\\rho_{A^m},\\mathcal N^{\\otimes m}),\n \\qquad\n Q(\\mathcal N):=\\sup_{m\\geq1}Q^{(m)}(\\mathcal N).\n\\tag{2}\n\\end{equation} \nFor an ensemble $\\{p_x,\\rho_x^{A^m}\\}$, let its joint channel output be\n\n \\begin{equation}\n \\omega^{XB^mE^m}\n :=\\sum_x p_x|x\\rangle\\!\\langle x|^X\\otimes\n V^{\\otimes m}\\rho_x^{A^m}(V^\\dagger)^{\\otimes m}.\n\\tag{3}\n\\end{equation} \nIn terms of the state in Eq. (3), define\n\n \\begin{equation}\n P^{(m)}(\\mathcal N)\n :=\\frac1m\\max_{\\{p_x,\\rho_x^{A^m}\\}}\n \\bigl[I(X;B^m)_\\omega-I(X;E^m)_\\omega\\bigr],\n \\qquad\n P(\\mathcal N):=\\sup_{m\\geq1}P^{(m)}(\\mathcal N).\n\\tag{4}\n\\end{equation} \nThe question is whether there exist finite integers $m_Q$ and $m_P$, independent of the channel dimensions and of $\\mathcal N$, such that\n\n \\begin{equation}\n Q(\\mathcal N)=Q^{(m_Q)}(\\mathcal N)\n \\quad\\text{and}\\quad\n P(\\mathcal N)=P^{(m_P)}(\\mathcal N)\n \\quad\\text{for every finite-dimensional }\\mathcal N.\n\\tag{5}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_3ea0de34a1fe6e0b/",
      "json": "https://qiqc-op.com/api/problems/op_3ea0de34a1fe6e0b.json",
      "tex": "https://qiqc-op.com/problem/op_3ea0de34a1fe6e0b/op_3ea0de34a1fe6e0b.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "fad7c8eafc8f8890f188968a9211f44b1a8e847ca65a68c1c42a1c5799126b89"
    },
    {
      "id": "op_54aa8f0f61ecc5b1",
      "ulid": "01M1Q787QRH2ASM04Q5SDG2H88",
      "aliases": [
        "op_54aa8f0f61ecc5b1",
        "01M1Q787QRH2ASM04Q5SDG2H88",
        "op-54aa8f0f61ecc5b1"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-algorithm"
        ],
        "topicIds": [
          "quantum-capacity",
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Computability of ordinary quantum capacity",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum algorithm"
      ],
      "topics": [
        "Quantum capacity",
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum algorithm",
        "Quantum capacity",
        "Computational complexity and computability"
      ],
      "statement": "Is the ordinary unassisted quantum capacity a computable function of a finite description of a finite-dimensional quantum channel? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be specified either by an exact finite list of Kraus matrices with computable algebraic entries (including rational entries) or by a finite quantum circuit over a fixed algebraic gate set, allowing state preparation and partial trace. For a complementary channel $\\mathcal N^c$, define coherent information and quantum capacity by\n\n \\begin{equation}\n I_c(\\rho,\\mathcal N)\n :=S(\\mathcal N(\\rho))-S(\\mathcal N^c(\\rho)),\n \\qquad\n Q(\\mathcal N)\n :=\\sup_{n\\geq1}\\frac1n\n \\max_{\\rho_{A^{\\otimes n}}}\n I_c(\\rho_{A^{\\otimes n}},\\mathcal N^{\\otimes n}),\n\\tag{1}\n\\end{equation} \nwhere $S(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau)$. Writing $\\langle\\mathcal N\\rangle$ for either finite encoding above, the question is whether there exists a Turing machine $T$ satisfying\n\n \\begin{equation}\n \\forall\\,\\langle\\mathcal N\\rangle\\ \\forall k\\in\\mathbb N:\n \\quad\n T(\\langle\\mathcal N\\rangle,k)=q_{\\mathcal N,k}\\in\\mathbb Q,\n \\qquad\n |q_{\\mathcal N,k}-Q(\\mathcal N)|\\leq2^{-k}.\n\\tag{2}\n\\end{equation} \nDetermine whether the algorithm in Eq. (2) exists for the capacity in Eq. (1), without imposing a running-time bound.",
      "url": "https://qiqc-op.com/problem/op_54aa8f0f61ecc5b1/",
      "json": "https://qiqc-op.com/api/problems/op_54aa8f0f61ecc5b1.json",
      "tex": "https://qiqc-op.com/problem/op_54aa8f0f61ecc5b1/op_54aa8f0f61ecc5b1.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "e1ee1dbd22f362ddf98fa8dd7ca0506946602215fc0728adb659573e1128ac62"
    },
    {
      "id": "op_87c77263c8bab523",
      "ulid": "01M1Q787QRJ5ASJACQYA9R7N35",
      "aliases": [
        "op_87c77263c8bab523",
        "01M1Q787QRJ5ASJACQYA9R7N35",
        "op-87c77263c8bab523"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-recovery",
          "matrix-and-entropy-inequalities"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Ordinary-Petz recovery bound for conditional mutual information",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum recovery",
        "Matrix and entropy inequalities"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum recovery",
        "Matrix and entropy inequalities"
      ],
      "statement": "Does the ordinary, unrotated Petz map universally recover a tripartite state with fidelity controlled by its conditional mutual information? For a finite-dimensional state $\\rho_{ABC}$, define\n\n \\begin{equation}\n I(A;B\\mid C)_\\rho\n :=S(AC)_\\rho+S(BC)_\\rho-S(C)_\\rho-S(ABC)_\\rho,\n \\qquad\n S(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau).\n\\tag{1}\n\\end{equation} \nThe ordinary Petz map associated with $\\rho_{AC}$ and the channel $\\operatorname{Tr}_A:AC\\to C$ is\n\n \\begin{equation}\n \\mathcal P^{\\rho}_{C\\to AC}(X_C)\n :=\\rho_{AC}^{1/2}\\!\\left[\n I_A\\otimes\\rho_C^{-1/2}X_C\\rho_C^{-1/2}\n \\right]\\rho_{AC}^{1/2},\n\\tag{2}\n\\end{equation} \nwhere the inverse is taken on $\\operatorname{supp}\\rho_C$. With squared fidelity $F(\\tau,\\omega):=\\lVert\\sqrt\\tau\\sqrt\\omega\\rVert_1^2$, determine whether the quantity in Eq. (1) always satisfies\n\n \\begin{equation}\n I(A;B\\mid C)_\\rho\n \\stackrel{?}{\\geq}\n -\\log_2 F\\!\\left(\n \\rho_{ABC},\n (\\operatorname{id}_B\\otimes\\mathcal P^{\\rho}_{C\\to AC})(\\rho_{BC})\n \\right),\n\\tag{3}\n\\end{equation} \nwith the recovered systems ordered canonically as $ABC$.",
      "url": "https://qiqc-op.com/problem/op_87c77263c8bab523/",
      "json": "https://qiqc-op.com/api/problems/op_87c77263c8bab523.json",
      "tex": "https://qiqc-op.com/problem/op_87c77263c8bab523/op_87c77263c8bab523.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "0e0ebb0fb82908d8dde3f973f16f573e9eda0922419aabc757ac822a57f8f059"
    },
    {
      "id": "op_89fb664ba06ba5de",
      "ulid": "01M1Q787QRHF57Y6BJ3D34S0H6",
      "aliases": [
        "op_89fb664ba06ba5de",
        "01M1Q787QRHF57Y6BJ3D34S0H6",
        "op-89fb664ba06ba5de"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "bosonic-channels",
          "classical-capacity",
          "one-shot-and-finite-blocklength-bounds"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Candidate pure-loss second-order converse",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Bosonic channels",
        "Classical capacity",
        "One-shot and finite-blocklength bounds"
      ],
      "tags": [
        "Quantum Communication",
        "Bosonic channels",
        "Classical capacity",
        "One-shot and finite-blocklength bounds"
      ],
      "statement": "Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint? Let $\\mathcal N_\\eta$ be the single-mode pure-loss channel with transmissivity $0<\\eta<1$, defined in the Heisenberg picture by\n\n \\begin{equation}\n \\hat b=\\sqrt{\\eta}\\,\\hat a+\\sqrt{1-\\eta}\\,\\hat e,\n\\tag{1}\n\\end{equation} \nwhere the environment mode $E$ is in the vacuum. In an $n$-use code, the channel in Eq. (1) is used to transmit one of $M$ input states $\\rho_m^{A^n}$, with a decoding POVM $\\{\\Lambda_m^{B^n}\\}_{m=1}^M$. Write $\\overline\\rho_{A^n}:=M^{-1}\\sum_m\\rho_m^{A^n}$ and let $\\Pi_{\\lceil nN_S\\rceil}$ project onto the $n$-mode subspace of total photon number at most $\\lceil nN_S\\rceil$. For fixed $N_S>0$, $\\varepsilon\\in(0,1)$, and $c>0$, impose\n\n \\begin{equation}\n \\frac1M\\sum_{m=1}^M\n \\operatorname{Tr}\\!\\left[\n \\Lambda_m\\mathcal N_\\eta^{\\otimes n}(\\rho_m)\n \\right]\n \\geq1-\\varepsilon,\n \\qquad\n \\operatorname{Tr}\\!\\left[\n \\Pi_{\\lceil nN_S\\rceil}\\overline\\rho_{A^n}\n \\right]\n \\geq1-\\delta_n,\n \\qquad\n 0\\leq\\delta_n\\leq2^{-cn}.\n\\tag{2}\n\\end{equation} \nLet $M^*_{\\rm occ}(n,\\eta,N_S,\\varepsilon,c)$ be the largest $M$ satisfying Eq. (2). Define the thermal entropy and its entropy variance by\n\n \\begin{equation}\n g(x):=(x+1)\\log_2(x+1)-x\\log_2x,\n \\qquad\n v(x):=x(x+1)\n \\left[\\log_2(x+1)-\\log_2x\\right]^2,\n\\tag{3}\n\\end{equation} \nwhere $0\\log_2 0:=0$. With the functions in Eq. (3), is the following upper bound valid as $n\\to\\infty$?\n\n \\begin{equation}\n \\log_2 M^*_{\\rm occ}(n,\\eta,N_S,\\varepsilon,c)\n \\leq ng(\\eta N_S)\n +\\sqrt{n\\,v(\\eta N_S)}\\,\\Phi^{-1}(\\varepsilon)\n +O(\\log n),\n\\tag{4}\n\\end{equation} \nHere $\\Phi^{-1}$ in Eq. (4) is the inverse standard-normal cumulative distribution function, and the implicit constant may depend on $\\eta,N_S,\\varepsilon,$ and $c$, but not on $n$.",
      "url": "https://qiqc-op.com/problem/op_89fb664ba06ba5de/",
      "json": "https://qiqc-op.com/api/problems/op_89fb664ba06ba5de.json",
      "tex": "https://qiqc-op.com/problem/op_89fb664ba06ba5de/op_89fb664ba06ba5de.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "693ec46842252e7f5a504d06c3f1dbfce4ba53febfc13a2adb03adfe6727f267"
    },
    {
      "id": "op_a4600b38b94042a8",
      "ulid": "01M1Q787QRN9XH5T5717HHCXHG",
      "aliases": [
        "op_a4600b38b94042a8",
        "01M1Q787QRN9XH5T5717HHCXHG",
        "op-a4600b38b94042a8"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "channel-discrimination",
          "quantum-hypothesis-testing",
          "superchannels-and-quantum-combs"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Parallel versus nested-adaptive superchannel discrimination",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Channel discrimination",
        "Quantum hypothesis testing",
        "Superchannels and quantum combs"
      ],
      "tags": [
        "Quantum metrology",
        "Channel discrimination",
        "Quantum hypothesis testing",
        "Superchannels and quantum combs"
      ],
      "statement": "Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels? A superchannel maps channels $\\mathcal N:A\\to B$ to channels $C\\to D$ and admits a realization\n\n \\begin{equation}\n \\Theta(\\mathcal N)\n =\\mathcal D\\circ(\\mathcal N\\otimes\\operatorname{id}_S)\\circ\\mathcal E,\n \\qquad\n \\mathcal E:C\\to A S,\n \\quad \\mathcal D:B S\\to D.\n\\tag{1}\n\\end{equation} \nThe memory system $S$ in Eq. (1) is internal to the superchannel. In a fully parallel $n$-use strategy, one applies $\\Theta_i^{\\otimes n}$ to a joint $n$-partite inserted channel and then tests the resulting output on a joint input state. A nested-adaptive strategy may instead recursively insert the channel produced by one tested use into the channel slot of another, with arbitrary compatible CPTP maps between uses and a final binary measurement. For $\\mathsf S\\in\\{\\mathrm{par},\\mathrm{nest}\\}$, let\n\n \\begin{equation}\n \\begin{aligned}\n \\beta_{\\varepsilon,n}^{\\mathsf S}(\\Theta_1\\|\\Theta_2)\n &:=\\inf\\{\\beta_n(P):P\\in\\mathsf S_n,\\ \\alpha_n(P)\\leq\\varepsilon\\},\\\\\n \\zeta_{\\mathsf S}(\\Theta_1\\|\\Theta_2)\n &:=\\lim_{\\varepsilon\\downarrow0}\\liminf_{n\\to\\infty}\n -\\frac1n\\log_2\n \\beta_{\\varepsilon,n}^{\\mathsf S}(\\Theta_1\\|\\Theta_2),\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\alpha_n$ and $\\beta_n$ are the type-I and type-II errors. Is\n\n \\begin{equation}\n \\zeta_{\\mathrm{nest}}(\\Theta_1\\|\\Theta_2)\n =\\zeta_{\\mathrm{par}}(\\Theta_1\\|\\Theta_2)\n\\tag{3}\n\\end{equation} \nfor every pair $\\Theta_1,\\Theta_2$?",
      "url": "https://qiqc-op.com/problem/op_a4600b38b94042a8/",
      "json": "https://qiqc-op.com/api/problems/op_a4600b38b94042a8.json",
      "tex": "https://qiqc-op.com/problem/op_a4600b38b94042a8/op_a4600b38b94042a8.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "091b68406b7ba6dddc89e3dd1dddd52edf68409b4a41d011c6bc05e1a2a23b98"
    },
    {
      "id": "op_a59d7cc1c843edb5",
      "ulid": "01M1Q787QRJWENG45M2E70WZXW",
      "aliases": [
        "op_a59d7cc1c843edb5",
        "01M1Q787QRJWENG45M2E70WZXW",
        "op-a59d7cc1c843edb5"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "channel-degradability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Exactly solvable nondegradable quantum channels",
      "status": "Solved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Channel degradability"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Channel degradability"
      ],
      "statement": "Does there exist a finite-dimensional quantum channel that is neither degradable nor antidegradable and whose unassisted quantum capacity is known exactly? For a channel $\\mathcal N_{A\\to B}$ with complementary channel $\\mathcal N^c_{A\\to E}$, degradability and antidegradability mean, respectively, that there is a channel $\\mathcal D$ or $\\mathcal A$ satisfying\n\n \\begin{equation}\n \\mathcal N^c=\\mathcal D_{B\\to E}\\circ\\mathcal N,\n \\qquad\\text{or}\\qquad\n \\mathcal N=\\mathcal A_{E\\to B}\\circ\\mathcal N^c.\n\\tag{1}\n\\end{equation} \nThe channel sought must satisfy neither identity in Eq. (1). Its quantum capacity is defined by the coherent-information regularization\n\n \\begin{equation}\n Q(\\mathcal N)\n :=\\sup_{n\\geq1}\\frac1n\\max_{\\rho_{A^n}}\n \\left[\n S(\\mathcal N^{\\otimes n}(\\rho_{A^n}))\n -S((\\mathcal N^c)^{\\otimes n}(\\rho_{A^n}))\n \\right].\n\\tag{2}\n\\end{equation} \nThe question asks for an explicit channel outside both classes in Eq. (1) together with an exact evaluation of Eq. (2).",
      "url": "https://qiqc-op.com/problem/op_a59d7cc1c843edb5/",
      "json": "https://qiqc-op.com/api/problems/op_a59d7cc1c843edb5.json",
      "tex": "https://qiqc-op.com/problem/op_a59d7cc1c843edb5/op_a59d7cc1c843edb5.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "4398cabe343f7e86d434895c97f0099a09bf5d6d51663dced39cb3821b6c93d9"
    },
    {
      "id": "op_ad12295d8fdfb02a",
      "ulid": "01M1Q787QRDHHK1971H9D8YPN9",
      "aliases": [
        "op_ad12295d8fdfb02a",
        "01M1Q787QRDHHK1971H9D8YPN9",
        "op-ad12295d8fdfb02a"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "classical-capacity",
          "entanglement-assisted-communication",
          "additivity-and-regularization"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Structural criterion for classical–entanglement trade-off advantage",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Classical capacity",
        "Entanglement-assisted communication",
        "Additivity and regularization"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Resource Theory",
        "Classical capacity",
        "Entanglement-assisted communication",
        "Additivity and regularization"
      ],
      "statement": "Characterize the finite-dimensional quantum channels for which joint classical–entanglement coding strictly outperforms time sharing between unassisted and unlimited-entanglement classical communication. For a channel $\\mathcal N:A'\\to B$, let $C_{\\rm CE}(\\mathcal N,e)$ be the supremum of asymptotically achievable classical rates when at most $e$ ebits per channel use are consumed. Operationally,\n\n \\begin{equation}\n C_{\\rm CE}(\\mathcal N,e)\n :=\\sup\\left\\{\n R:\\begin{array}{l}\n \\text{there are length-$n$ classical-message codes with}\\\\\n n^{-1}\\log_2M_n\\to R,\\quad\n \\limsup_{n\\to\\infty}n^{-1}\\log_2K_n\\leq e,\\quad\n P_{\\rm err}^{(n)}\\to0\n \\end{array}\n \\right\\},\n\\tag{1}\n\\end{equation} \nwhere $K_n$ in Eq. (1) is the Schmidt rank of the preshared maximally entangled resource. Define the unassisted and unlimited-entanglement endpoints by\n\n \\begin{equation}\n C_0(\\mathcal N):=C_{\\rm CE}(\\mathcal N,0),\n \\qquad\n C_{\\rm EA}(\\mathcal N):=\\sup_{e\\geq0}C_{\\rm CE}(\\mathcal N,e),\n \\qquad\n e_{\\rm EA}(\\mathcal N)\n :=\\inf\\{e:C_{\\rm CE}(\\mathcal N,e)=C_{\\rm EA}(\\mathcal N)\\}.\n\\tag{2}\n\\end{equation} \nFor $e_{\\rm EA}(\\mathcal N)>0$, endpoint time sharing gives\n\n \\begin{equation}\n C_{\\rm TS}(\\mathcal N,e)\n :=\\begin{cases}\n \\left(1-\\dfrac{e}{e_{\\rm EA}(\\mathcal N)}\\right)C_0(\\mathcal N)\n +\\dfrac{e}{e_{\\rm EA}(\\mathcal N)}C_{\\rm EA}(\\mathcal N),\n &0\\leq e\\leq e_{\\rm EA}(\\mathcal N),\\\\[2mm]\n C_{\\rm EA}(\\mathcal N),&e\\geq e_{\\rm EA}(\\mathcal N).\n \\end{cases}\n\\tag{3}\n\\end{equation} \nIf the infimum in Eq. (2) is not attained, interpret Eq. (3) through its operational closure; if $e_{\\rm EA}(\\mathcal N)=0$, set $C_{\\rm TS}(\\mathcal N,e)=C_{\\rm EA}(\\mathcal N)$. Give intrinsic necessary and sufficient conditions for strict suboptimality of this benchmark:\n\n \\begin{equation}\n \\exists e>0:\\qquad\n C_{\\rm CE}(\\mathcal N,e)>C_{\\rm TS}(\\mathcal N,e).\n\\tag{4}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_ad12295d8fdfb02a/",
      "json": "https://qiqc-op.com/api/problems/op_ad12295d8fdfb02a.json",
      "tex": "https://qiqc-op.com/problem/op_ad12295d8fdfb02a/op_ad12295d8fdfb02a.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "c48afafbecb46a1f5d1a5d8acee5e239476beca5bd2baea68d088962448deb78"
    },
    {
      "id": "op_c9b532d3a7389c77",
      "ulid": "01M1Q787QRSHGZH7NDSMFG88GH",
      "aliases": [
        "op_c9b532d3a7389c77",
        "01M1Q787QRSHGZH7NDSMFG88GH",
        "op-c9b532d3a7389c77"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-source-coding",
          "strong-converses"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Strong converse for general mixed-state quantum compression",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum source coding",
        "Strong converses"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum source coding",
        "Strong converses"
      ],
      "statement": "Does general finite-dimensional i.i.d. mixed-state quantum compression obey an unrestricted strong converse? Let $\\rho^{AR}$ be a finite-dimensional source state, where only $A$ is available to the encoder and $R$ is an inaccessible reference. Fix a Koashi–Imoto isometry $U:A\\to CNQ$ for which the source has the form\n\n \\begin{equation}\n \\omega^{CNQR}\n :=(U\\otimes I_R)\\rho^{AR}(U^\\dagger\\otimes I_R)\n =\\sum_j p_j|j\\rangle\\!\\langle j|^C\n \\otimes\\omega_j^N\\otimes\\rho_j^{QR},\n\\tag{1}\n\\end{equation} \nwhere $C$ is classical, $N$ is redundant relative to $R$ conditioned on $C$, and $Q$ carries the remaining source–reference correlations. At blocklength $n$, allow arbitrary encoding and decoding channels $\\mathcal E_n:A^{\\otimes n}\\to M_n$ and $\\mathcal D_n:M_n\\to\\widehat A^{\\otimes n}$, with $\\widehat A\\cong A$. Their reference-preserving squared fidelity is\n\n \\begin{equation}\n F_n:=F\\!\\left(\n (\\rho^{AR})^{\\otimes n},\n \\left[(\\mathcal D_n\\circ\\mathcal E_n)\n \\otimes\\operatorname{id}_{R^{\\otimes n}}\\right]\n ((\\rho^{AR})^{\\otimes n})\n \\right),\n \\qquad\n F(\\tau,\\zeta):=\\lVert\\sqrt\\tau\\sqrt\\zeta\\rVert_1^2.\n\\tag{2}\n\\end{equation} \nThe optimal first-order qubit rate is $S(CQ)_\\omega$. Determine whether every sequence of unrestricted channels defining Eq. (2) satisfies the strong-converse implication\n\n \\begin{equation}\n \\limsup_{n\\to\\infty}\\frac1n\\log_2|M_n|<S(CQ)_\\omega\n \\quad\\Longrightarrow\\quad\n \\lim_{n\\to\\infty}F_n=0.\n\\tag{3}\n\\end{equation} \nEquation (3) imposes no unitality, isometry, or dimension-expansion condition on the encoder or decoder.",
      "url": "https://qiqc-op.com/problem/op_c9b532d3a7389c77/",
      "json": "https://qiqc-op.com/api/problems/op_c9b532d3a7389c77.json",
      "tex": "https://qiqc-op.com/problem/op_c9b532d3a7389c77/op_c9b532d3a7389c77.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "d44fe42224126ac5316e858e2cdc125806e90326ba6ebacd6537e4d02820fc10"
    },
    {
      "id": "op_cbc0bf88b109b122",
      "ulid": "01M1Q787QR701HKYB3YDFJ15TK",
      "aliases": [
        "op_cbc0bf88b109b122",
        "01M1Q787QR701HKYB3YDFJ15TK",
        "op-cbc0bf88b109b122"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-recovery",
          "quantum-relative-entropy",
          "matrix-and-entropy-inequalities"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Ordinary-Petz fidelity remainder for relative-entropy data processing",
      "status": "Solved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum recovery",
        "Quantum relative entropy",
        "Matrix and entropy inequalities"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum recovery",
        "Quantum relative entropy",
        "Matrix and entropy inequalities"
      ],
      "statement": "Does the ordinary Petz recovery map give a universal fidelity remainder for monotonicity of quantum relative entropy? Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be a finite-dimensional quantum channel, and let $\\rho,\\sigma\\in\\mathcal D(A)$ satisfy $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$. With inverses taken on the relevant supports, define the Petz map by\n\n \\begin{equation}\n \\mathcal P_{\\sigma,\\mathcal N}(X)\n :=\\sigma^{1/2}\\mathcal N^{\\dagger}\\!\\left(\n \\mathcal N(\\sigma)^{-1/2}X\n \\mathcal N(\\sigma)^{-1/2}\n \\right)\\sigma^{1/2},\n\\tag{1}\n\\end{equation} \nwhere $\\mathcal N^{\\dagger}$ is the Hilbert–Schmidt adjoint. Write \\(D(\\tau\\Vert\\omega):=\\operatorname{Tr}[\\tau(\\log_2\\tau-\n\\log_2\\omega)]\\) and use squared fidelity $F(\\tau,\\omega):=\\lVert\\sqrt\\tau\\sqrt\\omega\\rVert_1^2$. The proposed remainder bound for the map in Eq. (1) is\n\n \\begin{equation}\n D(\\rho\\Vert\\sigma)\n -D\\!\\left(\\mathcal N(\\rho)\\middle\\Vert\\mathcal N(\\sigma)\\right)\n \\stackrel{?}{\\geq}\n -\\log_2 F\\!\\left(\n \\rho,\n \\mathcal P_{\\sigma,\\mathcal N}(\\mathcal N(\\rho))\n \\right).\n\\tag{2}\n\\end{equation} \nDetermine whether Eq. (2) holds for every such triple $(\\rho,\\sigma,\\mathcal N)$.",
      "url": "https://qiqc-op.com/problem/op_cbc0bf88b109b122/",
      "json": "https://qiqc-op.com/api/problems/op_cbc0bf88b109b122.json",
      "tex": "https://qiqc-op.com/problem/op_cbc0bf88b109b122/op_cbc0bf88b109b122.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "eae70bec77e763b8a5814a1ad948be3f75f534b0ae65f232b49a1f9d5b3ced7b"
    },
    {
      "id": "op_cdad869fc8c3c0ea",
      "ulid": "01M1Q787QRY0AKF1CQTFSQME12",
      "aliases": [
        "op_cdad869fc8c3c0ea",
        "01M1Q787QRY0AKF1CQTFSQME12",
        "op-cdad869fc8c3c0ea"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "channel-degradability",
          "strong-converses"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "All-code exponential strong converse for degradable channels",
      "status": "Solved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Channel degradability",
        "Strong converses"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Channel degradability",
        "Strong converses"
      ],
      "statement": "Does every finite-dimensional degradable quantum channel satisfy an all-code exponential strong converse for quantum communication at its quantum capacity? Let $V:A\\to B\\otimes E$ be a Stinespring isometry, and define the channel and one complementary channel by\n\n \\begin{equation}\n \\mathcal N(\\rho):=\\operatorname{Tr}_E(V\\rho V^\\dagger),\n \\qquad\n \\mathcal N^c(\\rho):=\\operatorname{Tr}_B(V\\rho V^\\dagger).\n\\tag{1}\n\\end{equation} \nThe channel in Eq. (1) is degradable when there is a completely positive trace-preserving map $\\mathcal D:\\mathcal L(B)\\to\\mathcal L(E)$ such that\n\n \\begin{equation}\n \\mathcal N^c=\\mathcal D\\circ\\mathcal N.\n\\tag{2}\n\\end{equation} \nFor a channel satisfying Eq. (2), its quantum capacity is the single-letter coherent information\n\n \\begin{equation}\n Q(\\mathcal N)=Q^{(1)}(\\mathcal N)\n :=\\max_{\\rho_A}\n \\left[S(\\mathcal N(\\rho_A))-S(\\mathcal N^c(\\rho_A))\\right],\n \\qquad\n S(\\tau):=-\\operatorname{Tr}(\\tau\\log_2\\tau).\n\\tag{3}\n\\end{equation} \nFor an arbitrary $n$-use entanglement-transmission code, let the encoder and decoder be CPTP maps $\\mathcal E_n:\\mathcal L(S_n)\\to\\mathcal L(A^{\\otimes n})$ and $\\mathcal R_n:\\mathcal L(B^{\\otimes n})\\to\\mathcal L(\\widehat S_n)$, with $\\dim S_n=\\dim\\widehat S_n=M_n$. If $\\Phi_{M_n}^{R_nS_n}$ is maximally entangled, define the code rate and entanglement fidelity by\n\n \\begin{equation}\n r_n:=\\frac{1}{n}\\log_2 M_n,\n \\qquad\n F_n:=\\operatorname{Tr}\\!\\left[\n \\Phi_{M_n}^{R_n\\widehat S_n}\n \\bigl(\\operatorname{id}_{R_n}\\otimes\n \\mathcal R_n\\circ\\mathcal N^{\\otimes n}\\circ\\mathcal E_n\\bigr)\n (\\Phi_{M_n}^{R_nS_n})\n \\right].\n\\tag{4}\n\\end{equation} \nThe all-code exponential strong converse asks whether, for every $R>Q(\\mathcal N)$, there are constants $\\gamma_R>0$ and $n_R$ such that every code in Eq. (4) obeys\n\n \\begin{equation}\n r_n\\geq R,\\qquad n\\geq n_R\n \\quad\\Longrightarrow\\quad\n F_n\\leq 2^{-\\gamma_R n}.\n\\tag{5}\n\\end{equation} \nEquation (5) requires a bound for every encoder–decoder pair, rather than for almost every member of a random code ensemble.",
      "url": "https://qiqc-op.com/problem/op_cdad869fc8c3c0ea/",
      "json": "https://qiqc-op.com/api/problems/op_cdad869fc8c3c0ea.json",
      "tex": "https://qiqc-op.com/problem/op_cdad869fc8c3c0ea/op_cdad869fc8c3c0ea.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "aad5d7328f35e16847fe43d1d2a86d79a945d12a0be35d619f270746b36a0494"
    },
    {
      "id": "op_d754e0d170c01f86",
      "ulid": "01M1Q787QR780435R6682GE26Y",
      "aliases": [
        "op_d754e0d170c01f86",
        "01M1Q787QR780435R6682GE26Y",
        "op-d754e0d170c01f86"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-relative-entropy",
          "one-shot-and-finite-blocklength-bounds"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Diamond-smoothed max-relative-entropy AEP for quantum channels",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum relative entropy",
        "One-shot and finite-blocklength bounds"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum relative entropy",
        "One-shot and finite-blocklength bounds"
      ],
      "statement": "Does the max-relative entropy of finite-dimensional quantum channels satisfy an asymptotic equipartition property under uniform diamond-norm smoothing? Let $\\mathcal N,\\mathcal M:\\mathcal L(A)\\to\\mathcal L(B)$ be quantum channels with $D_{\\max}(\\mathcal N\\|\\mathcal M)<\\infty$. Using unnormalized Choi operators, define the channel max-relative entropy and its smoothed version by\n\n \\begin{equation}\n \\begin{aligned}\n D_{\\max}(\\mathcal N\\|\\mathcal M)\n &:=\\inf\\{\\lambda:J_{\\mathcal N}\\leq2^\\lambda J_{\\mathcal M}\\},\\\\\n D_{\\max}^{\\varepsilon}(\\mathcal N\\|\\mathcal M)\n &:=\\inf_{\\substack{\\widetilde{\\mathcal N}\\ {\\rm CPTP}:\\\\\n \\frac12\\|\\widetilde{\\mathcal N}-\\mathcal N\\|_\\diamond\n \\leq\\varepsilon}}\n D_{\\max}(\\widetilde{\\mathcal N}\\|\\mathcal M).\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nThe smoothing in Eq. (1) requires one channel $\\widetilde{\\mathcal N}$ that approximates $\\mathcal N$ uniformly over all ancilla-assisted inputs. Define\n\n \\begin{equation}\n \\begin{aligned}\n D_{\\max}^{\\varepsilon,\\infty}(\\mathcal N\\|\\mathcal M)\n &:=\\limsup_{n\\to\\infty}\\frac1n\n D_{\\max}^{\\varepsilon}\n (\\mathcal N^{\\otimes n}\\|\\mathcal M^{\\otimes n}),\\\\\n D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M)\n &:=\\lim_{n\\to\\infty}\\frac1n\n \\sup_{\\psi_{R A^n}}\n D\\!\\left((\\operatorname{id}_R\\otimes\\mathcal N^{\\otimes n})(\\psi)\n \\middle\\|\n (\\operatorname{id}_R\\otimes\\mathcal M^{\\otimes n})(\\psi)\\right),\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $R\\simeq A^{\\otimes n}$ suffices and $D$ is quantum relative entropy. Is the following identity valid for every such channel pair, and can the $\\limsup$ in Eq. (2) be replaced by a limit?\n\n \\begin{equation}\n \\sup_{\\varepsilon>0}\n D_{\\max}^{\\varepsilon,\\infty}(\\mathcal N\\|\\mathcal M)\n =D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M).\n\\tag{3}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_d754e0d170c01f86/",
      "json": "https://qiqc-op.com/api/problems/op_d754e0d170c01f86.json",
      "tex": "https://qiqc-op.com/problem/op_d754e0d170c01f86/op_d754e0d170c01f86.tex",
      "created": "2026-09-03",
      "updated": "2026-09-04",
      "createdAt": "2026-09-03T02:22:57.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "1d6611834a7823837741fab7e0c43adad3bf970e0ca0c5d4dceed9d63ae84e6f"
    },
    {
      "id": "op_1e7f431ed9013d1e",
      "ulid": "01M1Q787QR71ZZJVC3XC65A5F3",
      "aliases": [
        "op_1e7f431ed9013d1e",
        "01M1Q787QR71ZZJVC3XC65A5F3",
        "op-1e7f431ed9013d1e"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "bosonic-channels"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Quantum capacity of the Gaussian random-displacement channel",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Bosonic channels"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Bosonic channels"
      ],
      "statement": "What is the unconstrained, unassisted quantum capacity of the single-mode Gaussian random-displacement channel? For a noise standard deviation $\\sigma>0$, define the channel by\n\n \\begin{equation}\n \\mathcal N_\\sigma(\\rho)\n :=\\frac{1}{\\pi\\sigma^2}\\int_{\\mathbb C}\n e^{-\\lvert\\alpha\\rvert^2/\\sigma^2}\n D(\\alpha)\\rho D(\\alpha)^\\dagger\\,d^2\\alpha,\n \\qquad\n D(\\alpha):=e^{\\alpha a^\\dagger-\\alpha^*a},\n\\tag{1}\n\\end{equation} \nwhere $a$ is the mode annihilation operator. Equivalently, writing $\\alpha=(x+iy)/\\sqrt2$, the channel in Eq. (1) adds independent classical Gaussian shifts $x,y\\sim\\mathcal N(0,\\sigma^2)$ to the two canonical quadratures. With $\\hat n_j:=a_j^\\dagger a_j$, define its energy-unconstrained capacity as\n\n \\begin{equation}\n \\begin{aligned}\n \\mathcal Q(\\mathcal N_\\sigma)\n &:={\\sup}_{0<N_{\\rm S}<\\infty}\\,\n \\lim_{n\\to\\infty}\\frac1n\n {\\sup}_{\\substack{\\rho_{A^n}:\\\\\n \\operatorname{Tr}[\\rho_{A^n}\\sum_{j=1}^n\\hat n_j]\n \\leq nN_{\\rm S}}}\n I_{\\rm c}(\\rho_{A^n},\\mathcal N_\\sigma^{\\otimes n}),\\\\\n I_{\\rm c}(\\rho,\\mathcal M)\n &:=S(\\mathcal M(\\rho))-S(\\mathcal M^{\\rm c}(\\rho)),\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\mathcal M^{\\rm c}$ is a complementary channel and logarithms in the entropy are base two. Determine Eq. (2) for every $\\sigma>0$.",
      "url": "https://qiqc-op.com/problem/op_1e7f431ed9013d1e/",
      "json": "https://qiqc-op.com/api/problems/op_1e7f431ed9013d1e.json",
      "tex": "https://qiqc-op.com/problem/op_1e7f431ed9013d1e/op_1e7f431ed9013d1e.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T23:47:21.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "325229a90df0f6248aa6ea011eaac8e4cb5827930823d14b6abe06f223f3ce9e"
    },
    {
      "id": "op_76e284219621a785",
      "ulid": "01M1Q787QR0M0RT7TK205931W8",
      "aliases": [
        "op_76e284219621a785",
        "01M1Q787QR0M0RT7TK205931W8",
        "op-76e284219621a785"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "quantum-state-discrimination"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Convergence of the JRF iteration for mixed-state discrimination",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Quantum state discrimination"
      ],
      "tags": [
        "Quantum metrology",
        "Quantum state discrimination"
      ],
      "statement": "Does the Ježek–Řeháček–Fiurášek (JRF) iteration, initialized by the uniform POVM, converge to a globally optimal minimum-error measurement for every finite ensemble containing mixed quantum states? Let $m\\geq2$, let $p_i>0$ be prior probabilities, and let $\\rho_i$ be density operators on a finite-dimensional Hilbert space. Define\n\n \\begin{equation}\n \\xi_i:=p_i\\rho_i,\n \\qquad\n \\operatorname{Tr}\\rho_i=1,\n \\qquad\n \\sum_{i=1}^{m}p_i=1,\n \\qquad\n \\mathcal H_0:=\\operatorname{supp}\\!\\left(\\sum_{i=1}^{m}\\xi_i\\right),\n\\tag{1}\n\\end{equation} \nwhere at least one $\\rho_i$ has rank greater than one. On the signal space $\\mathcal H_0$ in Eq. (1), the optimal guessing probability is\n\n \\begin{equation}\n P_{\\mathrm{opt}}\n :=\\max_{\\substack{\\Pi_i\\succeq0\\\\\n \\sum_{i=1}^{m}\\Pi_i=I_{\\mathcal H_0}}}\n \\sum_{i=1}^{m}\\operatorname{Tr}(\\xi_i\\Pi_i).\n\\tag{2}\n\\end{equation} \nStarting from $\\Pi_i^{(0)}:=I_{\\mathcal H_0}/m$, define the JRF iterates by\n\n \\begin{equation}\n \\Lambda_k\n :=\\left(\\sum_{j=1}^{m}\n \\xi_j\\Pi_j^{(k)}\\xi_j\\right)^{1/2},\n \\qquad\n \\Pi_i^{(k+1)}\n :=\\Lambda_k^{-1}\\xi_i\\Pi_i^{(k)}\\xi_i\\Lambda_k^{-1}\n \\quad (i=1,\\ldots,m).\n\\tag{3}\n\\end{equation} \nFor this initialization, $\\Lambda_k$ is inverted on $\\mathcal H_0$ and the operators in Eq. (3) form a POVM at every step. Determine whether there always exists an optimal POVM $\\{\\Pi_i^\\star\\}_{i=1}^{m}$ attaining Eq. (2) such that\n\n \\begin{equation}\n \\lim_{k\\to\\infty}\n \\sum_{i=1}^{m}\\lVert\\Pi_i^{(k)}-\\Pi_i^\\star\\rVert_2=0,\n \\qquad\n \\lim_{k\\to\\infty}\n \\sum_{i=1}^{m}\\operatorname{Tr}(\\xi_i\\Pi_i^{(k)})\n =P_{\\mathrm{opt}},\n\\tag{4}\n\\end{equation} \nwhere $\\lVert\\cdot\\rVert_2$ is the Hilbert–Schmidt norm. If Eq. (4) fails, construct an explicit mixed-state counterexample and characterize its limiting behavior.",
      "url": "https://qiqc-op.com/problem/op_76e284219621a785/",
      "json": "https://qiqc-op.com/api/problems/op_76e284219621a785.json",
      "tex": "https://qiqc-op.com/problem/op_76e284219621a785/op_76e284219621a785.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T23:12:23.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "3ba94d2d57fe2e5745bd0c93b9413df3e77a6e6537417711c5e274ef1212735e"
    },
    {
      "id": "op_c37650bfb81dbfc6",
      "ulid": "01M1Q787QR08CREPZSZYDXBTGN",
      "aliases": [
        "op_c37650bfb81dbfc6",
        "01M1Q787QR08CREPZSZYDXBTGN",
        "op-c37650bfb81dbfc6"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "local-unitary-equivalence"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Minimum LU–LC counterexample for graph states",
      "status": "Solved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Local unitary equivalence"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Local unitary equivalence"
      ],
      "statement": "What is the least number of qubits for which two graph states can be locally unitary equivalent without being locally Clifford equivalent? For a simple graph $G=(V,E)$ with $V=\\{1,\\ldots,n\\}$, define its graph state by\n\n \\begin{equation}\n \\lvert G\\rangle\n :=\\left(\\prod_{\\{u,v\\}\\in E}\\mathrm{CZ}_{uv}\\right)\n \\lvert+\\rangle^{\\otimes n},\n \\qquad\n \\lvert+\\rangle:=\\frac{\\lvert0\\rangle+\\lvert1\\rangle}{\\sqrt2}.\n\\tag{1}\n\\end{equation} \nFor graphs $G$ and $H$ on $n$ vertices, use the state convention in Eq. (1) and write\n\n \\begin{equation}\n \\begin{aligned}\n G\\sim_{\\mathrm{LU}}H\n &\\iff\n \\lvert H\\rangle=e^{i\\phi}\n \\left(\\bigotimes_{j=1}^{n}U_j\\right)\\lvert G\\rangle\n &&\\text{for some }U_j\\in U(2),\\ \\phi\\in\\mathbb R,\\\\\n G\\sim_{\\mathrm{LC}}H\n &\\iff\n \\lvert H\\rangle=e^{i\\theta}\n \\left(\\bigotimes_{j=1}^{n}C_j\\right)\\lvert G\\rangle\n &&\\text{for some }C_j\\in\\mathcal C_1,\\ \\theta\\in\\mathbb R,\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\mathcal C_1$ is the single-qubit Clifford group. Since $\\mathcal C_1\\subset U(2)$, LC equivalence implies LU equivalence. Define the minimum counterexample size by\n\n \\begin{equation}\n n_{\\min}\n :=\\min\\left\\{n:\\text{there exist $n$-vertex graphs $G,H$ with\n $G\\sim_{\\mathrm{LU}}H$ and $G\\not\\sim_{\\mathrm{LC}}H$}\\right\\}.\n\\tag{3}\n\\end{equation} \nDetermine the integer in Eq. (3).",
      "url": "https://qiqc-op.com/problem/op_c37650bfb81dbfc6/",
      "json": "https://qiqc-op.com/api/problems/op_c37650bfb81dbfc6.json",
      "tex": "https://qiqc-op.com/problem/op_c37650bfb81dbfc6/op_c37650bfb81dbfc6.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T23:12:23.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "b192e3c0a2b91e3a7597bf526bc0908ece01a2a8e0e25967a7dbcc8fddf6d1f4"
    },
    {
      "id": "op_09b9fa91a1ac1a76",
      "ulid": "01M1HME780EGVAT19D7T4BGNB5",
      "aliases": [
        "op_09b9fa91a1ac1a76",
        "01M1HME780EGVAT19D7T4BGNB5",
        "op-09b9fa91a1ac1a76",
        "mothe-2023-indefinite-causal-order-asymptotic-metrology",
        "gaugeforge-quantum-0055"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "quantum-estimation",
          "indefinite-causal-order"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Asymptotic metrology with quantum-controlled causal order",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Quantum estimation",
        "Indefinite causal order"
      ],
      "tags": [
        "Quantum metrology",
        "Quantum Resource Theory",
        "Quantum estimation",
        "Indefinite causal order"
      ],
      "statement": "Does quantum control of causal order yield a persistent asymptotic metrological advantage over parallel access for some smooth finite-dimensional channel family? Let $\\Lambda_\\theta:\\mathcal L(A)\\to\\mathcal L(B)$ be a smooth one-parameter family, and let $\\mathcal F_{\\mathrm{PAR}}^{(N)}(\\theta)$ and $\\mathcal F_{\\mathrm{QCQC}}^{(N)}(\\theta)$ be the largest output-state quantum Fisher information attainable from $N$ black-box uses by, respectively, a parallel strategy and a quantum circuit with quantum control of causal order (QC-QC). In a QC-QC strategy, a coherent control can dynamically select which unused black-box call occurs next; this is more general than coherently superposing fixed causal orders. At every regular parameter value for which $\\mathcal F_{\\mathrm{PAR}}^{(N)}(\\theta)>0$ for all sufficiently large $N$, decide whether every channel family satisfies\n\n \\begin{equation}\n \\limsup_{N\\to\\infty}\n \\frac{\\mathcal F_{\\mathrm{QCQC}}^{(N)}(\\theta)}\n {\\mathcal F_{\\mathrm{PAR}}^{(N)}(\\theta)}\n =1.\n\\tag{1}\n\\end{equation} \nEquivalently, either prove Eq. (1) or construct a noisy finite-dimensional family and physical QC-QC strategies with a persistent constant-factor advantage or a larger scaling exponent.",
      "url": "https://qiqc-op.com/problem/op_09b9fa91a1ac1a76/",
      "json": "https://qiqc-op.com/api/problems/op_09b9fa91a1ac1a76.json",
      "tex": "https://qiqc-op.com/problem/op_09b9fa91a1ac1a76/op_09b9fa91a1ac1a76.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "61a0d2c5138e381eaf2167628e504ff7a034f3a95b98b839c2f51562f801cfa1"
    },
    {
      "id": "op_523ed75735cfe6c3",
      "ulid": "01M1HME78053BRQ9RY6RY1YHSW",
      "aliases": [
        "op_523ed75735cfe6c3",
        "01M1HME78053BRQ9RY6RY1YHSW",
        "op-523ed75735cfe6c3",
        "ruskai-2007-convex-decompositions-cpt-maps"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Equal-weight low-Choi-rank decompositions of quantum channels",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum channel structure"
      ],
      "statement": "Can every finite-dimensional quantum channel be written as the uniform mixture of $d_B$ channels whose Choi ranks are at most the input dimension? Let $A$ and $B$ have dimensions $d_A$ and $d_B$, respectively, and let $\\Phi:\\mathcal L(A)\\to\\mathcal L(B)$ be completely positive and trace preserving. In a fixed orthonormal basis of $A$, define its Choi operator by\n\n \\begin{equation}\n J(\\Phi)\n :=\\sum_{i,j=1}^{d_A}\n \\lvert i\\rangle\\!\\langle j\\rvert_A\\otimes\n \\Phi\\!\\left(\\lvert i\\rangle\\!\\langle j\\rvert_A\\right),\n\\tag{1}\n\\end{equation} \nThe rank of the operator in Eq. (1) is independent of the chosen basis. Determine whether every $\\Phi$ admits completely positive trace-preserving maps $\\Phi_1,\\ldots,\\Phi_{d_B}:\\mathcal L(A)\\to\\mathcal L(B)$ satisfying\n\n \\begin{equation}\n \\Phi=\\frac1{d_B}\\sum_{r=1}^{d_B}\\Phi_r,\n \\qquad\n \\operatorname{rank}J(\\Phi_r)\\leq d_A\n \\quad\\text{for every }r\\in\\{1,\\ldots,d_B\\}.\n\\tag{2}\n\\end{equation} \nThus Eq. (2) requires both the prescribed input-dimension rank bound and exactly equal mixing weights.",
      "url": "https://qiqc-op.com/problem/op_523ed75735cfe6c3/",
      "json": "https://qiqc-op.com/api/problems/op_523ed75735cfe6c3.json",
      "tex": "https://qiqc-op.com/problem/op_523ed75735cfe6c3/op_523ed75735cfe6c3.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "cb3400470092b23678eb57e92b232d75c9edb61038a1e81f7faa2ffeb97b024b"
    },
    {
      "id": "op_7920f48995bc8511",
      "ulid": "01M1HME780EK3RBP2STMGR8JS5",
      "aliases": [
        "op_7920f48995bc8511",
        "01M1HME780EK3RBP2STMGR8JS5",
        "op-7920f48995bc8511",
        "ruskai-2007-mutually-degradable-channels"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "channel-degradability",
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Nontrivial mutually degradable channel pairs",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Channel degradability",
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Communication",
        "Channel degradability",
        "Quantum channel structure"
      ],
      "statement": "Does there exist an integer $d\\geq2$ and a pair of distinct channels $\\mathcal M,\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$, with $A\\simeq B\\simeq\\mathbb C^d$, that both have Choi rank exactly $d$, are mutually degradable, and are each nondegradable? Let $E\\simeq\\mathbb C^d$ and choose minimal Stinespring isometries $V_{\\mathcal M},V_{\\mathcal N}:A\\to B\\otimes E$ defining the channels and their complements by\n\n \\begin{equation}\n \\begin{aligned}\n \\mathcal M(\\rho)&=\\operatorname{Tr}_E\n (V_{\\mathcal M}\\rho V_{\\mathcal M}^{\\dagger}),\n &\\mathcal M^c(\\rho)&=\\operatorname{Tr}_B\n (V_{\\mathcal M}\\rho V_{\\mathcal M}^{\\dagger}),\\\\\n \\mathcal N(\\rho)&=\\operatorname{Tr}_E\n (V_{\\mathcal N}\\rho V_{\\mathcal N}^{\\dagger}),\n &\\mathcal N^c(\\rho)&=\\operatorname{Tr}_B\n (V_{\\mathcal N}\\rho V_{\\mathcal N}^{\\dagger}).\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nEquation (1) fixes representatives of the complementary channels; changing a minimal dilation only applies an output unitary to a complement.\n\nFor $\\lvert\\Omega_d\\rangle:=\\sum_{j=1}^d\\lvert j\\rangle_{A'}\\lvert j\\rangle_A$, the required Choi-rank condition is\n\n \\begin{equation}\n J(\\mathcal T):=(\\operatorname{id}_{A'}\\otimes\\mathcal T)\n (\\lvert\\Omega_d\\rangle\\!\\langle\\Omega_d\\rvert),\n \\qquad\n \\operatorname{rank}J(\\mathcal M)\n =\\operatorname{rank}J(\\mathcal N)=d.\n\\tag{2}\n\\end{equation} \nThe equality in Eq. (2) makes the environment dimension in Eq. (1) minimal.\n\nMutual degradability requires channels $\\mathcal X,\\mathcal Y:\\mathcal L(B)\\to\\mathcal L(E)$ such that\n\n \\begin{equation}\n \\mathcal X\\circ\\mathcal M=\\mathcal N^c,\n \\qquad\n \\mathcal Y\\circ\\mathcal N=\\mathcal M^c.\n\\tag{3}\n\\end{equation} \nIn addition to Eq. (3), neither channel may admit its own degrading map:\n\n \\begin{equation}\n \\begin{aligned}\n &\\nexists\\ \\mathcal D_{\\mathcal M}:\\mathcal L(B)\\to\\mathcal L(E)\n \\quad\\text{CPTP with}\\quad\n \\mathcal M^c=\\mathcal D_{\\mathcal M}\\circ\\mathcal M,\\\\\n &\\nexists\\ \\mathcal D_{\\mathcal N}:\\mathcal L(B)\\to\\mathcal L(E)\n \\quad\\text{CPTP with}\\quad\n \\mathcal N^c=\\mathcal D_{\\mathcal N}\\circ\\mathcal N.\n \\end{aligned}\n\\tag{4}\n\\end{equation} \nEquation (4), together with $\\mathcal M\\neq\\mathcal N$, excludes the known degenerate constructions.",
      "url": "https://qiqc-op.com/problem/op_7920f48995bc8511/",
      "json": "https://qiqc-op.com/api/problems/op_7920f48995bc8511.json",
      "tex": "https://qiqc-op.com/problem/op_7920f48995bc8511/op_7920f48995bc8511.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "642ff389a919c0d38f4536361a02edb594bc1e393b7b04d6b57ebdcd33cf39f6"
    },
    {
      "id": "op_8c6e6d0cc3d28e86",
      "ulid": "01M1Q787QRMK8JJ7BH5J7A8VXS",
      "aliases": [
        "op_8c6e6d0cc3d28e86",
        "01M1Q787QRMK8JJ7BH5J7A8VXS",
        "op-8c6e6d0cc3d28e86"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-recovery",
          "quantum-relative-entropy",
          "matrix-and-entropy-inequalities"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Umegaki relative entropy of local recovery",
      "status": "Solved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum recovery",
        "Quantum relative entropy",
        "Matrix and entropy inequalities"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum recovery",
        "Quantum relative entropy",
        "Matrix and entropy inequalities"
      ],
      "statement": "Does the conditional mutual information of every finite-dimensional tripartite state dominate its Umegaki relative entropy of local recovery? For a state $\\rho_{ABC}$, define\n\n \\begin{equation}\n I(A:C\\mid B)_\\rho\n :=S(AB)_\\rho+S(BC)_\\rho-S(B)_\\rho-S(ABC)_\\rho.\n\\tag{1}\n\\end{equation} \nWith $D(\\tau\\Vert\\omega):=\\operatorname{Tr}[\\tau(\\log\\tau-\\log\\omega)]$ when $\\operatorname{supp}\\tau\\subseteq\\operatorname{supp}\\omega$, the question is whether the quantity in Eq. (1) always satisfies\n\n \\begin{equation}\n I(A:C\\mid B)_\\rho\n \\stackrel{?}{\\ge}\n \\min_{\\mathcal R_{B\\to BC}}\n D\\!\\left(\n \\rho_{ABC}\n \\middle\\Vert\n (\\operatorname{id}_A\\otimes\\mathcal R_{B\\to BC})(\\rho_{AB})\n \\right),\n\\tag{2}\n\\end{equation} \nwhere the minimum in Eq. (2) is over all completely positive trace-preserving recovery maps $\\mathcal R_{B\\to BC}$.",
      "url": "https://qiqc-op.com/problem/op_8c6e6d0cc3d28e86/",
      "json": "https://qiqc-op.com/api/problems/op_8c6e6d0cc3d28e86.json",
      "tex": "https://qiqc-op.com/problem/op_8c6e6d0cc3d28e86/op_8c6e6d0cc3d28e86.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "bf81b93ccfc414106425ab6ded6509fa1815d1a412e2b75142725992e7e86a68"
    },
    {
      "id": "op_a3a8680c50800797",
      "ulid": "01M1Q787QRBXA9T9KBKMSKZBMY",
      "aliases": [
        "op_a3a8680c50800797",
        "01M1Q787QRBXA9T9KBKMSKZBMY",
        "op-a3a8680c50800797"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 1,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-measures",
          "bell-diagonal-states"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Entanglement of formation of generalized Bell-diagonal states",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement measures",
        "Bell-diagonal states"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement measures",
        "Bell-diagonal states"
      ],
      "statement": "For every local dimension $d\\geq3$, determine the entanglement of formation of an arbitrary Weyl–Bell-diagonal state. Let $\\omega_d:=\\exp(2\\pi i/d)$ and define the generalized Pauli operators and their associated Bell basis by\n\n \\begin{equation}\n X\\lvert j\\rangle:=\\lvert j+1\\!\\!\\pmod d\\rangle,\n \\qquad\n Z\\lvert j\\rangle:=\\omega_d^j\\lvert j\\rangle,\n \\qquad\n \\lvert\\Phi_{a,b}\\rangle\n :=(I\\otimes X^aZ^b)\\lvert\\Phi_d\\rangle,\n \\qquad\n \\lvert\\Phi_d\\rangle:=\\frac1{\\sqrt d}\\sum_{j=0}^{d-1}\\lvert j,j\\rangle,\n\\tag{1}\n\\end{equation} \nwhere $a,b\\in\\mathbb Z_d$. In this problem, “Pauli-diagonal” means diagonal in the generalized Bell basis in Eq. (1). Thus the state is\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n :=\\sum_{a,b\\in\\mathbb Z_d}p_{a,b}\n \\lvert\\Phi_{a,b}\\rangle\\!\\langle\\Phi_{a,b}\\rvert,\n \\qquad\n p_{a,b}\\geq0,\n \\qquad\n \\sum_{a,b\\in\\mathbb Z_d}p_{a,b}=1.\n\\tag{2}\n\\end{equation} \nFor every probability array $\\mathbf p$ in Eq. (2), determine an evaluable exact formula and an optimal pure-state ensemble for\n\n \\begin{equation}\n E_F(\\rho_{\\mathbf p})\n :=\\inf_{\\rho_{\\mathbf p}=\\sum_i q_i\n \\lvert\\psi_i\\rangle\\!\\langle\\psi_i\\rvert}\n \\sum_i q_i\\,\n S\\!\\left(\\operatorname{Tr}_B\n \\lvert\\psi_i\\rangle\\!\\langle\\psi_i\\rvert\\right),\n \\qquad\n S(\\sigma):=-\\operatorname{Tr}(\\sigma\\log_2\\sigma).\n\\tag{3}\n\\end{equation} \nThe infimum in Eq. (3) is over finite pure-state ensembles with $q_i\\geq0$ and $\\sum_iq_i=1$.",
      "url": "https://qiqc-op.com/problem/op_a3a8680c50800797/",
      "json": "https://qiqc-op.com/api/problems/op_a3a8680c50800797.json",
      "tex": "https://qiqc-op.com/problem/op_a3a8680c50800797/op_a3a8680c50800797.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "8499e1449ce77981cf215cc4d29be94997f3b46a3131503cead9e6980ff9c54a"
    },
    {
      "id": "op_ad05396ff490713c",
      "ulid": "01M1HME780XSRZ7K9HQJSZ176R",
      "aliases": [
        "op_ad05396ff490713c",
        "01M1HME780XSRZ7K9HQJSZ176R",
        "op-ad05396ff490713c",
        "ruskai-2007-werner-holevo-channel-multiplicativity"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "additivity-and-regularization",
          "quantum-channel-structure",
          "matrix-and-entropy-inequalities"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Multiplicativity for polarized Werner–Holevo channels",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Additivity and regularization",
        "Quantum channel structure",
        "Matrix and entropy inequalities"
      ],
      "tags": [
        "Quantum Communication",
        "Additivity and regularization",
        "Quantum channel structure",
        "Matrix and entropy inequalities"
      ],
      "statement": "For every integer $d\\geq3$, every $x\\in(0,1)$, and every $1<p<2$, is the maximal output Schatten $p$-norm of the polarized Werner–Holevo channel multiplicative on two identical copies? Define the Werner–Holevo channel and its polarized interpolation with the identity channel by\n\n \\begin{equation}\n \\mathcal W_d(X):=\\frac{\\operatorname{Tr}(X)I_d-X^{\\mathsf T}}{d-1},\n \\qquad\n \\Phi_{x,d}:=x\\,\\operatorname{id}_d+(1-x)\\mathcal W_d,\n\\tag{1}\n\\end{equation} \nwhere the transpose in Eq. (1) is taken in a fixed basis. For a channel $\\Phi$ with $d$-dimensional input, set\n\n \\begin{equation}\n \\lVert A\\rVert_p:=\\bigl(\\operatorname{Tr}\\lvert A\\rvert^p\\bigr)^{1/p},\n \\qquad\n \\nu_p(\\Phi):=\\max_{\\rho\\in\\mathcal D(\\mathbb C^d)}\n \\lVert\\Phi(\\rho)\\rVert_p.\n\\tag{2}\n\\end{equation} \nWith the convention in Eq. (2), determine whether\n\n \\begin{equation}\n \\nu_p(\\Phi_{x,d}\\otimes\\Phi_{x,d})\n =\\nu_p(\\Phi_{x,d})^2\n\\tag{3}\n\\end{equation} \nholds throughout the stated parameter range.",
      "url": "https://qiqc-op.com/problem/op_ad05396ff490713c/",
      "json": "https://qiqc-op.com/api/problems/op_ad05396ff490713c.json",
      "tex": "https://qiqc-op.com/problem/op_ad05396ff490713c/op_ad05396ff490713c.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "94fc54a06d76a11f9fa8eee5c16c5a1cc163ab510c15ba963a684c1562dd4f63"
    },
    {
      "id": "op_bca77ec42ddd1d5c",
      "ulid": "01M1HME780S5JZKCQN8X0RR8TG",
      "aliases": [
        "op_bca77ec42ddd1d5c",
        "01M1HME780S5JZKCQN8X0RR8TG",
        "op-bca77ec42ddd1d5c",
        "theoremdb-p42-quantum-pcp-conjecture",
        "theoremdb-p42"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "hamiltonian-complexity",
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "The quantum PCP conjecture",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Hamiltonian complexity",
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Hamiltonian complexity",
        "Computational complexity and computability"
      ],
      "statement": "Is the constant-relative-gap local Hamiltonian problem QMA-hard? More precisely, do there exist fixed integers $q,k\\geq2$ and constants $0\\leq a<b\\leq1$ such that the following promise problem is QMA-hard? An instance consists of $n$ subsystems of local dimension $q$ and $m=\\operatorname{poly}(n)$ positive semidefinite terms, each specified with polynomially many bits, for which\n\n \\begin{equation}\n H:=\\sum_{i=1}^{m}H_i,\n \\qquad\n 0\\preceq H_i\\preceq I,\n \\qquad\n \\lvert\\operatorname{supp}(H_i)\\rvert\\leq k.\n\\tag{1}\n\\end{equation} \nGiven the Hamiltonian in Eq. (1), distinguish the promised alternatives\n\n \\begin{equation}\n \\lambda_{\\min}(H)\\leq am\n \\qquad\\text{and}\\qquad\n \\lambda_{\\min}(H)\\geq bm.\n\\tag{2}\n\\end{equation} \nThus the gap in Eq. (2) is a fixed positive fraction $(b-a)m$ of the number of local terms, independent of $n$.",
      "url": "https://qiqc-op.com/problem/op_bca77ec42ddd1d5c/",
      "json": "https://qiqc-op.com/api/problems/op_bca77ec42ddd1d5c.json",
      "tex": "https://qiqc-op.com/problem/op_bca77ec42ddd1d5c/op_bca77ec42ddd1d5c.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "8ed4c4391c5a86233bb8552a6df1388fe72b2067f897d8b26b9d44b7e6e38733"
    },
    {
      "id": "op_c0b1045a614d2353",
      "ulid": "01M1HME78010TTEQK6NFPRCGZT",
      "aliases": [
        "op_c0b1045a614d2353",
        "01M1HME78010TTEQK6NFPRCGZT",
        "op-c0b1045a614d2353",
        "ruskai-2007-additivity-violation-power-m"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "additivity-and-regularization",
          "matrix-and-entropy-inequalities",
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Delayed-onset additivity violation for minimum output Rényi entropy",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Additivity and regularization",
        "Matrix and entropy inequalities",
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Communication",
        "Additivity and regularization",
        "Matrix and entropy inequalities",
        "Quantum channel structure"
      ],
      "statement": "Does there exist a finite-dimensional quantum channel whose minimum output Rényi entropy is additive for every tensor power below some order and first becomes strictly subadditive at that order? Let $\\Phi:\\mathcal L(A)\\to\\mathcal L(B)$ be completely positive and trace preserving. For $p>0$, define the Rényi entropy and the corresponding minimum output entropy by\n\n \\begin{equation}\n S_p(\\sigma)\n :=\\begin{cases}\n \\displaystyle\\frac{1}{1-p}\\log_2\\operatorname{Tr}(\\sigma^p),\n &p\\neq1,\\\\[2mm]\n -\\operatorname{Tr}(\\sigma\\log_2\\sigma),&p=1,\n \\end{cases}\n \\qquad\n S_{p,\\min}(\\Phi)\n :=\\min_{\\substack{\\rho\\succeq0\\\\\\operatorname{Tr}\\rho=1}}\n S_p\\!\\left(\\Phi(\\rho)\\right).\n\\tag{1}\n\\end{equation} \nWith the quantities in Eq. (1), determine whether there are $p>0$, an integer $m\\geq3$, and a channel $\\Phi$ such that\n\n \\begin{equation}\n S_{p,\\min}(\\Phi^{\\otimes n})\n =nS_{p,\\min}(\\Phi)\n \\quad\\text{for every }1\\leq n<m,\n \\qquad\n S_{p,\\min}(\\Phi^{\\otimes m})\n <mS_{p,\\min}(\\Phi).\n\\tag{2}\n\\end{equation} \nThe restriction $m\\geq3$ in Eq. (2) makes the lower-power requirement nontrivial: the equality at $n=1$ is automatic, whereas equality at $n=2$ is required.",
      "url": "https://qiqc-op.com/problem/op_c0b1045a614d2353/",
      "json": "https://qiqc-op.com/api/problems/op_c0b1045a614d2353.json",
      "tex": "https://qiqc-op.com/problem/op_c0b1045a614d2353/op_c0b1045a614d2353.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T21:32:33.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "cc736f216b48177dbb56a269455617133c4e68b96d6129317549b02cb1a7adcc"
    },
    {
      "id": "op_414fbcc7d5dc0c64",
      "ulid": "01M1HME780WGEQBEXGETXMBSCQ",
      "aliases": [
        "op_414fbcc7d5dc0c64",
        "01M1HME780WGEQBEXGETXMBSCQ",
        "op-414fbcc7d5dc0c64",
        "v2-exponential-strong-converse-for-transpose-degradable-channels",
        "open-problem-v2-problem-55"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "channel-degradability",
          "strong-converses"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Exponential strong converse for transpose-degradable channels",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Channel degradability",
        "Strong converses"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Channel degradability",
        "Strong converses"
      ],
      "statement": "Does every finite-dimensional transpose-degradable channel satisfy an exponential strong converse for quantum communication at its single-letter quantum capacity? Let $V:A\\to B\\otimes E$ be an isometry and suppose that\n\n \\begin{equation}\n \\Phi(X):=\\operatorname{Tr}_E(VXV^\\dagger),\n \\qquad\n \\Phi^c(X):=\\operatorname{Tr}_B(VXV^\\dagger),\n \\qquad\n \\mathsf T_E\\circ\\Phi^c=\\mathcal G\\circ\\Phi,\n\\tag{1}\n\\end{equation} \nwhere $\\mathsf T_E$ is transpose in a fixed basis of $E$ and $\\mathcal G:\\mathcal L(B)\\to\\mathcal L(E)$ is completely positive and trace preserving. For the channel in Eq. (1), transpose degradability gives\n\n \\begin{equation}\n Q(\\Phi)=Q^{(1)}(\\Phi)\n :=\\max_{\\rho_A}\n \\left[S(\\Phi(\\rho_A))-S(\\Phi^c(\\rho_A))\\right],\n \\qquad\n S(\\sigma):=-\\operatorname{Tr}(\\sigma\\log_2\\sigma).\n\\tag{2}\n\\end{equation} \nEquation (2) fixes the rate threshold.\n\nAt blocklength $n$, let $\\mathcal E_n:\\mathcal L(S_n)\\to\\mathcal L(A^{\\otimes n})$ and $\\mathcal R_n:\\mathcal L(B^{\\otimes n})\\to\\mathcal L(\\widehat S_n)$ be arbitrary encoder and decoder channels, with $\\dim R_n=\\dim S_n=\\dim\\widehat S_n=M_n$. Define the maximally entangled target by\n\n \\begin{equation}\n \\varphi_{M_n}:=\n |\\varphi_{M_n}\\rangle\\!\\langle\\varphi_{M_n}|,\n \\qquad\n |\\varphi_{M_n}\\rangle\n :=\\frac1{\\sqrt{M_n}}\\sum_{i=1}^{M_n}|i\\rangle_{R_n}|i\\rangle_{S_n}.\n\\tag{3}\n\\end{equation} \nUsing the target in Eq. (3), let\n\n \\begin{equation}\n \\omega_n:=\n \\left(\\operatorname{id}_{R_n}\\otimes\n \\mathcal R_n\\circ\\Phi^{\\otimes n}\\circ\\mathcal E_n\\right)(\\varphi_{M_n}),\n \\qquad\n r_n:=\\frac1n\\log_2M_n,\n \\qquad\n F_n:=\\operatorname{Tr}(\\varphi_{M_n}^{R_n\\widehat S_n}\\omega_n).\n\\tag{4}\n\\end{equation} \nThe problem is whether, for every channel in Eq. (1), the quantities in Eq. (4) satisfy\n\n \\begin{equation}\n \\forall R>Q(\\Phi)\\ \\exists\\,\\gamma_R>0,\\ n_R\\in\\mathbb N:\n \\quad\n r_n\\geq R,\\ n\\geq n_R\n \\ \\Longrightarrow\\\n F_n\\leq2^{-\\gamma_R n}\n\\tag{5}\n\\end{equation} \nfor every encoder–decoder sequence. Equation (5) is the all-code exponential strong-converse property at the threshold in Eq. (2).",
      "url": "https://qiqc-op.com/problem/op_414fbcc7d5dc0c64/",
      "json": "https://qiqc-op.com/api/problems/op_414fbcc7d5dc0c64.json",
      "tex": "https://qiqc-op.com/problem/op_414fbcc7d5dc0c64/op_414fbcc7d5dc0c64.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T00:20:51.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "d9472a7fbc2f3ba052c3fe1524a8fb3e5f5da0c45ccdc2f0cdfb39b85e3f70fe"
    },
    {
      "id": "op_7e7e4a25fef5c994",
      "ulid": "01M1HME780DC6ZS9XG3V0R6V1A",
      "aliases": [
        "op_7e7e4a25fef5c994",
        "01M1HME780DC6ZS9XG3V0R6V1A",
        "op-7e7e4a25fef5c994",
        "v2-transpose-degradability-beyond-degradability",
        "open-problem-v2-problem-53"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "channel-degradability",
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Transpose degradability beyond degradability",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Channel degradability",
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Communication",
        "Channel degradability",
        "Quantum channel structure"
      ],
      "statement": "Does there exist a finite-dimensional transpose-degradable quantum channel that is not degradable? Let $V:A\\to B\\otimes E$ be an isometry defining a channel and a complementary channel by\n\n \\begin{equation}\n \\Phi(X):=\\operatorname{Tr}_E(VXV^\\dagger),\n \\qquad\n \\Phi^c(X):=\\operatorname{Tr}_B(VXV^\\dagger).\n\\tag{1}\n\\end{equation} \nEquation (1) fixes the output space $B$ and environment space $E$. For the transpose $\\mathsf T_E$ in a fixed basis of $E$, transpose degradability means that a completely positive trace-preserving map $\\mathcal D:\\mathcal L(B)\\to\\mathcal L(E)$ satisfies\n\n \\begin{equation}\n \\mathsf T_E\\circ\\Phi^c=\\mathcal D\\circ\\Phi.\n\\tag{2}\n\\end{equation} \nOrdinary degradability instead requires a completely positive trace-preserving map $\\widetilde{\\mathcal D}:\\mathcal L(B)\\to\\mathcal L(E)$ satisfying\n\n \\begin{equation}\n \\Phi^c=\\widetilde{\\mathcal D}\\circ\\Phi.\n\\tag{3}\n\\end{equation} \nThe question is whether Eq. (2) can hold while no map satisfying Eq. (3) exists.",
      "url": "https://qiqc-op.com/problem/op_7e7e4a25fef5c994/",
      "json": "https://qiqc-op.com/api/problems/op_7e7e4a25fef5c994.json",
      "tex": "https://qiqc-op.com/problem/op_7e7e4a25fef5c994/op_7e7e4a25fef5c994.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T00:20:51.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "dccd2c06d72b4fb936cc78f7321f4f93d829d9ae7e50e9c22901f284d1316742"
    },
    {
      "id": "op_b315f0d0b6ddbdee",
      "ulid": "01M1HME7803QE7KXJDNM1ACKBP",
      "aliases": [
        "op_b315f0d0b6ddbdee",
        "01M1HME7803QE7KXJDNM1ACKBP",
        "op-b315f0d0b6ddbdee",
        "v2-minimal-dimensions-for-strict-transpose-degradability",
        "open-problem-v2-problem-54"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "channel-degradability",
          "quantum-channel-structure"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Minimal dimensions for strict transpose degradability",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Channel degradability",
        "Quantum channel structure"
      ],
      "tags": [
        "Quantum Communication",
        "Channel degradability",
        "Quantum channel structure"
      ],
      "statement": "What are the componentwise-minimal dimension triples $(d_A,d_B,d_E)$ that admit a transpose-degradable but nondegradable channel? Let $d_X:=\\dim X$ and let $V:A\\to B\\otimes E$ be a support-minimal isometry, meaning that the channels\n\n \\begin{equation}\n \\Phi_V(X):=\\operatorname{Tr}_E(VXV^\\dagger),\n \\qquad\n \\Phi_V^c(X):=\\operatorname{Tr}_B(VXV^\\dagger)\n\\tag{1}\n\\end{equation} \nsatisfy $\\operatorname{supp}(\\Phi_V(I_A))=B$ and $\\operatorname{supp}(\\Phi_V^c(I_A))=E$. Equation (1) is strictly transpose degradable when, for a fixed-basis transpose $\\mathsf T_E$, its factorization properties are\n\n \\begin{equation}\n \\begin{aligned}\n &\\exists\\ \\mathcal D:\\mathcal L(B)\\to\\mathcal L(E)\\ \\text{CPTP},\n &&\\mathsf T_E\\circ\\Phi_V^c=\\mathcal D\\circ\\Phi_V,\\\\\n &\\nexists\\ \\widetilde{\\mathcal D}:\\mathcal L(B)\\to\\mathcal L(E)\\\n \\text{CPTP},\n &&\\Phi_V^c=\\widetilde{\\mathcal D}\\circ\\Phi_V.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nA feasible triple is componentwise minimal if no distinct feasible $(d'_A,d'_B,d'_E)$ satisfies $d'_X\\leq d_X$ for every $X\\in\\{A,B,E\\}$. Determine all minimal triples satisfying Eq. (2).",
      "url": "https://qiqc-op.com/problem/op_b315f0d0b6ddbdee/",
      "json": "https://qiqc-op.com/api/problems/op_b315f0d0b6ddbdee.json",
      "tex": "https://qiqc-op.com/problem/op_b315f0d0b6ddbdee/op_b315f0d0b6ddbdee.tex",
      "created": "2026-09-02",
      "updated": "2026-09-04",
      "createdAt": "2026-09-02T00:20:51.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "9b5efabcf507dba54a031fa745aed279428df404a687c8224cc40f251003eedb"
    },
    {
      "id": "op_08387c140f552732",
      "ulid": "01M1HME7803DZWKPJRHYYKHX0C",
      "aliases": [
        "op_08387c140f552732",
        "01M1HME7803DZWKPJRHYYKHX0C",
        "op-08387c140f552732",
        "v2-fixed-error-parallel-stein-lemma-for-quantum-channels",
        "open-problem-v2-problem-46"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology",
          "quantum-communication"
        ],
        "topicIds": [
          "channel-discrimination",
          "quantum-hypothesis-testing",
          "quantum-relative-entropy",
          "strong-converses"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Fixed-error parallel Stein lemma for quantum channels",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology",
        "Quantum Communication"
      ],
      "topics": [
        "Channel discrimination",
        "Quantum hypothesis testing",
        "Quantum relative entropy",
        "Strong converses"
      ],
      "tags": [
        "Quantum metrology",
        "Quantum Communication",
        "Channel discrimination",
        "Quantum hypothesis testing",
        "Quantum relative entropy",
        "Strong converses"
      ],
      "statement": "Let $\\mathcal N,\\mathcal M:\\mathcal L(A)\\to\\mathcal L(B)$ be quantum channels on finite-dimensional systems. For states $\\rho$ and $\\sigma$, define the relative entropy and the hypothesis-testing divergence by\n\n \\begin{equation}\n \\begin{aligned}\n D(\\rho\\|\\sigma)\n &:={\\rm Tr}\\!\\left[\\rho(\\log_2\\rho-\\log_2\\sigma)\\right],\\\\\n D_H^\\varepsilon(\\rho\\|\\sigma)\n &:=-\\log_2\\inf_{\\substack{0\\leq Q\\leq I\\\\\n {\\rm Tr}(Q\\rho)\\geq1-\\varepsilon}}\n {\\rm Tr}(Q\\sigma),\n \\end{aligned}\n\\tag{1}\n\\end{equation} \nwhere $D(\\rho\\|\\sigma)=+\\infty$ unless $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$, and $\\varepsilon\\in(0,1)$. For either divergence $\\mathbf D$ in Eq. (1), its stabilized channel extension is\n\n \\begin{equation}\n \\mathbf D_{\\rm ch}(\\mathcal N\\|\\mathcal M)\n :=\\sup_{\\psi_{RA}\\in\\mathcal D(R\\otimes A)}\n \\mathbf D\\!\\left(\n (\\operatorname{id}_R\\otimes\\mathcal N)(\\psi)\n \\middle\\|\n (\\operatorname{id}_R\\otimes\\mathcal M)(\\psi)\n \\right),\n \\qquad R\\simeq A.\n\\tag{2}\n\\end{equation} \nUsing Eq. (2), define the regularized channel relative entropy by\n\n \\begin{equation}\n D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M)\n :=\\lim_{n\\to\\infty}\\frac1n\n D_{\\rm ch}(\\mathcal N^{\\otimes n}\\|\\mathcal M^{\\otimes n}).\n\\tag{3}\n\\end{equation} \nWhenever Eq. (3) is finite, does the fixed-error parallel Stein limit exist and satisfy\n\n \\begin{equation}\n \\lim_{n\\to\\infty}\\frac1n\n D_{H,{\\rm ch}}^\\varepsilon\n (\\mathcal N^{\\otimes n}\\|\\mathcal M^{\\otimes n})\n =D_{\\rm ch}^{\\infty}(\\mathcal N\\|\\mathcal M)\n \\qquad\\text{for every }\\varepsilon\\in(0,1)?\n\\tag{4}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_08387c140f552732/",
      "json": "https://qiqc-op.com/api/problems/op_08387c140f552732.json",
      "tex": "https://qiqc-op.com/problem/op_08387c140f552732/op_08387c140f552732.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "6dc10eb6bfd091f4fbde5e33c967912872f3f11429c3b8397658a20b58dce2c4"
    },
    {
      "id": "op_0c86d9293aba3b01",
      "ulid": "01M1HME7809XJE4T6RFQKPGJVF",
      "aliases": [
        "op_0c86d9293aba3b01",
        "01M1HME7809XJE4T6RFQKPGJVF",
        "op-0c86d9293aba3b01",
        "v2-finite-nontrivial-lu-moduli-of-ame-states",
        "open-problem-v2-problem-43"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "absolutely-maximally-entangled-states",
          "local-unitary-equivalence"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Finite nontrivial LU moduli of AME states",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Absolutely maximally entangled states",
        "Local unitary equivalence"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Absolutely maximally entangled states",
        "Local unitary equivalence"
      ],
      "statement": "Do there exist integers $N,d\\geq 2$ for which the absolutely maximally entangled states of $N$ qudits of local dimension $d$ form finitely many, but more than one, local-unitary equivalence classes? A normalized vector $|\\psi\\rangle\\in(\\mathbb{C}^{d})^{\\otimes N}$ is an $\\operatorname{AME}(N,d)$ state when every subsystem of at most half the parties is maximally mixed:\n\n \\begin{equation}\n \\operatorname{Tr}_{S^{c}}\\!\\left(|\\psi\\rangle\\!\\langle\\psi|\\right)\n =\\frac{I_{d^{|S|}}}{d^{|S|}}\n \\quad\\text{for every }S\\subseteq\\{1,\\ldots,N\\}\n \\text{ with }|S|\\leq\\left\\lfloor\\frac{N}{2}\\right\\rfloor .\n\\tag{1}\n\\end{equation} \nFor normalized states satisfying Eq. (1), define local-unitary equivalence by\n\n \\begin{equation}\n |\\psi\\rangle\\sim_{\\mathrm{LU}}|\\phi\\rangle\n \\quad\\Longleftrightarrow\\quad\n |\\psi\\rangle=(U_1\\otimes\\cdots\\otimes U_N)|\\phi\\rangle\n \\quad\\text{for some }U_1,\\ldots,U_N\\in U(d).\n\\tag{2}\n\\end{equation} \nIf $\\mathcal{A}_{N,d}$ denotes the set specified by Eq. (1), determine whether the quotient under Eq. (2) can satisfy\n\n \\begin{equation}\n \\mathfrak{M}_{N,d}:=\\mathcal{A}_{N,d}/\\!\\sim_{\\mathrm{LU}},\n \\qquad\n 1<\\lvert\\mathfrak{M}_{N,d}\\rvert<\\infty .\n\\tag{3}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_0c86d9293aba3b01/",
      "json": "https://qiqc-op.com/api/problems/op_0c86d9293aba3b01.json",
      "tex": "https://qiqc-op.com/problem/op_0c86d9293aba3b01/op_0c86d9293aba3b01.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "77856fe21201e4f5e00c4b4157271f27f5f77239f5498598aa7f37cd87a71a59"
    },
    {
      "id": "op_2579e084f37ac18c",
      "ulid": "01M1HME780RHDHC0HWTHTESKBH",
      "aliases": [
        "op_2579e084f37ac18c",
        "01M1HME780RHDHC0HWTHTESKBH",
        "op-2579e084f37ac18c",
        "v2-generalized-stein-lemma-for-fully-quantum-channel-resources",
        "open-problem-v2-problem-47"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-metrology"
        ],
        "topicIds": [
          "quantum-hypothesis-testing",
          "channel-discrimination",
          "quantum-relative-entropy"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Generalized Stein lemma for fully quantum channel resources",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum metrology"
      ],
      "topics": [
        "Quantum hypothesis testing",
        "Channel discrimination",
        "Quantum relative entropy"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum metrology",
        "Quantum hypothesis testing",
        "Channel discrimination",
        "Quantum relative entropy"
      ],
      "statement": "Let $\\mathcal N:\\mathcal L(A)\\to\\mathcal L(B)$ be a finite-dimensional quantum channel. For each $n$, let $\\mathfrak F_n$ be a nonempty compact, convex, permutation-invariant set of channels from $A^{\\otimes n}$ to $B^{\\otimes n}$, closed under tensor products and containing $\\mathcal R_\\omega^{\\otimes n}$ for one full-rank state $\\omega$, where $\\mathcal R_\\omega(X):=\\operatorname{Tr}(X)\\omega$. Define\n\n \\begin{equation}\n D_{\\rm ch}(\\mathcal N^{\\otimes n}\\|\\mathcal M_n)\n :=\\sup_{\\psi_{R_nA^n}}\n D\\!\\left(\n (\\operatorname{id}_{R_n}\\otimes\\mathcal N^{\\otimes n})(\\psi)\n \\middle\\|\n (\\operatorname{id}_{R_n}\\otimes\\mathcal M_n)(\\psi)\n \\right),\n \\qquad R_n\\simeq A^{\\otimes n},\n\\tag{1}\n\\end{equation} \nwhere the supremum in Eq. (1) is over density operators and $D$ is the quantum relative entropy. The distance to the free set is\n\n \\begin{equation}\n E_n^{\\rm QQ}(\\mathcal N\\|\\mathfrak F_n)\n :=\\inf_{\\mathcal M_n\\in\\mathfrak F_n}\n D_{\\rm ch}(\\mathcal N^{\\otimes n}\\|\\mathcal M_n).\n\\tag{2}\n\\end{equation} \nEquation (2) is the $n$-use relative-entropy distance to the free channel set. For $\\varepsilon\\in(0,1)$, define the optimal worst-case type-II error of a parallel quantum-input/quantum-output test by\n\n \\begin{equation}\n \\begin{aligned}\n \\beta_{\\varepsilon,n}^{\\rm QQ}(\\mathcal N\\|\\mathfrak F_n)\n :=\\inf_{\\substack{\\psi_{R_nA^n},\\ 0\\leq Q\\leq I\\\\\n \\operatorname{Tr}[Q(\\operatorname{id}_{R_n}\\otimes\n \\mathcal N^{\\otimes n})(\\psi)]\\geq1-\\varepsilon}}\n \\ \\sup_{\\mathcal M_n\\in\\mathfrak F_n}\n \\operatorname{Tr}\\!\\left[\n Q(\\operatorname{id}_{R_n}\\otimes\\mathcal M_n)(\\psi)\n \\right].\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nUnder what additional structural assumptions on $(\\mathfrak F_n)_{n\\geq1}$, if any, do both limits exist and obey the fully quantum generalized Stein identity\n\n \\begin{equation}\n \\lim_{n\\to\\infty}-\\frac1n\\log_2\n \\beta_{\\varepsilon,n}^{\\rm QQ}(\\mathcal N\\|\\mathfrak F_n)\n =\\lim_{n\\to\\infty}\\frac1n\n E_n^{\\rm QQ}(\\mathcal N\\|\\mathfrak F_n)\n \\qquad\\text{for every }\\varepsilon\\in(0,1)?\n\\tag{4}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_2579e084f37ac18c/",
      "json": "https://qiqc-op.com/api/problems/op_2579e084f37ac18c.json",
      "tex": "https://qiqc-op.com/problem/op_2579e084f37ac18c/op_2579e084f37ac18c.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "6cb81cd55f2052cde0e1fc20eb0020f0920a11f0cbf2a8c9d561c49c5a9a7bed"
    },
    {
      "id": "op_2982ddd94453b5b6",
      "ulid": "01M1HME780WBAHVDTT361D6NNF",
      "aliases": [
        "op_2982ddd94453b5b6",
        "01M1HME780WBAHVDTT361D6NNF",
        "op-2982ddd94453b5b6",
        "v2-povm-steering-threshold-of-higher-dimensional-werner-states",
        "open-problem-v2-problem-33"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "quantum-steering"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "POVM steering threshold of higher-dimensional Werner states",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Quantum steering"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum steering"
      ],
      "statement": "What is the exact steering threshold for arbitrary POVMs on a higher-dimensional Werner state? For $d\\geq3$, let $F$ be the swap operator on $\\mathbb C^d\\otimes\\mathbb C^d$ and define\n\n \\begin{equation}\n \\rho_f^{(d)}\n :=\\frac{(d-f)I+(df-1)F}{d(d^2-1)},\n \\qquad -1\\leq f\\leq1.\n\\tag{1}\n\\end{equation} \nIf Alice applies a POVM $\\{M_{a\\mid x}\\}_a$ to the state in Eq. (1), Bob’s subnormalized conditional states are\n\n \\begin{equation}\n \\sigma_{a\\mid x}\n :=\\operatorname{Tr}_A\\!\\left[\n (M_{a\\mid x}\\otimes I)\\rho_f^{(d)}\n \\right].\n\\tag{2}\n\\end{equation} \nThe assemblage in Eq. (2) is unsteerable when it admits a local-hidden-state decomposition\n\n \\begin{equation}\n \\sigma_{a\\mid x}\n =\\int_\\Lambda\\mu(d\\lambda)\\,\n p(a\\mid x,\\lambda)\\tau_\\lambda,\n\\tag{3}\n\\end{equation} \nwhere $\\mu$ is a probability measure, $p(a\\mid x,\\lambda)$ are response functions, and $\\tau_\\lambda$ are density operators. Determine the critical value $f_{\\mathrm{POVM}}(d)$ characterized by\n\n \\begin{equation}\n \\rho_f^{(d)}\\ \\text{is unsteerable from Alice to Bob for every POVM}\n \\quad\\Longleftrightarrow\\quad\n f\\geq f_{\\mathrm{POVM}}(d).\n\\tag{4}\n\\end{equation} \nIn particular, decide whether the threshold in Eq. (4) equals the exact projective-measurement threshold\n\n \\begin{equation}\n f_{\\mathrm{PVM}}(d)=-1+\\frac1d+\\frac1{d^2}\n\\tag{5}\n\\end{equation} \nfor every $d\\geq3$. Equation (3) fixes the notion of unsteerability used in both threshold statements.",
      "url": "https://qiqc-op.com/problem/op_2982ddd94453b5b6/",
      "json": "https://qiqc-op.com/api/problems/op_2982ddd94453b5b6.json",
      "tex": "https://qiqc-op.com/problem/op_2982ddd94453b5b6/op_2982ddd94453b5b6.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "360ab87a88415fd0a563ea50c095a2ef67f4e9414e6285135f3cb56080d6b773"
    },
    {
      "id": "op_299c3becfd760123",
      "ulid": "01M1HME780VVDD86Q565CV668W",
      "aliases": [
        "op_299c3becfd760123",
        "01M1HME780VVDD86Q565CV668W",
        "op-299c3becfd760123",
        "v2-secret-key-from-every-bell-nonlocal-behavior",
        "open-problem-v2-problem-30"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-cryptography",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "secret-key-distillation",
          "device-independent-cryptography",
          "bell-nonlocality"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Secret key from every Bell-nonlocal behavior",
      "status": "Unsolved",
      "fields": [
        "Quantum Cryptography",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Secret-key distillation",
        "Device-independent cryptography",
        "Bell nonlocality"
      ],
      "tags": [
        "Quantum Cryptography",
        "Quantum Resource Theory",
        "Secret-key distillation",
        "Device-independent cryptography",
        "Bell nonlocality"
      ],
      "statement": "Does every finite-alphabet Bell-nonlocal behavior have a strictly positive asymptotic secret-key rate against arbitrary individual nonsignalling attacks? Let $P(a,b\\mid x,y)$ be bipartite and nonsignalling, and suppose that it has no local decomposition of the form\n\n \\begin{equation}\n P(a,b\\mid x,y)\n =\\int_\\Lambda \\mu(d\\lambda)\\,\n P_A(a\\mid x,\\lambda)P_B(b\\mid y,\\lambda).\n\\tag{1}\n\\end{equation} \nThus Eq. (1) fails for every probability measure $\\mu$ and local response functions $P_A,P_B$. Alice and Bob receive independent copies of $P$; on each copy, an adversary may hold an arbitrary nonsignalling extension. The honest parties may choose their inputs, process all outputs locally, and communicate publicly. The question asks whether some such protocol always extracts secret key at a nonzero asymptotic rate.",
      "url": "https://qiqc-op.com/problem/op_299c3becfd760123/",
      "json": "https://qiqc-op.com/api/problems/op_299c3becfd760123.json",
      "tex": "https://qiqc-op.com/problem/op_299c3becfd760123/op_299c3becfd760123.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "84d3ebe2c72576cb58c9dbf17614df7f860a46f33b7a61368a9d8735d7d5c73a"
    },
    {
      "id": "op_2beed65d248be57a",
      "ulid": "01M1HME7803ZFMDQWV9SVP8FH8",
      "aliases": [
        "op_2beed65d248be57a",
        "01M1HME7803ZFMDQWV9SVP8FH8",
        "op-2beed65d248be57a",
        "v2-positivity-threshold-for-thermal-attenuator-quantum-capacity",
        "open-problem-v2-problem-50"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "bosonic-channels"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Positivity threshold for thermal-attenuator quantum capacity",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Bosonic channels"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Bosonic channels"
      ],
      "statement": "For $0<\\eta<1$ and $0<\\nu<\\infty$, let the single-mode bosonic thermal attenuator be\n\n \\begin{equation}\n \\Phi_{\\eta,\\nu}(\\rho_A)\n :=\\operatorname{Tr}_{E'}\\!\\left[\n U_\\eta(\\rho_A\\otimes\\tau_{\\nu,E})U_\\eta^\\dagger\n \\right],\n \\qquad\n \\tau_{\\nu,E}:=\\sum_{k=0}^{\\infty}\n \\frac{\\nu^k}{(\\nu+1)^{k+1}}\n |k\\rangle_E\\!\\langle k|_E,\n\\tag{1}\n\\end{equation} \nwhere $U_\\eta:AE\\to BE'$ is a beam-splitter unitary and $\\tau_{\\nu,E}$ acts on the environment mode $E$. For an environment mode of angular frequency $\\omega_E$ at temperature $T$, $\\nu=(e^{\\hbar\\omega_E/(k_{\\rm B}T)}-1)^{-1}$. Thus $0<T<\\infty$ is equivalent to $0<\\nu<\\infty$ for fixed $\\omega_E>0$; $\\nu=0$ corresponds to $T=0$, while $\\nu\\to\\infty$ as $T\\to\\infty$. Define its unassisted quantum capacity and its critical transmissivity by\n\n \\begin{equation}\n \\mathcal Q(\\Phi_{\\eta,\\nu})\n :=\\lim_{n\\to\\infty}\\frac1n\\sup_{\\rho_{A^n}}\n I_{\\rm c}(\\rho_{A^n},\\Phi_{\\eta,\\nu}^{\\otimes n}),\n \\qquad\n \\eta_{\\rm c}(\\nu)\n :=\\inf\\{\\eta\\in(0,1):\\mathcal Q(\\Phi_{\\eta,\\nu})>0\\},\n\\tag{2}\n\\end{equation} \nwhere \\(I_{\\rm c}(\\rho,\\mathcal N)\n:=S(\\mathcal N(\\rho))-S(\\mathcal N^{\\rm c}(\\rho))\\). Determine the exact threshold curve in Eq. (2) for the channel in Eq. (1).",
      "url": "https://qiqc-op.com/problem/op_2beed65d248be57a/",
      "json": "https://qiqc-op.com/api/problems/op_2beed65d248be57a.json",
      "tex": "https://qiqc-op.com/problem/op_2beed65d248be57a/op_2beed65d248be57a.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "19ca802eceac776d26a3a3ad6e685f2858cc9a29dc4153a32b6ebaf1a98565aa"
    },
    {
      "id": "op_308ac6c848756630",
      "ulid": "01M1HME780GHC51FDW8ZSTJHK9",
      "aliases": [
        "op_308ac6c848756630",
        "01M1HME780GHC51FDW8ZSTJHK9",
        "op-308ac6c848756630",
        "v2-sic-povm-existence-in-every-dimension",
        "open-problem-v2-problem-17"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "symmetric-informationally-complete-measurements"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "SIC-POVM existence in every dimension",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Symmetric informationally complete measurements"
      ],
      "tags": [
        "Quantum metrology",
        "Symmetric informationally complete measurements"
      ],
      "statement": "Does a symmetric informationally complete positive-operator-valued measure exist in every finite dimension $d\\ge2$? Equivalently, determine whether for every such $d$ there are $d^2$ unit vectors $\\lvert\\psi_1\\rangle,\\ldots,\\lvert\\psi_{d^2}\\rangle\\in\\mathbb{C}^d$ satisfying\n\n \\begin{equation}\n \\sum_{j=1}^{d^2}\n \\lvert\\psi_j\\rangle\\!\\langle\\psi_j\\rvert=dI_d,\n \\qquad\n \\left|\\langle\\psi_j\\vert\\psi_k\\rangle\\right|^2\n =\\frac{1}{d+1}\n \\quad\\text{for all }j\\ne k.\n\\tag{1}\n\\end{equation} \nWhen Eq. (1) holds, the effects $\\Pi_j=d^{-1}\\lvert\\psi_j\\rangle\\!\\langle\\psi_j\\rvert$ form the desired SIC-POVM. No covariance or additional symmetry is required.",
      "url": "https://qiqc-op.com/problem/op_308ac6c848756630/",
      "json": "https://qiqc-op.com/api/problems/op_308ac6c848756630.json",
      "tex": "https://qiqc-op.com/problem/op_308ac6c848756630/op_308ac6c848756630.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "898cc033d84ef57a632cf09a8077f3c7f31a3c514f5be02efc5240f22d4f8920"
    },
    {
      "id": "op_3770ad932d54692f",
      "ulid": "01M1HME780BM029QANMMXSYJXR",
      "aliases": [
        "op_3770ad932d54692f",
        "01M1HME780BM029QANMMXSYJXR",
        "op-3770ad932d54692f",
        "v2-unconditional-classical-verification-with-one-quantum-prover",
        "open-problem-v2-problem-39"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "verification-of-quantum-computation",
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Unconditional classical verification with one quantum prover",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Verification of quantum computation",
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Verification of quantum computation",
        "Computational complexity and computability"
      ],
      "statement": "Does every language $L\\in\\mathsf{BQP}$ admit a single-prover interactive proof with a fully classical verifier, an efficient quantum honest prover, and information-theoretic soundness? Precisely, require a probabilistic classical polynomial-time verifier exchanging only classical messages with one prover; a uniform quantum polynomial-time honest prover; acceptance probability at least $2/3$ for every yes-instance when the honest prover is used; and acceptance probability at most $1/3$ for every no-instance against every prover, including a computationally unbounded one. Equivalently, can one classically verify an arbitrary efficient quantum computation with one efficient quantum prover, no quantum capability for the verifier, and no cryptographic assumption?",
      "url": "https://qiqc-op.com/problem/op_3770ad932d54692f/",
      "json": "https://qiqc-op.com/api/problems/op_3770ad932d54692f.json",
      "tex": "https://qiqc-op.com/problem/op_3770ad932d54692f/op_3770ad932d54692f.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "22bb6ce147affba6d2fd7055695b531643ef5e996252a8372cde479a694618c5"
    },
    {
      "id": "op_3db9de0ddc492750",
      "ulid": "01M1HME7807PZA3MKVGSNTAP1V",
      "aliases": [
        "op_3db9de0ddc492750",
        "01M1HME7807PZA3MKVGSNTAP1V",
        "op-3db9de0ddc492750",
        "v2-finite-alphabet-nonsignalling-simulation-of-entangled-qubits",
        "open-problem-v2-problem-29"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality",
          "resource-conversion"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Finite-alphabet nonsignalling simulation of entangled qubits",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality",
        "Resource conversion"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Bell nonlocality",
        "Resource conversion"
      ],
      "statement": "For every partially entangled two-qubit pure state, does some fixed finite-alphabet nonsignalling resource give an exact noncommunicating simulation of all local projective measurements? Fix\n\n \\begin{equation}\n \\lvert\\psi_\\theta\\rangle\n =\\cos\\theta\\,\\lvert00\\rangle+\\sin\\theta\\,\\lvert11\\rangle,\n \\qquad 0<\\theta<\\frac{\\pi}{4}.\n\\tag{1}\n\\end{equation} \nThe target correlations for Bloch directions $\\mathbf x,\\mathbf y\\in S^2$ are\n\n \\begin{equation}\n P_\\theta(a,b\\mid\\mathbf x,\\mathbf y)\n =\\operatorname{Tr}\\!\\left[\n \\lvert\\psi_\\theta\\rangle\\!\\langle\\psi_\\theta\\rvert\n \\bigl(M_{a\\mid\\mathbf x}\\otimes M_{b\\mid\\mathbf y}\\bigr)\n \\right],\n \\qquad a,b\\in\\{0,1\\},\n\\tag{2}\n\\end{equation} \nwhere $M_{a\\mid\\mathbf x}=(I+(-1)^a\\mathbf x\\cdot\\boldsymbol\\sigma)/2$ and similarly for Bob. For each $\\theta$ in Eq. (1), one may choose a nonsignalling box $R(u,v\\mid s,t)$ with finite input and output alphabets and a finite number of copies of it. The box, the number of copies, and the local wiring may depend on $\\theta$ but not on $\\mathbf x$ or $\\mathbf y$, and the wiring may use unlimited shared randomness but no communication. It must reproduce Eq. (2) exactly.",
      "url": "https://qiqc-op.com/problem/op_3db9de0ddc492750/",
      "json": "https://qiqc-op.com/api/problems/op_3db9de0ddc492750.json",
      "tex": "https://qiqc-op.com/problem/op_3db9de0ddc492750/op_3db9de0ddc492750.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "4345c406edd9192310d3eab550174dac49f5a671955bcaffa3719659e5f52f61"
    },
    {
      "id": "op_439ae5e7e9b3b043",
      "ulid": "01M1HME780TFDMAP1RAR9PDCSJ",
      "aliases": [
        "op_439ae5e7e9b3b043",
        "01M1HME780TFDMAP1RAR9PDCSJ",
        "op-439ae5e7e9b3b043",
        "v2-unclassified-existence-parameters-for-homogeneous-ame-states",
        "open-problem-v2-problem-45"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-error-correction"
        ],
        "topicIds": [
          "absolutely-maximally-entangled-states",
          "quantum-coding-theory"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Unclassified existence parameters for homogeneous AME states",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum Error Correction"
      ],
      "topics": [
        "Absolutely maximally entangled states",
        "Quantum coding theory"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum Error Correction",
        "Absolutely maximally entangled states",
        "Quantum coding theory"
      ],
      "statement": "For which of the parameter pairs specified below does an absolutely maximally entangled state exist? A normalized vector $|\\psi\\rangle\\in(\\mathbb C^d)^{\\otimes n}$ is an $\\operatorname{AME}(n,d)$ state when\n\n \\begin{equation}\n \\operatorname{Tr}_{S^c}\\!\\left(|\\psi\\rangle\\!\\langle\\psi|\\right)\n =\\frac{I_{d^{|S|}}}{d^{|S|}}\n \\quad\\text{for every }S\\subseteq\\{1,\\ldots,n\\}\n \\text{ with }|S|\\leq\\left\\lfloor\\frac n2\\right\\rfloor.\n\\tag{1}\n\\end{equation} \nDetermine whether a state satisfying Eq. (1) exists for every pair in\n\n \\begin{equation}\n \\mathcal U:=\\bigl\\{(8,6),(8,10),(9,6),(9,10),(10,6),(10,10),\n (11,3),(11,6),(11,10),(12,6),(12,10)\\bigr\\}.\n\\tag{2}\n\\end{equation} \nThus the task is to classify every pair in Eq. (2) by existence or nonexistence, without restricting to stabilizer or minimal-support states.",
      "url": "https://qiqc-op.com/problem/op_439ae5e7e9b3b043/",
      "json": "https://qiqc-op.com/api/problems/op_439ae5e7e9b3b043.json",
      "tex": "https://qiqc-op.com/problem/op_439ae5e7e9b3b043/op_439ae5e7e9b3b043.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "046ce09dfe5326048a1b9a1fadb84560d14c38469fc05d4e78c43932eca26b72"
    },
    {
      "id": "op_43d4aca67bd52554",
      "ulid": "01M1HME7809M71BG24CSMYKA8A",
      "aliases": [
        "op_43d4aca67bd52554",
        "01M1HME7809M71BG24CSMYKA8A",
        "op-43d4aca67bd52554",
        "v2-quantum-capacity-of-a-qubit-pauli-channel",
        "open-problem-v2-problem-1"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Quantum capacity of a qubit Pauli channel",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity"
      ],
      "statement": "What is the quantum capacity $\\mathcal{Q}(\\Lambda_{\\mathbf p})$ of the qubit Pauli channel defined by\n\n \\begin{equation}\n \\Lambda_{\\mathbf p}(\\rho)\n =p_I\\rho+p_XX\\rho X+p_YY\\rho Y+p_ZZ\\rho Z,\n \\qquad p_I+p_X+p_Y+p_Z=1?\n\\tag{1}\n\\end{equation} \nEquation (1) fixes the channel and the error-probability convention used throughout this section.",
      "url": "https://qiqc-op.com/problem/op_43d4aca67bd52554/",
      "json": "https://qiqc-op.com/api/problems/op_43d4aca67bd52554.json",
      "tex": "https://qiqc-op.com/problem/op_43d4aca67bd52554/op_43d4aca67bd52554.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "2a88129d6224481700a9544f7e4574e42bfe7551743c137ac4667de514df2bdd"
    },
    {
      "id": "op_452fc7d8cc44b728",
      "ulid": "01M1HME780FM1RNT5S8HSBTDFF",
      "aliases": [
        "op_452fc7d8cc44b728",
        "01M1HME780FM1RNT5S8HSBTDFF",
        "op-452fc7d8cc44b728",
        "v2-universal-simulation-with-two-pr-boxes",
        "open-problem-v2-problem-28"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality",
          "resource-conversion"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Universal simulation with two PR boxes",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality",
        "Resource conversion"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Bell nonlocality",
        "Resource conversion"
      ],
      "statement": "Can shared randomness and at most two Popescu–Rohrlich boxes exactly simulate every pair of local projective measurements on every two-qubit state, without communication? A Popescu–Rohrlich box is the binary nonsignalling behavior\n\n \\begin{equation}\n P_{\\mathrm{PR}}(u,v\\mid s,t)\n =\\begin{cases}\n \\tfrac12,&u\\oplus v=st,\\\\\n 0,&u\\oplus v\\neq st,\n \\end{cases}\n \\qquad s,t,u,v\\in\\{0,1\\}.\n\\tag{1}\n\\end{equation} \nThe parties may use arbitrary local, possibly adaptive wirings of two independent copies of Eq. (1). By convexity and Schmidt decomposition, it suffices to solve the simulation for all pure states\n\n \\begin{equation}\n \\lvert\\psi_\\theta\\rangle\n =\\cos\\theta\\,\\lvert00\\rangle+\\sin\\theta\\,\\lvert11\\rangle,\n \\qquad 0<\\theta\\leq\\frac{\\pi}{4},\n\\tag{2}\n\\end{equation} \nand all local projective measurements on the state in Eq. (2). A complementary finite-scenario objective is to characterize the convex set generated by two-box wirings, for example through its facet inequalities.",
      "url": "https://qiqc-op.com/problem/op_452fc7d8cc44b728/",
      "json": "https://qiqc-op.com/api/problems/op_452fc7d8cc44b728.json",
      "tex": "https://qiqc-op.com/problem/op_452fc7d8cc44b728/op_452fc7d8cc44b728.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "36385dcb3303fc6137229754290fcb7a2f46f736ba98560a3767d2eb0f24aa67"
    },
    {
      "id": "op_4830398d0c2feb5f",
      "ulid": "01M1HME7800GAYCEBF3MNS33JF",
      "aliases": [
        "op_4830398d0c2feb5f",
        "01M1HME7800GAYCEBF3MNS33JF",
        "op-4830398d0c2feb5f",
        "v2-quantum-ldpc-codes-at-the-pauli-hashing-bound",
        "open-problem-v2-problem-6"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "editor-formulated",
        "posed": null,
        "areaIds": [
          "quantum-error-correction"
        ],
        "topicIds": [
          "quantum-ldpc-codes",
          "quantum-coding-theory",
          "decoding-algorithms"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Quantum LDPC codes at the Pauli hashing bound",
      "status": "Unsolved",
      "fields": [
        "Quantum Error Correction"
      ],
      "topics": [
        "Quantum LDPC codes",
        "Quantum coding theory",
        "Decoding algorithms"
      ],
      "tags": [
        "Quantum Error Correction",
        "Quantum LDPC codes",
        "Quantum coding theory",
        "Decoding algorithms"
      ],
      "statement": "For a probability vector $\\mathbf p=(p_I,p_X,p_Y,p_Z)$, the qubit Pauli channel is\n\n \\begin{equation}\n \\Lambda_{\\mathbf p}(\\rho)\n =p_I\\rho+p_XX\\rho X+p_YY\\rho Y+p_ZZ\\rho Z,\n \\qquad p_I+p_X+p_Y+p_Z=1,\n\\tag{1}\n\\end{equation} \nwhere every $p_i\\ge0$, $I$ is the identity, and $X,Y,Z$ are the Pauli operators. The qubit depolarizing channel is the specialization of Eq. (1) given by\n\n \\begin{equation}\n \\mathcal D_p(\\rho)\n =(1-p)\\rho+\\frac p3\\bigl(X\\rho X+Y\\rho Y+Z\\rho Z\\bigr),\n \\qquad 0\\le p\\le1.\n\\tag{2}\n\\end{equation} \nThus Eq. (2) has $\\mathbf p=(1-p,p/3,p/3,p/3)$. For the general channel in Eq. (1), define the hashing rate by\n\n \\begin{equation}\n R_{\\mathrm{hash}}(\\mathbf p)\n :=\\max\\{0,1-H(\\mathbf p)\\},\n \\qquad\n H(\\mathbf p):=-\\sum_{i\\in\\{I,X,Y,Z\\}}p_i\\log_2p_i,\n \\qquad 0\\log_2 0:=0.\n\\tag{3}\n\\end{equation} \nA family of stabilizer codes $[[n,k_n]]$ is quantum low-density parity-check (LDPC) if it has stabilizer generators of uniformly bounded weight and each qubit participates in a uniformly bounded number of generators. Does there exist, for a given $\\mathbf p$, such a family with decoding error tending to zero under $\\Lambda_{\\mathbf p}^{\\otimes n}$ and asymptotic transmission rate $R:=\\liminf_{n\\to\\infty}k_n/n$ satisfying $R\\ge R_{\\mathrm{hash}}(\\mathbf p)$ from Eq. (3)?",
      "url": "https://qiqc-op.com/problem/op_4830398d0c2feb5f/",
      "json": "https://qiqc-op.com/api/problems/op_4830398d0c2feb5f.json",
      "tex": "https://qiqc-op.com/problem/op_4830398d0c2feb5f/op_4830398d0c2feb5f.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "49fef23324a879b69815a1c51ea90a96fbc106989ab3ecbbd8a7f64203ede3b0"
    },
    {
      "id": "op_4a4434b5cc6e85e6",
      "ulid": "01M1HME78068MQY7E9KA81B7WX",
      "aliases": [
        "op_4a4434b5cc6e85e6",
        "01M1HME78068MQY7E9KA81B7WX",
        "op-4a4434b5cc6e85e6",
        "v2-gaussian-entanglement-of-formation-beyond-bisymmetry",
        "open-problem-v2-problem-26"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "gaussian-quantum-information",
          "entanglement-measures"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Gaussian entanglement of formation beyond bisymmetry",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Gaussian quantum information",
        "Entanglement measures"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Gaussian quantum information",
        "Entanglement measures"
      ],
      "statement": "Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every non-bisymmetric multimode Gaussian state? Let $\\rho_{AB}$ be a Gaussian state of $n_A+n_B\\geq3$ bosonic modes, with covariance matrix $V$. Its entanglement of formation is\n\n \\begin{equation}\n E_F(\\rho_{AB})\n :=\\inf_{\\rho_{AB}=\\sum_jp_j\\lvert\\psi_j\\rangle\\!\\langle\\psi_j\\rvert}\n \\sum_jp_j S\\!\\left(\\operatorname{Tr}_B\n \\lvert\\psi_j\\rangle\\!\\langle\\psi_j\\rvert\\right),\n\\tag{1}\n\\end{equation} \nwhere the infimum is over all pure-state ensembles. The Gaussian restriction of Eq. (1) is equivalently\n\n \\begin{equation}\n E_F^G(\\rho_{AB})\n :=\\inf_{\\substack{V_p\\preceq V\\\\V_p\\ \\mathrm{pure\\ Gaussian}}}\n E(V_p),\n\\tag{2}\n\\end{equation} \nwhere $E(V_p)$ is the entropy of either reduced state of the pure Gaussian state with covariance matrix $V_p$. A covariance matrix is bisymmetric when it is invariant under arbitrary permutations of Alice’s modes and, independently, of Bob’s modes. The question is whether $E_F(\\rho_{AB})=E_F^G(\\rho_{AB})$ outside this bisymmetric family; the two quantities are fixed by Eqs. (1) and (2).",
      "url": "https://qiqc-op.com/problem/op_4a4434b5cc6e85e6/",
      "json": "https://qiqc-op.com/api/problems/op_4a4434b5cc6e85e6.json",
      "tex": "https://qiqc-op.com/problem/op_4a4434b5cc6e85e6/op_4a4434b5cc6e85e6.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "6bd4a932c7aeda59d3e6f4573daae7b0bd0e95ae5a443ad5027bcf0b7159a8a1"
    },
    {
      "id": "op_4cf3e7b8663b1d41",
      "ulid": "01M1HME78004TME758T7JBWF1D",
      "aliases": [
        "op_4cf3e7b8663b1d41",
        "01M1HME78004TME758T7JBWF1D",
        "op-4cf3e7b8663b1d41",
        "v2-achievability-of-the-rains-bound-under-ppt-preserving-channels",
        "open-problem-v2-problem-4"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-distillation",
          "ppt-preserving-operations",
          "bell-diagonal-states"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Achievability of the Rains bound under PPT-preserving channels",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement distillation",
        "PPT-preserving operations",
        "Bell-diagonal states"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement distillation",
        "PPT-preserving operations",
        "Bell-diagonal states"
      ],
      "statement": "Consider distilling the bipartite Bell-diagonal state\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n =p_I\\lvert\\Phi^+\\rangle\\!\\langle\\Phi^+\\rvert\n +p_X\\lvert\\Psi^+\\rangle\\!\\langle\\Psi^+\\rvert\n +p_Y\\lvert\\Psi^-\\rangle\\!\\langle\\Psi^-\\rvert\n +p_Z\\lvert\\Phi^-\\rangle\\!\\langle\\Phi^-\\rvert,\n\\tag{1}\n\\end{equation} \nwhere $p_i>0$ and $p_I+p_X+p_Y+p_Z=1$, and $\\lvert\\Phi^\\pm\\rangle:=(\\lvert00\\rangle\\pm\\lvert11\\rangle)/\\sqrt2$ and $\\lvert\\Psi^\\pm\\rangle:=(\\lvert01\\rangle\\pm\\lvert10\\rangle)/\\sqrt2$. Is the Rains bound of the state in Eq. (1) achievable by a positive-partial-transpose-state-preserving (PPT-state-preserving, or PPT-preserving) quantum channel? If so, what is the constructive quantum channel? A channel is PPT-state-preserving if every PPT input state is mapped to a PPT output state.",
      "url": "https://qiqc-op.com/problem/op_4cf3e7b8663b1d41/",
      "json": "https://qiqc-op.com/api/problems/op_4cf3e7b8663b1d41.json",
      "tex": "https://qiqc-op.com/problem/op_4cf3e7b8663b1d41/op_4cf3e7b8663b1d41.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "8b0c769916289383fd1590c951cedddeae0b1078c8152a73d67505d1b029505d"
    },
    {
      "id": "op_56578531a2c610fe",
      "ulid": "01M1HME7809FZ29BV78FAZ01QX",
      "aliases": [
        "op_56578531a2c610fe",
        "01M1HME7809FZ29BV78FAZ01QX",
        "op-56578531a2c610fe",
        "v2-resources-for-implementing-gibbs-preserving-channels",
        "open-problem-v2-problem-37"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "quantum-thermodynamics",
          "channel-simulation",
          "one-shot-and-finite-blocklength-bounds",
          "resource-conversion"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Resources for implementing Gibbs-preserving channels",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Quantum thermodynamics",
        "Channel simulation",
        "One-shot and finite-blocklength bounds",
        "Resource conversion"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum thermodynamics",
        "Channel simulation",
        "One-shot and finite-blocklength bounds",
        "Resource conversion"
      ],
      "statement": "What are the exact one-shot coherence and work costs of implementing an arbitrary Gibbs-preserving channel by thermal operations? Let finite systems $S$ and $S'$ have Hamiltonians $H_S$ and $H_{S'}$ at inverse temperature $\\beta$, with Gibbs states\n\n \\begin{equation}\n \\gamma_X:=\\frac{e^{-\\beta H_X}}{\\operatorname{Tr}(e^{-\\beta H_X})},\n \\qquad X\\in\\{S,S'\\}.\n\\tag{1}\n\\end{equation} \nA channel $\\Phi:S\\to S'$ is Gibbs preserving when it maps the first state in Eq. (1) to the second. For approximation error $\\varepsilon\\geq0$, define its quantum-Fisher-information coherence cost by\n\n \\begin{equation}\n F_c^{\\varepsilon}(\\Phi)\n :=\\inf_{\\substack{(R,H_R),\\ \\eta_R\\in\\mathcal D(R)\\\\\n \\mathcal T:S\\otimes R\\to S'\\ \\mathrm{thermal}}}\n \\left\\{\n F_Q(\\eta_R,H_R):\n D_{\\mathrm{ch}}\\!\\left(\n \\Phi,\\mathcal T(\\,\\cdot\\,\\otimes\\eta_R)\n \\right)\\leq\\varepsilon\n \\right\\},\n\\tag{2}\n\\end{equation} \nwhere the infimum ranges over finite resource systems $R$ with Hamiltonian $H_R$, and $\\mathcal T$ is a thermal operation implemented with Gibbs ancillas and an energy-conserving unitary; set $\\inf\\varnothing:=+\\infty$. The distance $D_{\\mathrm{ch}}$ is the supremum of purified distance between the two channel outputs over inputs entangled with an arbitrary reference. If $\\eta_R=\\sum_j\\lambda_j\\lvert j\\rangle\\!\\langle j\\rvert$, the quantity in Eq. (2) is\n\n \\begin{equation}\n F_Q(\\eta_R,H_R)\n :=2\\!\\sum_{j,k:\\,\\lambda_j+\\lambda_k>0}\n \\frac{(\\lambda_j-\\lambda_k)^2}{\\lambda_j+\\lambda_k}\n \\left\\lvert\\langle j\\rvert H_R\\lvert k\\rangle\\right\\rvert^2.\n\\tag{3}\n\\end{equation} \nEquation (3) fixes the normalization of the coherence measure used in Eq. (2). Determine Eq. (2), up to matching bounds, for every Gibbs-preserving $\\Phi$ and every $\\varepsilon$. Give the analogous tight deterministic work cost when an ideal battery begins and ends in sharp energy states and the battery’s energy loss is charged as work. Which structural features of $\\Phi$ determine whether the exact costs are finite, and what is their sharp divergence as $\\varepsilon\\downarrow0$ when they are infinite at zero error?",
      "url": "https://qiqc-op.com/problem/op_56578531a2c610fe/",
      "json": "https://qiqc-op.com/api/problems/op_56578531a2c610fe.json",
      "tex": "https://qiqc-op.com/problem/op_56578531a2c610fe/op_56578531a2c610fe.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "390ae506e1c4e23a508a94227c27f382f1613c0575f4df1bcd9a3a7877a47b65"
    },
    {
      "id": "op_56cf60cdecde44f7",
      "ulid": "01M1HME780CC21XQAXBTWRJRCY",
      "aliases": [
        "op_56cf60cdecde44f7",
        "01M1HME780CC21XQAXBTWRJRCY",
        "op-56cf60cdecde44f7",
        "v2-collective-cost-of-tensor-power-state-preparation",
        "open-problem-v2-problem-16"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm"
        ],
        "topicIds": [
          "quantum-state-preparation",
          "quantum-circuit-complexity",
          "additivity-and-regularization"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Collective cost of tensor-power state preparation",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm"
      ],
      "topics": [
        "Quantum state preparation",
        "Quantum circuit complexity",
        "Additivity and regularization"
      ],
      "tags": [
        "Quantum algorithm",
        "Quantum state preparation",
        "Quantum circuit complexity",
        "Additivity and regularization"
      ],
      "statement": "Determine the asymptotic weighted circuit cost of preparing tensor powers of a known pure state, and characterize when collective preparation is cheaper per copy than independent preparation. On an arbitrary qubit register, allow Pauli-product rotations $e^{i\\phi P}$ with $\\phi\\in\\mathbb{R}$ and $P\\in\\{I,X,Y,Z\\}^{\\otimes q}$, assigning each such gate the cost $|\\phi|$. For a known $m$-qubit state $\\lvert\\psi\\rangle$, let $U(\\boldsymbol\\phi,\\boldsymbol P):=\\prod_{j=1}^r e^{i\\phi_jP_j}$ for a finite sequence of angles and Pauli products. Define the phase-independent exact $n$-copy cost by\n\n \\begin{equation}\n C_n(\\psi)\n :=\\inf\\left\\{\n \\sum_{j=1}^{r}|\\phi_j|:\n \\begin{array}{l}\n r\\in\\mathbb N_0,\\ \\phi_j\\in\\mathbb R,\\ \n P_j\\in\\{I,X,Y,Z\\}^{\\otimes mn}\\ (1\\le j\\le r),\\\\\n U(\\boldsymbol\\phi,\\boldsymbol P)\n \\lvert0^{mn}\\rangle\\!\\langle0^{mn}\\rvert\n U(\\boldsymbol\\phi,\\boldsymbol P)^\\dagger\n =(\\lvert\\psi\\rangle\\!\\langle\\psi\\rvert)^{\\otimes n}\n \\end{array}\n \\right\\}.\n\\tag{1}\n\\end{equation} \nDetermine the growth of Eq. (1) with $n$ and characterize the states for which the regularized cost\n\n \\begin{equation}\n C_\\infty(\\psi)\n :=\\lim_{n\\to\\infty}\\frac{C_n(\\psi)}{n}\n =\\inf_{n\\ge1}\\frac{C_n(\\psi)}{n}\n\\tag{2}\n\\end{equation} \nsatisfies $C_\\infty(\\psi)<C_1(\\psi)$. For an approximation tolerance $\\varepsilon\\ge0$, also determine the scaling of\n\n \\begin{equation}\n C_n^\\varepsilon(\\psi)\n :=\\inf_U\\left\\{\n \\operatorname{cost}(U):\n \\frac12\\left\\|\n U\\lvert0^{mn}\\rangle\\!\\langle0^{mn}\\rvert U^\\dagger\n -(\\lvert\\psi\\rangle\\!\\langle\\psi\\rvert)^{\\otimes n}\n \\right\\|_1\\le\\varepsilon\n \\right\\},\n\\tag{3}\n\\end{equation} \nwhere $U$ ranges over all finite circuits of Pauli-product rotations on the $mn$-qubit register and $\\operatorname{cost}(U)$ is the corresponding sum of absolute rotation angles, as in Eq. (1). The question for Eq. (3) includes fixed and vanishing error sequences.",
      "url": "https://qiqc-op.com/problem/op_56cf60cdecde44f7/",
      "json": "https://qiqc-op.com/api/problems/op_56cf60cdecde44f7.json",
      "tex": "https://qiqc-op.com/problem/op_56cf60cdecde44f7/op_56cf60cdecde44f7.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "9aba7f4e6dc4188fb4f2efe062ddc41d17eadda34683b7633f655595827f0ed2"
    },
    {
      "id": "op_64db785888803dd3",
      "ulid": "01M1HME780PY65PJ3HAY1NNW15",
      "aliases": [
        "op_64db785888803dd3",
        "01M1HME780PY65PJ3HAY1NNW15",
        "op-64db785888803dd3",
        "v2-one-bit-simulation-of-partially-entangled-qubits",
        "open-problem-v2-problem-27"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality",
          "quantum-communication-complexity"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "One-bit simulation of partially entangled qubits",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality",
        "Quantum communication complexity"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Resource Theory",
        "Bell nonlocality",
        "Quantum communication complexity"
      ],
      "statement": "Can shared randomness and one classical bit exactly simulate every pair of local projective measurements on every pure entangled two-qubit state? In Schmidt form the state is\n\n \\begin{equation}\n \\lvert\\psi_p\\rangle\n =\\sqrt p\\,\\lvert00\\rangle+\\sqrt{1-p}\\,\\lvert11\\rangle,\n \\qquad \\frac12<p<1.\n\\tag{1}\n\\end{equation} \nFor arbitrary Bloch vectors $\\mathbf x,\\mathbf y\\in S^2$ and outcomes $a,b\\in\\{0,1\\}$, the target distribution associated with Eq. (1) is\n\n \\begin{equation}\n \\begin{aligned}\n P_p(a,b\\mid\\mathbf x,\\mathbf y)\n &=\\operatorname{Tr}\\!\\left[\n \\lvert\\psi_p\\rangle\\!\\langle\\psi_p\\rvert\n \\bigl(M_{a\\mid\\mathbf x}\\otimes M_{b\\mid\\mathbf y}\\bigr)\n \\right],\\\\\n M_{a\\mid\\mathbf x}\n &=\\frac{I+(-1)^a\\mathbf x\\cdot\\boldsymbol\\sigma}{2},\n \\qquad\n M_{b\\mid\\mathbf y}\n =\\frac{I+(-1)^b\\mathbf y\\cdot\\boldsymbol\\sigma}{2}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nThe protocol may use unlimited shared randomness; after receiving $\\mathbf x$, Alice sends Bob one bit, and their local outputs must reproduce Eq. (2) for every pair of measurement directions. If no such universal protocol exists, find a finite-setting linear inequality satisfied by all one-bit protocols and violated by one of these target distributions.",
      "url": "https://qiqc-op.com/problem/op_64db785888803dd3/",
      "json": "https://qiqc-op.com/api/problems/op_64db785888803dd3.json",
      "tex": "https://qiqc-op.com/problem/op_64db785888803dd3/op_64db785888803dd3.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "96568db80445bf6e5a25fde85bb1434c69f83b923a6bc983f1ecad357bee6dce"
    },
    {
      "id": "op_660dbca2258b1a9b",
      "ulid": "01M1HME780W2G8FNTVYWK67QBN",
      "aliases": [
        "op_660dbca2258b1a9b",
        "01M1HME780W2G8FNTVYWK67QBN",
        "op-660dbca2258b1a9b",
        "v2-long-range-vacuum-chsh-violation",
        "open-problem-v2-problem-13"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Long-range vacuum CHSH violation",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Bell nonlocality"
      ],
      "statement": "Does the vacuum of a massive free scalar Bose field violate the CHSH inequality between bounded localization regions at arbitrarily large spacelike separation? Fix a bounded region $O$ in Minkowski spacetime, let $O_L$ be a congruent spacelike translate at separation $L$, and write $\\mathcal{A}(O)$ and $\\mathcal{A}(O_L)$ for the commuting local von Neumann algebras. For the vacuum state $\\omega_0$, define\n\n \\begin{equation}\n \\beta(L)\n :=\\frac12\\sup_{A,A',B,B'}\n \\left|\n \\omega_0\\!\\left(A(B+B')+A'(B-B')\\right)\n \\right|,\n\\tag{1}\n\\end{equation} \nwhere $A,A'\\in\\mathcal{A}(O)$ and $B,B'\\in\\mathcal{A}(O_L)$ range over self-adjoint contractions. With the normalization in Eq. (1), the local hidden-variable bound is $1$. The open question is whether\n\n \\begin{equation}\n \\left\\{L>0:\\beta(L)>1\\right\\}\n \\quad\\text{is unbounded}.\n\\tag{2}\n\\end{equation} \nEquation (2) asks for direct, unfiltered two-setting violation by the vacuum restrictions to the two local algebras.",
      "url": "https://qiqc-op.com/problem/op_660dbca2258b1a9b/",
      "json": "https://qiqc-op.com/api/problems/op_660dbca2258b1a9b.json",
      "tex": "https://qiqc-op.com/problem/op_660dbca2258b1a9b/op_660dbca2258b1a9b.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "f624d9bc39dc6bc675499dc23ae310014f496126226d4bc35ed0fed418774a9f"
    },
    {
      "id": "op_6ba929179cc40c0a",
      "ulid": "01M1HME780146X04XW01Y1DZHB",
      "aliases": [
        "op_6ba929179cc40c0a",
        "01M1HME780146X04XW01Y1DZHB",
        "op-6ba929179cc40c0a",
        "v2-weyl-heisenberg-covariant-sics-in-every-dimension",
        "open-problem-v2-problem-18"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "symmetric-informationally-complete-measurements"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Weyl–Heisenberg-covariant SICs in every dimension",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Symmetric informationally complete measurements"
      ],
      "tags": [
        "Quantum metrology",
        "Symmetric informationally complete measurements"
      ],
      "statement": "Does every finite dimension admit a symmetric informationally complete measurement that is a single Weyl–Heisenberg orbit? For an integer $d\\geq2$, let $\\omega_d=e^{2\\pi i/d}$ and define shift, phase, and displacement operators on the computational basis by\n\n \\begin{equation}\n X_d\\lvert j\\rangle=\\lvert j+1\\!\\!\\pmod d\\rangle,\n \\qquad\n Z_d\\lvert j\\rangle=\\omega_d^j\\lvert j\\rangle,\n \\qquad\n D_{p,q}=X_d^pZ_d^q,\n \\quad (p,q)\\in\\mathbb Z_d^2.\n\\tag{1}\n\\end{equation} \nEquation (1) fixes a phase convention that does not affect the orbit of rank-one projectors. The question is whether, for every $d\\geq2$, there is a unit vector $\\lvert\\phi\\rangle\\in\\mathbb C^d$ satisfying\n\n \\begin{equation}\n \\bigl|\\langle\\phi\\rvert D_{p,q}\\lvert\\phi\\rangle\\bigr|^2\n =\\frac{1}{d+1}\n \\qquad\n \\text{for every }(p,q)\\in\\mathbb Z_d^2\\setminus\\{(0,0)\\}.\n\\tag{2}\n\\end{equation} \nIf Eq. (2) holds, the $d^2$ projectors in the Weyl–Heisenberg orbit of $\\lvert\\phi\\rangle$ form a SIC.",
      "url": "https://qiqc-op.com/problem/op_6ba929179cc40c0a/",
      "json": "https://qiqc-op.com/api/problems/op_6ba929179cc40c0a.json",
      "tex": "https://qiqc-op.com/problem/op_6ba929179cc40c0a/op_6ba929179cc40c0a.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "f414ac7fbd2e2b80d51fbebe00ef2b99b762a7715c9065037c52ffd3c3a688ca"
    },
    {
      "id": "op_6cb323ea3ec0b70e",
      "ulid": "01M1HME7808X29G1W7P0AC8QWZ",
      "aliases": [
        "op_6cb323ea3ec0b70e",
        "01M1HME7808X29G1W7P0AC8QWZ",
        "op-6cb323ea3ec0b70e",
        "v2-trace-exponential-lower-bound-for-matrix-word-averages",
        "open-problem-v2-problem-34"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "matrix-and-entropy-inequalities"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Trace-exponential lower bound for matrix-word averages",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Matrix and entropy inequalities"
      ],
      "tags": [
        "Quantum Communication",
        "Matrix and entropy inequalities"
      ],
      "statement": "Is the normalized trace average of all words in two positive-definite matrices always bounded below by the corresponding trace exponential? Let $A,B\\in M_d(\\mathbb C)$ be positive definite, let $n,m\\geq1$, and let $\\mathcal W_{n,m}$ be the set of words containing exactly $n$ letters $A$ and $m$ letters $B$. Define the normalized average by\n\n \\begin{equation}\n p_{n,m}(A,B)\n :=\\frac{1}{\\binom{n+m}{n}}\n \\sum_{W\\in\\mathcal W_{n,m}}\\operatorname{Tr}W(A,B).\n\\tag{1}\n\\end{equation} \nThe question is whether the quantity in Eq. (1) satisfies\n\n \\begin{equation}\n p_{n,m}(A,B)\n \\geq \\operatorname{Tr}\\exp\\!\\bigl(n\\log A+m\\log B\\bigr)\n\\tag{2}\n\\end{equation} \nfor every finite $d$ and all $n,m\\geq1$. A positive-semidefinite extension is obtained, whenever the limit exists, by applying Eq. (2) to $A+\\varepsilon I$ and $B+\\varepsilon I$ and then taking $\\varepsilon\\downarrow0$.",
      "url": "https://qiqc-op.com/problem/op_6cb323ea3ec0b70e/",
      "json": "https://qiqc-op.com/api/problems/op_6cb323ea3ec0b70e.json",
      "tex": "https://qiqc-op.com/problem/op_6cb323ea3ec0b70e/op_6cb323ea3ec0b70e.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "19ded7c04baf36c08172309af36c2a3aab4ab7c7fabf6390d60b4db4686fe68b"
    },
    {
      "id": "op_74c4ffb5b042baf2",
      "ulid": "01M1HME780V1V27MDXXVA3SCW5",
      "aliases": [
        "op_74c4ffb5b042baf2",
        "01M1HME780V1V27MDXXVA3SCW5",
        "op-74c4ffb5b042baf2",
        "v2-npt-bound-entanglement-and-the-rank-five-frontier",
        "open-problem-v2-problem-10"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-distillation",
          "bound-entanglement",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "NPT bound entanglement and the rank-five frontier",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement distillation",
        "Bound entanglement",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement distillation",
        "Bound entanglement",
        "Local operations and classical communication"
      ],
      "statement": "Does there exist a finite-dimensional bipartite state with non-positive partial transpose (NPT) that is undistillable by local operations and classical communication (LOCC)? For a density operator $\\rho_{AB}$, partial transposition on $B$ is denoted by $\\rho_{AB}^{T_B}:=(\\operatorname{id}_A\\otimes T_B)(\\rho_{AB})$. The finite-copy criterion makes the counterexample sought by the general problem precise:\n\n \\begin{equation}\n \\exists\\,\\rho_{AB}:\\quad\n \\rho_{AB}^{T_B}\\not\\succeq0,\n \\qquad\n \\forall\\,n\\geq1\\ \\ \\forall\\,\\lvert\\psi_n\\rangle\n \\text{ with }\\operatorname{SR}_{A^n:B^n}(\\lvert\\psi_n\\rangle)\\leq2,\n \\quad\n \\langle\\psi_n\\rvert\n (\\rho_{AB}^{T_B})^{\\otimes n}\n \\lvert\\psi_n\\rangle\\geq0.\n\\tag{1}\n\\end{equation} \nHere $\\operatorname{SR}_{A^n:B^n}$ is Schmidt rank across the indicated cut. A state satisfying Eq. (1) would be NPT bound entangled; equivalently, the question is whether every NPT state instead has a negative expectation on some Schmidt-rank-at-most-two vector at some finite copy number.\n\nThe nested low-rank frontier asks whether the counterexample in Eq. (1) can additionally satisfy\n\n \\begin{equation}\n \\operatorname{rank}(\\rho_{AB})=5.\n\\tag{2}\n\\end{equation} \nThus Eq. (2) asks for the lowest rank not already excluded by known one-copy results.",
      "url": "https://qiqc-op.com/problem/op_74c4ffb5b042baf2/",
      "json": "https://qiqc-op.com/api/problems/op_74c4ffb5b042baf2.json",
      "tex": "https://qiqc-op.com/problem/op_74c4ffb5b042baf2/op_74c4ffb5b042baf2.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "b738b6b12332d7f6ed75eab94469b4749445e1d141f7beb17aa8e9c0ba039396"
    },
    {
      "id": "op_7a17786328198e24",
      "ulid": "01M1HME7805PVXXPJE60E83TJ5",
      "aliases": [
        "op_7a17786328198e24",
        "01M1HME7805PVXXPJE60E83TJ5",
        "op-7a17786328198e24",
        "v2-minimal-support-frontier-for-absolutely-maximally-entangled-states",
        "open-problem-v2-problem-42"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-error-correction"
        ],
        "topicIds": [
          "absolutely-maximally-entangled-states",
          "quantum-coding-theory"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Minimal-support frontier for absolutely maximally entangled states",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum Error Correction"
      ],
      "topics": [
        "Absolutely maximally entangled states",
        "Quantum coding theory"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum Error Correction",
        "Absolutely maximally entangled states",
        "Quantum coding theory"
      ],
      "statement": "Determine, for every integer $d\\geq 2$, the largest number $\\mathcal N(d)$ of $d$-level parties that admit an absolutely maximally entangled state of minimal computational-basis support. To make this precise, let $m=\\lfloor N/2\\rfloor$ and write a normalized state as \\(|\\psi\\rangle=\\sum_{\\boldsymbol{x}\\in[d]^N}c_{\\boldsymbol{x}}\n|\\boldsymbol{x}\\rangle\\), where $[d]=\\{0,\\ldots,d-1\\}$. The state is $\\operatorname{AME}(N,d)$ when every subsystem $S\\subseteq\\{1,\\ldots,N\\}$ with $|S|\\leq m$ has reduced state\n\n \\begin{equation}\n \\operatorname{Tr}_{S^{\\mathrm c}}|\\psi\\rangle\\!\\langle\\psi|\n =\\frac{I_{d^{|S|}}}{d^{|S|}}.\n\\tag{1}\n\\end{equation} \nCondition (1) implies the computational-basis support bound\n\n \\begin{equation}\n \\bigl|\\operatorname{supp}_{\\mathrm{comp}}(\\psi)\\bigr|\n :=\\bigl|\\{\\boldsymbol{x}:c_{\\boldsymbol{x}}\\neq0\\}\\bigr|\n \\geq d^{m}.\n\\tag{2}\n\\end{equation} \nMinimal support means equality in (2), and the frontier to be determined is\n\n \\begin{equation}\n \\mathcal N(d):=\\max\\bigl\\{N\\geq2:\\text{a minimal-support }\n \\operatorname{AME}(N,d)\\text{ state exists}\\bigr\\}.\n\\tag{3}\n\\end{equation} \nEquivalently, for any alphabet $\\mathcal A$ of size $d$, the existence event in (3) is characterized by\n\n \\begin{equation}\n \\begin{split}\n &\\text{a minimal-support }\\operatorname{AME}(N,d)\\text{ state exists}\n \\\\\n &\\quad\\Longleftrightarrow\\quad\n \\exists\\,\\mathcal C\\subseteq\\mathcal A^N:\\quad\n |\\mathcal C|=d^{\\lfloor N/2\\rfloor},\\qquad\n \\min_{\\substack{\\boldsymbol{x},\\boldsymbol{y}\\in\\mathcal C\\\\\n \\boldsymbol{x}\\neq\\boldsymbol{y}}}\n d_{\\mathrm H}(\\boldsymbol{x},\\boldsymbol{y})\n =\\left\\lceil\\frac N2\\right\\rceil+1.\n \\end{split}\n\\tag{4}\n\\end{equation} \nThus (4) asks for general, possibly nonlinear, MDS codes rather than only linear codes [GAL+15], [Ber19].",
      "url": "https://qiqc-op.com/problem/op_7a17786328198e24/",
      "json": "https://qiqc-op.com/api/problems/op_7a17786328198e24.json",
      "tex": "https://qiqc-op.com/problem/op_7a17786328198e24/op_7a17786328198e24.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "7db451a6105bad825c2c3ca66855dbfcbb77f05683d40f6162635fe8e5f5c2f3"
    },
    {
      "id": "op_7a9051ff6d0a1739",
      "ulid": "01M1HME780J76RC69YY1FTM06V",
      "aliases": [
        "op_7a9051ff6d0a1739",
        "01M1HME780J76RC69YY1FTM06V",
        "op-7a9051ff6d0a1739",
        "v2-entanglement-cost-of-an-amplitude-damping-channel-choi-state",
        "open-problem-v2-problem-8"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-cost",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Entanglement cost of an amplitude-damping-channel Choi state",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement cost",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement cost",
        "Local operations and classical communication"
      ],
      "statement": "What is the entanglement cost of the Choi state of the qubit amplitude-damping channel\n\n \\begin{equation}\n \\mathcal A_p(\\rho)=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\n \\qquad 0\\le p\\le1?\n\\tag{1}\n\\end{equation} \nThe Kraus operators in Eq. (1) are\n\n \\begin{equation}\n \\begin{aligned}\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert\n +\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}1&0\\\\0&\\sqrt{1-p}\\end{pmatrix},\\\\\n A_1&=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}0&\\sqrt p\\\\0&0\\end{pmatrix}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nIn Eq. (2), $p$ is the decay probability of the excited state. Let $\\lvert\\Phi^+\\rangle_{RA}=(\\lvert00\\rangle+\\lvert11\\rangle)/\\sqrt2$. The normalized Choi state of the channel in Eq. (1) is\n\n \\begin{equation}\n \\begin{aligned}\n \\omega_p^{RB}\n &:=(\\operatorname{id}_R\\otimes\\mathcal A_p)\n (\\lvert\\Phi^+\\rangle\\!\\langle\\Phi^+\\rvert_{RA})\\\\\n &=\\frac12\\Bigl[\n \\lvert00\\rangle\\!\\langle00\\rvert\n +\\sqrt{1-p}\\bigl(\\lvert00\\rangle\\!\\langle11\\rvert\n +\\lvert11\\rangle\\!\\langle00\\rvert\\bigr)\n +(1-p)\\lvert11\\rangle\\!\\langle11\\rvert\n +p\\lvert10\\rangle\\!\\langle10\\rvert\n \\Bigr].\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nHere the first and second entries in each ket in Eq. (3) label $R$ and $B$, respectively. Thus the question is to determine $E_C(\\omega_p)$, the asymptotic number of ebits per copy required to prepare many copies of $\\omega_p$ by local operations and classical communication.",
      "url": "https://qiqc-op.com/problem/op_7a9051ff6d0a1739/",
      "json": "https://qiqc-op.com/api/problems/op_7a9051ff6d0a1739.json",
      "tex": "https://qiqc-op.com/problem/op_7a9051ff6d0a1739/op_7a9051ff6d0a1739.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "28ab786be62a20f864947c7df4ba57650cd2b31d419b2a57d41579702cafd9e9"
    },
    {
      "id": "op_7ceb25c3fe5c9403",
      "ulid": "01M1HME780CAT6PDF6X8ZEV5VP",
      "aliases": [
        "op_7ceb25c3fe5c9403",
        "01M1HME780CAT6PDF6X8ZEV5VP",
        "op-7ceb25c3fe5c9403",
        "v2-lockability-of-two-way-distillable-entanglement",
        "open-problem-v2-problem-21"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-distillation",
          "entanglement-measures",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Lockability of two-way distillable entanglement",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement distillation",
        "Entanglement measures",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement distillation",
        "Entanglement measures",
        "Local operations and classical communication"
      ],
      "statement": "Can discarding one local qubit reduce two-way distillable entanglement by an arbitrarily large amount? Let $D_{\\leftrightarrow}(A:B)_\\rho$ denote the asymptotic singlet-distillation rate of $\\rho_{AB}$ under local operations and unrestricted two-way classical communication. The question is whether there are finite-dimensional states $\\rho^{(r)}_{A_ra:B_r}$, with $\\dim a=2$, such that\n\n \\begin{equation}\n D_{\\leftrightarrow}(A_ra:B_r)_{\\rho^{(r)}}\n -D_{\\leftrightarrow}(A_r:B_r)_{\\operatorname{Tr}_a\\rho^{(r)}}\n \\xrightarrow[r\\to\\infty]{}\\infty.\n\\tag{1}\n\\end{equation} \nEquation (1) requires an unbounded loss while the discarded subsystem has fixed dimension two.",
      "url": "https://qiqc-op.com/problem/op_7ceb25c3fe5c9403/",
      "json": "https://qiqc-op.com/api/problems/op_7ceb25c3fe5c9403.json",
      "tex": "https://qiqc-op.com/problem/op_7ceb25c3fe5c9403/op_7ceb25c3fe5c9403.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "9dc0355fd485edd655e472d447964573f7c861e1814188171762581d8dcd9e32"
    },
    {
      "id": "op_9209e4eb15586dec",
      "ulid": "01M1HME780AZ2GCKS76GDX97RQ",
      "aliases": [
        "op_9209e4eb15586dec",
        "01M1HME780AZ2GCKS76GDX97RQ",
        "op-9209e4eb15586dec",
        "v2-quantum-capacity-of-a-bosonic-thermal-attenuator",
        "open-problem-v2-problem-49"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "bosonic-channels"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Quantum capacity of a bosonic thermal attenuator",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Bosonic channels"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Bosonic channels"
      ],
      "statement": "Let $\\Phi_{\\eta,\\nu}$ be the single-mode bosonic thermal attenuator with transmissivity $0<\\eta<1$ and environmental mean photon number $0<\\nu<\\infty$, defined by\n\n \\begin{equation}\n \\Phi_{\\eta,\\nu}(\\rho_A)\n :=\\operatorname{Tr}_{E'}\\!\\left[\n U_\\eta(\\rho_A\\otimes\\tau_{\\nu,E})U_\\eta^\\dagger\n \\right],\n \\qquad\n \\tau_{\\nu,E}:=\\sum_{k=0}^{\\infty}\n \\frac{\\nu^k}{(\\nu+1)^{k+1}}\n |k\\rangle_E\\!\\langle k|_E,\n\\tag{1}\n\\end{equation} \nwhere $U_\\eta:AE\\to BE'$ is a beam-splitter unitary and $\\tau_{\\nu,E}$ acts on the environment mode $E$. For an environment mode of angular frequency $\\omega_E$ at temperature $T$, $\\nu=(e^{\\hbar\\omega_E/(k_{\\rm B}T)}-1)^{-1}$. Thus $0<T<\\infty$ is equivalent to $0<\\nu<\\infty$ for fixed $\\omega_E>0$; $\\nu=0$ corresponds to $T=0$, while $\\nu\\to\\infty$ as $T\\to\\infty$. Its unassisted quantum capacity is the regularized coherent information\n\n \\begin{equation}\n \\mathcal Q(\\Phi_{\\eta,\\nu})\n :=\\lim_{n\\to\\infty}\\frac1n\\sup_{\\rho_{A^n}}\n I_{\\rm c}(\\rho_{A^n},\\Phi_{\\eta,\\nu}^{\\otimes n}),\n \\qquad\n I_{\\rm c}(\\rho,\\mathcal N)\n :=S(\\mathcal N(\\rho))-S(\\mathcal N^{\\rm c}(\\rho)),\n\\tag{2}\n\\end{equation} \nwhere $S(\\sigma):=-\\operatorname{Tr}(\\sigma\\log_2\\sigma)$ and $\\mathcal N^{\\rm c}$ is any complementary channel. Determine Eq. (2) for every $0<\\eta<1$ and $0<\\nu<\\infty$, equivalently every strictly positive finite environment temperature.",
      "url": "https://qiqc-op.com/problem/op_9209e4eb15586dec/",
      "json": "https://qiqc-op.com/api/problems/op_9209e4eb15586dec.json",
      "tex": "https://qiqc-op.com/problem/op_9209e4eb15586dec/op_9209e4eb15586dec.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "b68cbed87fc7b5bfcb9802365db3b80120ac86c0e2da789e62a0da64833a05dc"
    },
    {
      "id": "op_96cf7aa1c1be9be2",
      "ulid": "01M1HME780TJ3Z7X332QY1GWFR",
      "aliases": [
        "op_96cf7aa1c1be9be2",
        "01M1HME780TJ3Z7X332QY1GWFR",
        "op-96cf7aa1c1be9be2",
        "v2-polynomial-shared-resource-lower-bounds-for-routing",
        "open-problem-v2-problem-38"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication",
          "quantum-cryptography"
        ],
        "topicIds": [
          "quantum-communication-complexity",
          "position-based-quantum-cryptography"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Polynomial shared-resource lower bounds for routing",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication",
        "Quantum Cryptography"
      ],
      "topics": [
        "Quantum communication complexity",
        "Position-based quantum cryptography"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum Cryptography",
        "Quantum communication complexity",
        "Position-based quantum cryptography"
      ],
      "statement": "Does an explicit total Boolean family require polynomial shared-state cost for bounded-error one-round $f$-routing? More precisely, do there exist constants $c>0$ and $\\varepsilon>0$ and a sequence\n\n \\begin{equation}\n f_n:\\{0,1\\}^n\\times\\{0,1\\}^n\\longrightarrow\\{0,1\\}\n\\tag{1}\n\\end{equation} \nsuch that one uniform deterministic algorithm computes $f_n(x,y)$ from $(n,x,y)$ in time polynomial in $n$, and every routing protocol for the map in Eq. (1) with worst-case diamond-norm error at most $\\varepsilon$ has cost at least $n^c$?\n\nIn an $f$-routing protocol, Alice receives $x$ and an unknown qubit $Q$, Bob receives $y$, and they may share an arbitrary state $\\rho_{LR}$ before the inputs arrive. After one simultaneous message in each direction, Alice must recover $Q$ when $f(x,y)=0$, and Bob must recover it when $f(x,y)=1$. Message sizes and local operations are unrestricted. Measure only the shared state by\n\n \\begin{equation}\n E_{\\mathrm{dim}}(\\rho_{LR})\n :=\\log_2\\min\\!\\left\\{\n \\operatorname{rank}\\rho_L,\\operatorname{rank}\\rho_R\n \\right\\}.\n\\tag{2}\n\\end{equation} \nThe target is a family in Eq. (1) for which every valid protocol satisfies $E_{\\mathrm{dim}}(\\rho_{LR})\\geq n^c$, with the cost defined in Eq. (2).",
      "url": "https://qiqc-op.com/problem/op_96cf7aa1c1be9be2/",
      "json": "https://qiqc-op.com/api/problems/op_96cf7aa1c1be9be2.json",
      "tex": "https://qiqc-op.com/problem/op_96cf7aa1c1be9be2/op_96cf7aa1c1be9be2.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "11b03aeb502213456dc98f282f8ace797e3ac3859f84de5fd519842dc3167244"
    },
    {
      "id": "op_a34f0e2d6489068f",
      "ulid": "01M1HME7804M1QPND87GJGK3MH",
      "aliases": [
        "op_a34f0e2d6489068f",
        "01M1HME7804M1QPND87GJGK3MH",
        "op-a34f0e2d6489068f",
        "v2-statistical-strength-of-cglmp-measurements",
        "open-problem-v2-problem-25"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-metrology"
        ],
        "topicIds": [
          "bell-nonlocality",
          "quantum-hypothesis-testing"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Statistical strength of CGLMP measurements",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum metrology"
      ],
      "topics": [
        "Bell nonlocality",
        "Quantum hypothesis testing"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum metrology",
        "Bell nonlocality",
        "Quantum hypothesis testing"
      ],
      "statement": "For every $d\\geq3$, do the standard CGLMP Fourier–phase measurements maximize the relative-entropy statistical strength against local realism among all projective $d$-outcome measurements on the fixed state $\\lvert\\Phi_d\\rangle=d^{-1/2}\\sum_{j=0}^{d-1}\\lvert j,j\\rangle$, when the setting distribution is also optimized? For a behavior $p(a,b\\mid x,y)$, a distribution $\\mu(x,y)$ on the four setting pairs, and the local polytope $\\mathcal L$, define\n\n \\begin{equation}\n S(p;\\mu)\n :=\\inf_{\\ell\\in\\mathcal L}\n \\sum_{a,b,x,y}\\mu(x,y)p(a,b\\mid x,y)\n \\log_2\\!\\frac{p(a,b\\mid x,y)}{\\ell(a,b\\mid x,y)},\n\\tag{1}\n\\end{equation} \nSet $S^\\star(p):=\\sup_{\\mu\\in\\Delta(\\{0,1\\}^2)}S(p;\\mu)$; by Eq. (1), this is the optimized asymptotic evidence rate against the best local model. The candidate bases are\n\n \\begin{equation}\n \\begin{aligned}\n \\lvert a;x\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(a+\\alpha_x)\\right)\\lvert j\\rangle,\\\\\n \\lvert b;y\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(-b+\\beta_y)\\right)\\lvert j\\rangle,\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nwhere $\\alpha_0=0$, $\\alpha_1=-1/2$, $\\beta_0=1/4$, and $\\beta_1=3/4$. Let $p_{\\mathrm{CGLMP}}$ denote the behavior produced on $\\lvert\\Phi_d\\rangle$ by the bases in Eq. (2), and write $p_M$ for the behavior produced by any other measurement choice $M$ on $\\lvert\\Phi_d\\rangle$. The conjectured optimality of Eq. (2) is\n\n \\begin{equation}\n S^\\star(p_{\\mathrm{CGLMP}})\n =\\sup_M S^\\star(p_M),\n\\tag{3}\n\\end{equation} \nwhere the supremum is over two projective $d$-outcome measurements per party. Equation (3) is the question to be resolved.",
      "url": "https://qiqc-op.com/problem/op_a34f0e2d6489068f/",
      "json": "https://qiqc-op.com/api/problems/op_a34f0e2d6489068f.json",
      "tex": "https://qiqc-op.com/problem/op_a34f0e2d6489068f/op_a34f0e2d6489068f.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "e2629974f55bd5504e4ea8c873620be6f35b4e6bf36b03fbbead4b8eb38ce423"
    },
    {
      "id": "op_a9d8fc135ea71ca1",
      "ulid": "01M1HME780PPH667FYKZK7VTV6",
      "aliases": [
        "op_a9d8fc135ea71ca1",
        "01M1HME780PPH667FYKZK7VTV6",
        "op-a9d8fc135ea71ca1",
        "v2-extensible-causality-and-process-purification",
        "open-problem-v2-problem-35"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "indefinite-causal-order"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Extensible causality and process purification",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Indefinite causal order"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Indefinite causal order"
      ],
      "statement": "Is every finite-dimensional extensibly causal process matrix purifiable? Call an $N$-laboratory process matrix $W$ extensibly causal if $W\\otimes\\rho_R$ generates only causal correlations for every ancillary input system $R$, every joint ancillary state $\\rho_R$, and every choice of local instruments. Call $W$ purifiable if there exist auxiliary global past and future systems, a fixed pure state on the auxiliary past, and a pure process $S$ such that inserting the fixed state into $S$ and discarding the auxiliary future yields $W$. If $\\mathrm{EC}_N$ and $\\mathrm{Pur}_N$ denote the corresponding classes, the question is whether\n\n \\begin{equation}\n \\mathrm{EC}_N\\subseteq\\mathrm{Pur}_N\n\\tag{1}\n\\end{equation} \nholds for every $N$ and every choice of finite local dimensions. In particular, does the inclusion in Eq. (1) hold for two laboratories?",
      "url": "https://qiqc-op.com/problem/op_a9d8fc135ea71ca1/",
      "json": "https://qiqc-op.com/api/problems/op_a9d8fc135ea71ca1.json",
      "tex": "https://qiqc-op.com/problem/op_a9d8fc135ea71ca1/op_a9d8fc135ea71ca1.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "46547037fd0882069ad911bffa12acaa3cee53f0c1f38d7cc2a9c66b532e85a0"
    },
    {
      "id": "op_aa21ac5ebca8b888",
      "ulid": "01M1HME780JH0D9Y0RQ750ZZPY",
      "aliases": [
        "op_aa21ac5ebca8b888",
        "01M1HME780JH0D9Y0RQ750ZZPY",
        "op-aa21ac5ebca8b888",
        "v2-entanglement-cost-of-a-qubit-bell-diagonal-state",
        "open-problem-v2-problem-7"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-cost",
          "bell-diagonal-states",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Entanglement cost of a qubit Bell-diagonal state",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement cost",
        "Bell-diagonal states",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement cost",
        "Bell-diagonal states",
        "Local operations and classical communication"
      ],
      "statement": "What is the entanglement cost of a qubit Bell-diagonal state for an arbitrary probability vector $\\mathbf p=(p_I,p_X,p_Y,p_Z)$? Let $\\lvert\\Phi^+\\rangle=(\\lvert00\\rangle+\\lvert11\\rangle)/\\sqrt2$ and $\\lvert\\Phi_P\\rangle=(I\\otimes P)\\lvert\\Phi^+\\rangle$ for $P\\in\\{I,X,Y,Z\\}$. The state is\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n :=\\sum_{P\\in\\{I,X,Y,Z\\}}p_P\n \\lvert\\Phi_P\\rangle\\!\\langle\\Phi_P\\rvert,\n \\qquad p_P\\geq0,\n \\qquad \\sum_{P\\in\\{I,X,Y,Z\\}}p_P=1.\n\\tag{1}\n\\end{equation} \nFor the state in Eq. (1), determine $E_C(\\rho_{\\mathbf p})$ for every $\\mathbf p$ in the probability simplex, where $E_C$ is the infimum asymptotic rate of ebits consumed by LOCC protocols that prepare $\\rho_{\\mathbf p}^{\\otimes n}$ with trace-norm error tending to zero.",
      "url": "https://qiqc-op.com/problem/op_aa21ac5ebca8b888/",
      "json": "https://qiqc-op.com/api/problems/op_aa21ac5ebca8b888.json",
      "tex": "https://qiqc-op.com/problem/op_aa21ac5ebca8b888/op_aa21ac5ebca8b888.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "ff88add61844d43755f2e73ac4fd3de01a7ad8797edacac97678cdd6d12859e9"
    },
    {
      "id": "op_b063bdae4363cda8",
      "ulid": "01M1HME780N8J8JBTFX6CSPACD",
      "aliases": [
        "op_b063bdae4363cda8",
        "01M1HME780N8J8JBTFX6CSPACD",
        "op-b063bdae4363cda8",
        "v2-absolute-separability-from-spectra",
        "open-problem-v2-problem-15"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "quantum-separability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Absolute separability from spectra",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Quantum separability"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum separability"
      ],
      "statement": "Characterize the spectra of bipartite states that remain separable under every global unitary, and decide whether absolute separability equals absolute positivity under partial transpose in higher local dimensions. Let $2\\le m\\le n$ and let $\\lambda=(\\lambda_1,\\ldots,\\lambda_{mn})$ be a decreasing probability vector. Define the absolutely separable spectra by\n\n \\begin{equation}\n \\operatorname{ASEP}_{m,n}\n :=\\left\\{\n \\lambda:\n U\\operatorname{diag}(\\lambda)U^\\dagger\n \\text{ is separable on }\\mathbb{C}^m\\otimes\\mathbb{C}^n\n \\text{ for every }U\\in\\mathrm{U}(mn)\n \\right\\}.\n\\tag{1}\n\\end{equation} \nThe relaxation by positivity under partial transpose is\n\n \\begin{equation}\n \\operatorname{APPT}_{m,n}\n :=\\left\\{\n \\lambda:\n \\left(U\\operatorname{diag}(\\lambda)U^\\dagger\\right)^{T_B}\\succeq0\n \\text{ for every }U\\in\\mathrm{U}(mn)\n \\right\\}.\n\\tag{2}\n\\end{equation} \nThe task is to characterize the set in Eq. (1) and, for $3\\le m\\le n$, decide whether it equals the set in Eq. (2).",
      "url": "https://qiqc-op.com/problem/op_b063bdae4363cda8/",
      "json": "https://qiqc-op.com/api/problems/op_b063bdae4363cda8.json",
      "tex": "https://qiqc-op.com/problem/op_b063bdae4363cda8/op_b063bdae4363cda8.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "ef98586ecee8e68161db1bfb2edc60d11d13db8d967880226e59c6a47d4dc9df"
    },
    {
      "id": "op_b08ad9d4371ed0cb",
      "ulid": "01M1HME780THG7M7MWH1AKVSRS",
      "aliases": [
        "op_b08ad9d4371ed0cb",
        "01M1HME780THG7M7MWH1AKVSRS",
        "op-b08ad9d4371ed0cb",
        "v2-existence-of-an-eight-ququart-perfect-tensor",
        "open-problem-v2-problem-40"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-error-correction"
        ],
        "topicIds": [
          "absolutely-maximally-entangled-states",
          "quantum-coding-theory"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Existence of an eight-ququart perfect tensor",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum Error Correction"
      ],
      "topics": [
        "Absolutely maximally entangled states",
        "Quantum coding theory"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum Error Correction",
        "Absolutely maximally entangled states",
        "Quantum coding theory"
      ],
      "statement": "Does an absolutely maximally entangled state of eight ququarts exist? More precisely, with $[8]=\\{1,\\ldots,8\\}$, determine whether there is a unit vector $|\\psi\\rangle\\in(\\mathbb{C}^{4})^{\\otimes 8}$ such that\n\n \\begin{equation}\n \\operatorname{Tr}_{S^{c}}\\!\\left(|\\psi\\rangle\\!\\langle\\psi|\\right)\n =\\frac{I_{4^{4}}}{4^{4}}\n \\qquad\\text{for every }S\\subseteq[8]\\text{ with }|S|=4.\n\\tag{1}\n\\end{equation} \nCondition (1) is equivalently the existence of a rank-eight perfect tensor of bond dimension $4$: every flattening across a $4|4$ partition is proportional to a unitary [PYHP15]. Under the pure-code correspondence, it is also equivalent to a pure quantum MDS code with parameters $\\bigl[\\!\\bigl[8,0,5\\bigr]\\!\\bigr]_{4}$ [SZZL26].",
      "url": "https://qiqc-op.com/problem/op_b08ad9d4371ed0cb/",
      "json": "https://qiqc-op.com/api/problems/op_b08ad9d4371ed0cb.json",
      "tex": "https://qiqc-op.com/problem/op_b08ad9d4371ed0cb/op_b08ad9d4371ed0cb.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "6f45ffe22a0dcef9a49a76f27b2123d501e403dd25a7711de33c976b729e7e75"
    },
    {
      "id": "op_b1b41f737f9b4aeb",
      "ulid": "01M1HME780G3PDHGCZMWV9MP71",
      "aliases": [
        "op_b1b41f737f9b4aeb",
        "01M1HME780G3PDHGCZMWV9MP71",
        "op-b1b41f737f9b4aeb",
        "v2-tough-error-models",
        "open-problem-v2-problem-14"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-error-correction"
        ],
        "topicIds": [
          "quantum-coding-theory"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Tough error models",
      "status": "Unsolved",
      "fields": [
        "Quantum Error Correction"
      ],
      "topics": [
        "Quantum coding theory"
      ],
      "tags": [
        "Quantum Error Correction",
        "Quantum coding theory"
      ],
      "statement": "Determine $c(e,n)$ and construct tough error models in the sense of [KW05]. Let $H=\\mathbb{C}^n$, and let an $e$-dimensional error model be a complex linear subspace $E\\subseteq\\operatorname{End}(H)$ with $1\\le e\\le n^2$. A linear subspace $C\\subseteq H$ corrects $E$ exactly when, for every $A,B\\in E$, there is a scalar $\\lambda(A,B)\\in\\mathbb{C}$ such that\n\n \\begin{equation}\n P_C A^\\dagger B P_C=\\lambda(A,B)P_C,\n\\tag{1}\n\\end{equation} \nwhere $P_C$ is the orthogonal projector onto $C$. Define the guaranteed code dimension by\n\n \\begin{equation}\n c(e,n)\n :=\\min_{\\substack{E\\subseteq\\operatorname{End}(H)\\\\\n \\dim E=e}}\n \\max\\left\\{\n \\dim C:\n \\begin{array}{l}\n C\\subseteq H\\text{ is a linear subspace, and}\\\\\n \\forall A,B\\in E\\ \\exists\\lambda(A,B)\\in\\mathbb{C}:\\\n P_C A^\\dagger B P_C=\\lambda(A,B)P_C\n \\end{array}\n \\right\\}.\n\\tag{2}\n\\end{equation} \nDetermine $c(e,n)$ in Eq. (2), exactly or with asymptotically matching bounds, and exhibit explicit tough error models whose largest correcting code is close to this value.",
      "url": "https://qiqc-op.com/problem/op_b1b41f737f9b4aeb/",
      "json": "https://qiqc-op.com/api/problems/op_b1b41f737f9b4aeb.json",
      "tex": "https://qiqc-op.com/problem/op_b1b41f737f9b4aeb/op_b1b41f737f9b4aeb.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "d78663d5aee0484987f372be5e421ca735b9301c5ff421763cb18cec2caf8463"
    },
    {
      "id": "op_b54ea1af90e24eaa",
      "ulid": "01M1HME7802V93CEFVEFG215MP",
      "aliases": [
        "op_b54ea1af90e24eaa",
        "01M1HME7802V93CEFVEFG215MP",
        "op-b54ea1af90e24eaa",
        "v2-semialgebraicity-of-closed-quantum-correlations",
        "open-problem-v2-problem-31"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Semialgebraicity of closed quantum correlations",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Bell nonlocality"
      ],
      "statement": "Is the closure of the finite-dimensional tensor-product correlation set semialgebraic in every fixed bipartite Bell scenario? Fix positive integers $n_A,n_B,m_A,m_B$. Inputs satisfy $1\\leq x\\leq n_A$ and $1\\leq y\\leq n_B$, while outputs satisfy $1\\leq a\\leq m_A$ and $1\\leq b\\leq m_B$. The set $\\mathcal C_q(n_A,n_B,m_A,m_B)$ consists of behaviors\n\n \\begin{equation}\n p(a,b\\mid x,y)\n =\\operatorname{Tr}\\!\\left[\n \\rho\\bigl(E_a^x\\otimes F_b^y\\bigr)\n \\right]\n\\tag{1}\n\\end{equation} \nobtained from arbitrary finite-dimensional local Hilbert spaces, a density operator $\\rho$, and local POVMs satisfying\n\n \\begin{equation}\n E_a^x\\succeq0,\\qquad F_b^y\\succeq0,\\qquad\n \\sum_{a=1}^{m_A}E_a^x=I,\\qquad\n \\sum_{b=1}^{m_B}F_b^y=I.\n\\tag{2}\n\\end{equation} \nEquation (2) fixes the measurement class in Eq. (1). Define the closed correlation set by\n\n \\begin{equation}\n \\mathcal C_{qa}(n_A,n_B,m_A,m_B)\n :=\\overline{\\mathcal C_q(n_A,n_B,m_A,m_B)}.\n\\tag{3}\n\\end{equation} \nFor every fixed tuple, is the set in Eq. (3) a finite Boolean combination of polynomial equalities and inequalities with real coefficients? The coefficients may be arbitrary real numbers, and the description need not be one conjunction of weak inequalities.",
      "url": "https://qiqc-op.com/problem/op_b54ea1af90e24eaa/",
      "json": "https://qiqc-op.com/api/problems/op_b54ea1af90e24eaa.json",
      "tex": "https://qiqc-op.com/problem/op_b54ea1af90e24eaa/op_b54ea1af90e24eaa.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "c87c816a95d3e81cec6bbd651c4711ca68dd775face0ae3594197a336c677806"
    },
    {
      "id": "op_b65cf15705065d81",
      "ulid": "01M1HME78030A51WENEAWKSA90",
      "aliases": [
        "op_b65cf15705065d81",
        "01M1HME78030A51WENEAWKSA90",
        "op-b65cf15705065d81",
        "v2-energy-constrained-quantum-capacity-of-a-thermal-attenuator",
        "open-problem-v2-problem-51"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "bosonic-channels"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Energy-constrained quantum capacity of a thermal attenuator",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Bosonic channels"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Bosonic channels"
      ],
      "statement": "For $0<\\eta<1$ and $0<\\nu<\\infty$, let the single-mode bosonic thermal attenuator be\n\n \\begin{equation}\n \\Phi_{\\eta,\\nu}(\\rho_A)\n :=\\operatorname{Tr}_{E'}\\!\\left[\n U_\\eta(\\rho_A\\otimes\\tau_{\\nu,E})U_\\eta^\\dagger\n \\right],\n \\qquad\n \\tau_{\\nu,E}:=\\sum_{k=0}^{\\infty}\n \\frac{\\nu^k}{(\\nu+1)^{k+1}}\n |k\\rangle_E\\!\\langle k|_E,\n\\tag{1}\n\\end{equation} \nwhere $U_\\eta:AE\\to BE'$ is a beam-splitter unitary and $\\tau_{\\nu,E}$ acts on the environment mode $E$. For an environment mode of angular frequency $\\omega_E$ at temperature $T$, $\\nu=(e^{\\hbar\\omega_E/(k_{\\rm B}T)}-1)^{-1}$. Thus $0<T<\\infty$ is equivalent to $0<\\nu<\\infty$ for fixed $\\omega_E>0$; $\\nu=0$ corresponds to $T=0$, while $\\nu\\to\\infty$ as $T\\to\\infty$. For a finite mean input photon number $0<N_{\\rm S}<\\infty$, define the energy-constrained unassisted quantum capacity by\n\n \\begin{equation}\n \\mathcal Q(\\Phi_{\\eta,\\nu},N_{\\rm S})\n :=\\lim_{n\\to\\infty}\\frac1n\n \\sup_{\\substack{\\rho_{A^n}:\\\\\n \\operatorname{Tr}[\\rho_{A^n}\\sum_{j=1}^n\\hat n_j]\n \\leq nN_{\\rm S}}}\n I_{\\rm c}(\\rho_{A^n},\\Phi_{\\eta,\\nu}^{\\otimes n}),\n\\tag{2}\n\\end{equation} \nwhere $\\hat n_j$ is the photon-number operator of the $j$th mode and \\(I_{\\rm c}(\\rho,\\mathcal N)\n:=S(\\mathcal N(\\rho))-S(\\mathcal N^{\\rm c}(\\rho))\\). Determine Eq. (2) for the thermal channel in Eq. (1).",
      "url": "https://qiqc-op.com/problem/op_b65cf15705065d81/",
      "json": "https://qiqc-op.com/api/problems/op_b65cf15705065d81.json",
      "tex": "https://qiqc-op.com/problem/op_b65cf15705065d81/op_b65cf15705065d81.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "eb4cdeb4a1d9c10d17300382c6c79bc1e744e4fb1c92f042e1099faee0a0f1d0"
    },
    {
      "id": "op_b93eb7197c926d9e",
      "ulid": "01M1HME780NTMHKFB95TSXKKFW",
      "aliases": [
        "op_b93eb7197c926d9e",
        "01M1HME780NTMHKFB95TSXKKFW",
        "op-b93eb7197c926d9e",
        "v2-distillable-entanglement-of-bell-diagonal-states",
        "open-problem-v2-problem-3"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-distillation",
          "bell-diagonal-states",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Distillable entanglement of Bell-diagonal states",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement distillation",
        "Bell-diagonal states",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement distillation",
        "Bell-diagonal states",
        "Local operations and classical communication"
      ],
      "statement": "What is the distillable entanglement $D(\\rho_{\\mathbf p})$ under local operations and classical communication (LOCC) for the Bell-diagonal state\n\n \\begin{equation}\n \\rho_{\\mathbf p}\n =p_I\\lvert\\Phi^+\\rangle\\!\\langle\\Phi^+\\rvert\n +p_X\\lvert\\Psi^+\\rangle\\!\\langle\\Psi^+\\rvert\n +p_Y\\lvert\\Psi^-\\rangle\\!\\langle\\Psi^-\\rvert\n +p_Z\\lvert\\Phi^-\\rangle\\!\\langle\\Phi^-\\rvert?\n\\tag{1}\n\\end{equation} \nWhat is this LOCC protocol?\n\nThe Bell states in Eq. (1) are defined by\n\n \\begin{equation}\n \\lvert\\Phi^\\pm\\rangle:=\\frac{\\lvert00\\rangle\\pm\\lvert11\\rangle}{\\sqrt2},\n \\qquad\n \\lvert\\Psi^\\pm\\rangle:=\\frac{\\lvert01\\rangle\\pm\\lvert10\\rangle}{\\sqrt2}.\n\\tag{2}\n\\end{equation} \nEquation (2) fixes the phase convention. Assume $\\sum_i p_i=1$ and $p_I\\ge1/2\\ge p_i>0$ for $i\\in\\{X,Y,Z\\}$.",
      "url": "https://qiqc-op.com/problem/op_b93eb7197c926d9e/",
      "json": "https://qiqc-op.com/api/problems/op_b93eb7197c926d9e.json",
      "tex": "https://qiqc-op.com/problem/op_b93eb7197c926d9e/op_b93eb7197c926d9e.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "ea595505f1fa555a63361688bdd61a94113c5529d00cf875ec8755cba8070e6d"
    },
    {
      "id": "op_b952ec2dd8246a10",
      "ulid": "01M1HME780SNG2DQVEGCDB0XSK",
      "aliases": [
        "op_b952ec2dd8246a10",
        "01M1HME780SNG2DQVEGCDB0XSK",
        "op-b952ec2dd8246a10",
        "v2-secret-key-from-every-entangled-state",
        "open-problem-v2-problem-20"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-cryptography",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "secret-key-distillation",
          "bound-entanglement",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Secret key from every entangled state",
      "status": "Unsolved",
      "fields": [
        "Quantum Cryptography",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Secret-key distillation",
        "Bound entanglement",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Cryptography",
        "Quantum Resource Theory",
        "Secret-key distillation",
        "Bound entanglement",
        "Local operations and classical communication"
      ],
      "statement": "Does every finite-dimensional entangled bipartite state have positive asymptotic distillable secret key? For a state $\\rho_{AB}$, with an adversary holding a purification, define its distillable-key rate by\n\n \\begin{equation}\n K_D(\\rho_{AB})\n :=\\sup\\left\\{R\\geq0:\n \\begin{array}{l}\n \\text{there are LOPC protocols $\\Lambda_n$ and private states\n $\\gamma^{(n)}_{K_n}$, with $K_n=2^{\\lfloor nR\\rfloor}$, such that}\\\\[-1mm]\n \\displaystyle\n \\lim_{n\\to\\infty}\n \\left\\|\\Lambda_n(\\rho_{AB}^{\\otimes n})\n -\\gamma^{(n)}_{K_n}\\right\\|_1=0\n \\end{array}\n \\right\\},\n\\tag{1}\n\\end{equation} \nwhere each target $\\gamma^{(n)}_{K_n}$ may have its own shield system and has key registers of dimension $K_n$. With the operational convention in Eq. (1), the question is whether the implication\n\n \\begin{equation}\n \\rho_{AB}\\ \\text{entangled}\n \\quad\\Longrightarrow\\quad\n K_D(\\rho_{AB})>0\n\\tag{2}\n\\end{equation} \nholds for every finite-dimensional $\\rho_{AB}$.",
      "url": "https://qiqc-op.com/problem/op_b952ec2dd8246a10/",
      "json": "https://qiqc-op.com/api/problems/op_b952ec2dd8246a10.json",
      "tex": "https://qiqc-op.com/problem/op_b952ec2dd8246a10/op_b952ec2dd8246a10.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "253f6a421a284a7d140d87d0a05db8512ef0fe6f9e533a15930da1fb44461b48"
    },
    {
      "id": "op_c9c62042b15fcb06",
      "ulid": "01M1HME780CSV212H08EXN5XFK",
      "aliases": [
        "op_c9c62042b15fcb06",
        "01M1HME780CSV212H08EXN5XFK",
        "op-c9c62042b15fcb06",
        "v2-maximum-number-of-mutually-unbiased-bases",
        "open-problem-v2-problem-44"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "mutually-unbiased-bases"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Maximum number of mutually unbiased bases",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Mutually unbiased bases"
      ],
      "tags": [
        "Quantum metrology",
        "Mutually unbiased bases"
      ],
      "statement": "For each integer $d\\geq2$, determine the maximum number $\\mu(d)$ of pairwise mutually unbiased orthonormal bases of $\\mathbb C^d$. Two orthonormal bases $\\mathcal B_r=\\{|e_i^{(r)}\\rangle\\}_{i=1}^{d}$ and $\\mathcal B_s=\\{|e_j^{(s)}\\rangle\\}_{j=1}^{d}$ are mutually unbiased when\n\n \\begin{equation}\n \\bigl|\\langle e_i^{(r)}|e_j^{(s)}\\rangle\\bigr|^2=\\frac1d\n \\qquad\\text{for every }i,j\\in\\{1,\\ldots,d\\}.\n\\tag{1}\n\\end{equation} \nThus the extremal quantity defined by Eq. (1) is\n\n \\begin{equation}\n \\mu(d):=\\max\\left\\{m:\\text{there exist $m$ orthonormal bases of\n $\\mathbb C^d$ that are pairwise mutually unbiased}\\right\\}.\n\\tag{2}\n\\end{equation} \nDetermine Eq. (2) in the non-prime-power regime, in particular for $d\\in\\{6,10,12,14,15\\}$.",
      "url": "https://qiqc-op.com/problem/op_c9c62042b15fcb06/",
      "json": "https://qiqc-op.com/api/problems/op_c9c62042b15fcb06.json",
      "tex": "https://qiqc-op.com/problem/op_c9c62042b15fcb06/op_c9c62042b15fcb06.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "d1fe5e267bff02485fb928a865954ddb31215d1300555b00f2e9689cd33a5fda"
    },
    {
      "id": "op_ccd560469b815618",
      "ulid": "01M1HME780VWYDSN7KF3J06003",
      "aliases": [
        "op_ccd560469b815618",
        "01M1HME780VWYDSN7KF3J06003",
        "op-ccd560469b815618",
        "v2-optimal-cglmp-measurements-for-a-maximally-entangled-state",
        "open-problem-v2-problem-24"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-metrology"
        ],
        "topicIds": [
          "bell-nonlocality"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Optimal CGLMP measurements for a maximally entangled state",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum metrology"
      ],
      "topics": [
        "Bell nonlocality"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum metrology",
        "Bell nonlocality"
      ],
      "statement": "For every $d\\geq3$, do the standard Fourier–phase measurements optimize the CGLMP violation of the maximally entangled state among all projective $d$-outcome measurements? Fix the state\n\n \\begin{equation}\n \\lvert\\Phi_d\\rangle\n =\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\\lvert j,j\\rangle.\n\\tag{1}\n\\end{equation} \nEquation (1) is held fixed; the shared state is not part of the optimization.\n\nFor outcomes in $\\mathbb Z_d$, let $[t]_d\\in\\{0,\\ldots,d-1\\}$ be the residue of $t$ and define the CGLMP functional by\n\n \\begin{equation}\n \\begin{aligned}\n B_d={}&\\mathbb E([A_0-B_0]_d)+\\mathbb E([B_0-A_1]_d)\\\\\n &+\\mathbb E([A_1-B_1]_d)+\\mathbb E([B_1-A_0-1]_d).\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nLocal behaviors satisfy $B_d\\geq d-1$ under the convention in Eq. (2). The candidate measurements have bases\n\n \\begin{equation}\n \\begin{aligned}\n \\lvert a;x\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(a+\\alpha_x)\\right)\\lvert j\\rangle,\\\\\n \\lvert b;y\\rangle\n &=\\frac{1}{\\sqrt d}\\sum_{j=0}^{d-1}\n \\exp\\!\\left(\\frac{2\\pi i}{d}j(-b+\\beta_y)\\right)\\lvert j\\rangle,\n \\end{aligned}\n\\tag{3}\n\\end{equation} \nwith $\\alpha_0=0$, $\\alpha_1=-1/2$, $\\beta_0=1/4$, and $\\beta_1=3/4$. The question is whether the bases in Eq. (3) minimize Eq. (2) on Eq. (1), up to symmetries of the state and functional.",
      "url": "https://qiqc-op.com/problem/op_ccd560469b815618/",
      "json": "https://qiqc-op.com/api/problems/op_ccd560469b815618.json",
      "tex": "https://qiqc-op.com/problem/op_ccd560469b815618/op_ccd560469b815618.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "5c31df2d745e64f6745cb23846584fe916f1e7d228dc6a01fe7b7f63b9bd6ddb"
    },
    {
      "id": "op_dcea1e5e3032b8c5",
      "ulid": "01M1HME780803VM2A45MA86N41",
      "aliases": [
        "op_dcea1e5e3032b8c5",
        "01M1HME780803VM2A45MA86N41",
        "op-dcea1e5e3032b8c5",
        "v2-quantum-violations-of-bipartite-bell-facets",
        "open-problem-v2-problem-23"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Quantum violations of bipartite Bell facets",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Bell nonlocality"
      ],
      "statement": "Must every nontrivial facet Bell inequality in a finite bipartite scenario have a quantum violation? Let $\\mathcal L$ be the local polytope, and let $\\mathcal Q$ consist of behaviors realizable as\n\n \\begin{equation}\n p(a,b\\mid x,y)\n =\\operatorname{Tr}\\!\\left[\\rho_{AB}\n \\bigl(M_x^a\\otimes N_y^b\\bigr)\\right],\n\\tag{1}\n\\end{equation} \nwhere $\\rho_{AB}$ is a finite-dimensional state and $\\{M_x^a\\}_a$ and $\\{N_y^b\\}_b$ are local POVMs. Equation (1) defines the quantum set used below.\n\nFor a linear functional $F(p)=\\sum_{a,b,x,y}c_{abxy}p(a,b\\mid x,y)$, suppose $F(p)\\leq\\beta_{\\mathrm L}$ supports a facet of $\\mathcal L$ and is not a positivity facet within the normalization and no-signalling affine hull. Is it necessarily true that\n\n \\begin{equation}\n \\sup_{p\\in\\mathcal Q}F(p)>\\beta_{\\mathrm L}?\n\\tag{2}\n\\end{equation} \nEquation (2) asks whether the bipartite local and quantum sets can share a nontrivial facet-supporting hyperplane.",
      "url": "https://qiqc-op.com/problem/op_dcea1e5e3032b8c5/",
      "json": "https://qiqc-op.com/api/problems/op_dcea1e5e3032b8c5.json",
      "tex": "https://qiqc-op.com/problem/op_dcea1e5e3032b8c5/op_dcea1e5e3032b8c5.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "3c7a0c16421ccaafbebf573fa13f4c09e3436b62227ec2b1368bf468839fec00"
    },
    {
      "id": "op_e2149f4ced34d1a8",
      "ulid": "01M1HME780JCWZMCJMARNSAXH3",
      "aliases": [
        "op_e2149f4ced34d1a8",
        "01M1HME780JCWZMCJMARNSAXH3",
        "op-e2149f4ced34d1a8",
        "v2-relative-entropy-of-entanglement-for-two-qubits",
        "open-problem-v2-problem-12"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "entanglement-measures",
          "quantum-separability",
          "quantum-relative-entropy"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Relative entropy of entanglement for two qubits",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Entanglement measures",
        "Quantum separability",
        "Quantum relative entropy"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Entanglement measures",
        "Quantum separability",
        "Quantum relative entropy"
      ],
      "statement": "Find a closed formula for the relative entropy of entanglement of every two-qubit density operator $\\rho$, including an explicit closest separable state. With $\\operatorname{Sep}(\\mathbb{C}^2:\\mathbb{C}^2)$ denoting the two-qubit separable states, the quantity is\n\n \\begin{equation}\n E_R(\\rho)\n :=\\min_{\\sigma\\in\\operatorname{Sep}(\\mathbb{C}^2:\\mathbb{C}^2)}\n D(\\rho\\Vert\\sigma),\n \\qquad\n D(\\rho\\Vert\\sigma)\n :=\\begin{cases}\n \\operatorname{Tr}\\!\\left[\\rho(\\log\\rho-\\log\\sigma)\\right],\n &\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma,\\\\\n +\\infty,&\\text{otherwise}.\n \\end{cases}\n\\tag{1}\n\\end{equation} \nIn the finite branch of Eq. (1), the trace is evaluated on $\\operatorname{supp}\\rho$, with $0\\log 0:=0$. The formula sought must determine at least one minimizing state $\\sigma_\\rho$ for every $\\rho$.",
      "url": "https://qiqc-op.com/problem/op_e2149f4ced34d1a8/",
      "json": "https://qiqc-op.com/api/problems/op_e2149f4ced34d1a8.json",
      "tex": "https://qiqc-op.com/problem/op_e2149f4ced34d1a8/op_e2149f4ced34d1a8.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "6849f44dc32ae3cf3dbe2a2ab33680b0ef3f828ec233dd0a9ece0ce10bb4d899"
    },
    {
      "id": "op_e490c9462b37a548",
      "ulid": "01M1HME780DDSDKPH6BERTWRWB",
      "aliases": [
        "op_e490c9462b37a548",
        "01M1HME780DDSDKPH6BERTWRWB",
        "op-e490c9462b37a548",
        "v2-two-way-quantum-capacity-amplitude-damping-channel",
        "open-problem-v2-problem-5"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Two-way quantum capacity: amplitude-damping channel",
      "status": "Unsolved",
      "fields": [
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Communication",
        "Quantum capacity",
        "Local operations and classical communication"
      ],
      "statement": "What is the two-way quantum capacity $\\mathcal{Q}_2(\\mathcal A_p)$ of the qubit amplitude-damping channel\n\n \\begin{equation}\n \\mathcal A_p(\\rho)=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\n \\qquad 0\\le p\\le1?\n\\tag{1}\n\\end{equation} \nThe operators in Eq. (1) are\n\n \\begin{equation}\n \\begin{aligned}\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert\n +\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}1&0\\\\0&\\sqrt{1-p}\\end{pmatrix},\\\\\n A_1&=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}0&\\sqrt p\\\\0&0\\end{pmatrix}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nEquation (2) uses $p$ as the decay probability of the excited state. Here $\\mathcal{Q}_2$ permits adaptive local operations and unlimited two-way classical communication between uses of the channel.",
      "url": "https://qiqc-op.com/problem/op_e490c9462b37a548/",
      "json": "https://qiqc-op.com/api/problems/op_e490c9462b37a548.json",
      "tex": "https://qiqc-op.com/problem/op_e490c9462b37a548/op_e490c9462b37a548.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "56ebb41bd861d884e7f41d5297efe24656ce0ae3d7db78c9df5b4cede8f2fd53"
    },
    {
      "id": "op_e4a8ae208470f288",
      "ulid": "01M1HME780CWYHZHQFF0KN89BM",
      "aliases": [
        "op_e4a8ae208470f288",
        "01M1HME780CWYHZHQFF0KN89BM",
        "op-e4a8ae208470f288",
        "v2-three-dimensional-self-correcting-quantum-memory",
        "open-problem-v2-problem-9"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-error-correction"
        ],
        "topicIds": [
          "self-correcting-quantum-memories",
          "quantum-thermodynamics",
          "quantum-coding-theory"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Three-dimensional self-correcting quantum memory",
      "status": "Unsolved",
      "fields": [
        "Quantum Error Correction"
      ],
      "topics": [
        "Self-correcting quantum memories",
        "Quantum thermodynamics",
        "Quantum coding theory"
      ],
      "tags": [
        "Quantum Error Correction",
        "Self-correcting quantum memories",
        "Quantum thermodynamics",
        "Quantum coding theory"
      ],
      "statement": "Does there exist a passive self-correcting quantum memory in three spatial dimensions? Consider finite-dimensional spins on a three-dimensional lattice $\\Lambda_L$ of linear size $L$ and a Hamiltonian $H_L=\\sum_{X\\subseteq\\Lambda_L}h_X$ whose interaction range, local strength $\\lVert h_X\\rVert$, and number of terms incident on each spin are bounded independently of $L$. Let its ground space $\\mathcal C_L$ encode a logical space $\\mathcal Q_L$ with $\\dim\\mathcal Q_L\\ge2$, and let $\\mathcal V_L:\\mathcal S(\\mathcal Q_L)\\to\\mathcal S(\\mathcal C_L)$ be the channel induced by an isometric encoding into $\\mathcal C_L$.\n\nDuring storage no active control or error correction is permitted. The encoded state evolves under a local thermal channel $\\mathcal E_{t,\\beta}^{(L)}$ at inverse temperature $\\beta$, such as a Davies semigroup, followed only at readout by a decoder $\\mathcal D_L$. For a fixed $0<\\varepsilon<1$, define the worst-case storage time by\n\n \\begin{equation}\n \\tau_L(\\beta,\\varepsilon)\n :=\\sup\\!\\left\\{t\\ge0:\n \\sup_{0\\le s\\le t}\\ \\sup_{\\rho\\in\\mathcal S(\\mathcal Q_L)}\n \\left\\|\n \\bigl(\\mathcal D_L\\circ\\mathcal E_{s,\\beta}^{(L)}\n \\circ\\mathcal V_L\\bigr)(\\rho)-\\rho\n \\right\\|_1\n \\le\\varepsilon\\right\\},\n\\tag{1}\n\\end{equation} \nwhere $\\mathcal S(\\mathcal Q_L)$ is the set of logical states. The problem is to construct such a Hamiltonian family with efficient decoders and a finite critical inverse temperature $\\beta_c$ for which\n\n \\begin{equation}\n \\beta>\\beta_c\n \\quad\\Longrightarrow\\quad\n \\lim_{L\\to\\infty}\\tau_L(\\beta,\\varepsilon)=\\infty.\n\\tag{2}\n\\end{equation} \nEquation (2) requires the lifetime in Eq. (1) to diverge at fixed nonzero temperature as $L\\to\\infty$; growth that terminates at a temperature-dependent system size is only partial self-correction.",
      "url": "https://qiqc-op.com/problem/op_e4a8ae208470f288/",
      "json": "https://qiqc-op.com/api/problems/op_e4a8ae208470f288.json",
      "tex": "https://qiqc-op.com/problem/op_e4a8ae208470f288/op_e4a8ae208470f288.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "5dfcd811b5c605bea6fc1213ab5f31046bffd8d2fe7a8ecbae3594f36eb423ad"
    },
    {
      "id": "op_eb5ca2d40deb7a38",
      "ulid": "01M1HME780637N8XEVFA4V7B4N",
      "aliases": [
        "op_eb5ca2d40deb7a38",
        "01M1HME780637N8XEVFA4V7B4N",
        "op-eb5ca2d40deb7a38",
        "v2-square-root-remainder-in-generalized-quantum-equipartition",
        "open-problem-v2-problem-48"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory",
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-hypothesis-testing",
          "quantum-relative-entropy",
          "one-shot-and-finite-blocklength-bounds"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Square-root remainder in generalized quantum equipartition",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory",
        "Quantum Communication"
      ],
      "topics": [
        "Quantum hypothesis testing",
        "Quantum relative entropy",
        "One-shot and finite-blocklength bounds"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Quantum Communication",
        "Quantum hypothesis testing",
        "Quantum relative entropy",
        "One-shot and finite-blocklength bounds"
      ],
      "statement": "Let $H$ be a $d$-dimensional Hilbert space. For each $n$, let $\\mathcal A_n\\subseteq\\mathcal D(H^{\\otimes n})$ and $\\mathcal B_n\\subseteq\\mathcal L(H^{\\otimes n})_+$ be nonempty compact, convex, permutation-invariant sets, each closed under tensor products. For a positive-operator set $\\mathcal C$, define its positive polar by\n\n \\begin{equation}\n \\mathcal C_+^\\circ\n :=\\{X\\geq0:\\operatorname{Tr}(XY)\\leq1\n \\text{ for every }Y\\in\\mathcal C\\}.\n\\tag{1}\n\\end{equation} \nAssume that both families of polars from Eq. (1) satisfy\n\n \\begin{equation}\n (\\mathcal A_m)_+^\\circ\\otimes(\\mathcal A_n)_+^\\circ\n \\subseteq(\\mathcal A_{m+n})_+^\\circ,\n \\qquad\n (\\mathcal B_m)_+^\\circ\\otimes(\\mathcal B_n)_+^\\circ\n \\subseteq(\\mathcal B_{m+n})_+^\\circ.\n\\tag{2}\n\\end{equation} \nIn addition to Eq. (2), assume that some constant $C<\\infty$ satisfies\n\n \\begin{equation}\n D_{\\max}(\\rho_n\\|\\sigma_n)\\leq Cn,\n \\qquad \\log_2\\operatorname{Tr}\\sigma_n\\leq Cn\n\\tag{3}\n\\end{equation} \nfor all $\\rho_n\\in\\mathcal A_n$ and $\\sigma_n\\in\\mathcal B_n$. The linear growth condition in Eq. (3) uses $D_{\\max}(\\rho\\|\\sigma):=\\inf\\{\\lambda:\\rho\\leq2^\\lambda\\sigma\\}$. For positive operators $\\rho$ and $\\sigma$, define\n\n \\begin{equation}\n \\begin{aligned}\n D(\\rho\\|\\sigma)\n &:=\\operatorname{Tr}[\\rho(\\log_2\\rho-\\log_2\\sigma)],\\\\\n D_H^\\varepsilon(\\rho\\|\\sigma)\n &:=-\\log_2\\inf_{\\substack{0\\leq Q\\leq I\\\\\n \\operatorname{Tr}(Q\\rho)\\geq1-\\varepsilon}}\n \\operatorname{Tr}(Q\\sigma),\n \\end{aligned}\n\\tag{4}\n\\end{equation} \nwhere $D(\\rho\\|\\sigma)=+\\infty$ unless $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$. Define the divergences between the two sets from Eq. (4) by\n\n \\begin{equation}\n \\begin{aligned}\n D(\\mathcal A_n\\|\\mathcal B_n)\n &:=\\inf_{\\rho_n\\in\\mathcal A_n,\\,\\sigma_n\\in\\mathcal B_n}\n D(\\rho_n\\|\\sigma_n),\\\\\n D_H^\\varepsilon(\\mathcal A_n\\|\\mathcal B_n)\n &:=\\inf_{\\rho_n\\in\\mathcal A_n,\\,\\sigma_n\\in\\mathcal B_n}\n D_H^\\varepsilon(\\rho_n\\|\\sigma_n),\n \\end{aligned}\n\\tag{5}\n\\end{equation} \nThe two quantities in Eq. (5) determine the first- and finite-blocklength orders of interest. Set\n\n \\begin{equation}\n D^\\infty(\\mathcal A\\|\\mathcal B)\n :=\\lim_{n\\to\\infty}\\frac1nD(\\mathcal A_n\\|\\mathcal B_n).\n\\tag{6}\n\\end{equation} \nDoes every fixed $\\varepsilon\\in(0,1)$ admit a constant $K_\\varepsilon<\\infty$ such that, for all sufficiently large $n$,\n\n \\begin{equation}\n \\left|D_H^\\varepsilon(\\mathcal A_n\\|\\mathcal B_n)\n -nD^\\infty(\\mathcal A\\|\\mathcal B)\\right|\n \\leq K_\\varepsilon\\sqrt n?\n\\tag{7}\n\\end{equation}",
      "url": "https://qiqc-op.com/problem/op_eb5ca2d40deb7a38/",
      "json": "https://qiqc-op.com/api/problems/op_eb5ca2d40deb7a38.json",
      "tex": "https://qiqc-op.com/problem/op_eb5ca2d40deb7a38/op_eb5ca2d40deb7a38.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "171da1fd1884e54c156b6b4cd9636d47bbe1eaf8bddaa87ccf3b961e7ac87627"
    },
    {
      "id": "op_ebee7d5c442d81a4",
      "ulid": "01M1HME780M4B4RRABEG3RDDCZ",
      "aliases": [
        "op_ebee7d5c442d81a4",
        "01M1HME780M4B4RRABEG3RDDCZ",
        "op-ebee7d5c442d81a4",
        "v2-complete-facet-descriptions-for-bell-polytopes",
        "open-problem-v2-problem-11"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "bell-nonlocality"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Complete facet descriptions for Bell polytopes",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Bell nonlocality"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Bell nonlocality"
      ],
      "statement": "Determine complete facet descriptions for local-behavior polytopes beyond the presently solved Bell scenarios, either for a specified unresolved finite scenario or for a nontrivial infinite family with additional structure. For positive integers $N$, $M$, and $K$, let $\\mathbf{x}\\in\\{1,\\ldots,M\\}^{N}$ denote the measurement settings and $\\mathbf{a}\\in\\{1,\\ldots,K\\}^{N}$ the outcomes. The relevant local polytope is\n\n \\begin{equation}\n \\mathcal{L}_{N,M,K}\n :=\\operatorname{conv}\\!\\left\\{\n p_{\\mathbf f}:p_{\\mathbf f}(\\mathbf a\\mid\\mathbf x)\n =\\prod_{i=1}^{N}\\mathbf{1}\\!\\left\\{a_i=f_i(x_i)\\right\\},\\quad\n f_i:\\{1,\\ldots,M\\}\\to\\{1,\\ldots,K\\}\n \\right\\}.\n\\tag{1}\n\\end{equation} \nHere $\\mathbf 1\\{\\cdot\\}$ is the indicator function. The task is to characterize all facet-defining inequalities of the polytope in Eq. (1) in a chosen unresolved regime, modulo permutations of parties, settings, and outcomes and modulo liftings obtained by adjoining redundant settings or outcomes. A solution must carry a proof of completeness, not merely generate a large collection of facets.",
      "url": "https://qiqc-op.com/problem/op_ebee7d5c442d81a4/",
      "json": "https://qiqc-op.com/api/problems/op_ebee7d5c442d81a4.json",
      "tex": "https://qiqc-op.com/problem/op_ebee7d5c442d81a4/op_ebee7d5c442d81a4.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "58ea4a157158c131a77d1c47aac98ed1934977369a4e5a1b6a56c35b30f3e3ba"
    },
    {
      "id": "op_ec184fc49232a9c0",
      "ulid": "01M1HME78078BW0JG3X252BVX5",
      "aliases": [
        "op_ec184fc49232a9c0",
        "01M1HME78078BW0JG3X252BVX5",
        "op-ec184fc49232a9c0",
        "v2-constant-trace-distance-separability-testing",
        "open-problem-v2-problem-36"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-algorithm",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "quantum-separability",
          "computational-complexity-and-computability"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Constant trace-distance separability testing",
      "status": "Unsolved",
      "fields": [
        "Quantum algorithm",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Quantum separability",
        "Computational complexity and computability"
      ],
      "tags": [
        "Quantum algorithm",
        "Quantum Resource Theory",
        "Quantum separability",
        "Computational complexity and computability"
      ],
      "statement": "What is the computational complexity of testing bipartite separability with a constant trace-distance promise gap? For local dimension $d$, define\n\n \\begin{equation}\n \\operatorname{Sep}(d,d)\n :=\\operatorname{conv}\\!\\left\\{\n \\alpha\\otimes\\beta:\n \\alpha,\\beta\\in\\mathcal D(\\mathbb C^d)\n \\right\\}.\n\\tag{1}\n\\end{equation} \nFix a constant $0<\\varepsilon_0<1$. Given a rational description of $\\rho\\in\\mathcal D(\\mathbb C^d\\otimes\\mathbb C^d)$, promised that exactly one of the following alternatives holds,\n\n \\begin{equation}\n \\begin{aligned}\n \\text{YES:}\\quad&\\rho\\in\\operatorname{Sep}(d,d),\\\\\n \\text{NO:}\\quad&\n \\inf_{\\sigma\\in\\operatorname{Sep}(d,d)}\n \\frac12\\lVert\\rho-\\sigma\\rVert_1\\geq\\varepsilon_0,\n \\end{aligned}\n\\tag{2}\n\\end{equation} \ndecide which case in Eq. (2) holds. Is there an algorithm polynomial or quasipolynomial in $d$? More generally, when the gap $\\varepsilon$ is part of the input, determine the optimal dependence of the complexity on $d$ and $\\varepsilon$.",
      "url": "https://qiqc-op.com/problem/op_ec184fc49232a9c0/",
      "json": "https://qiqc-op.com/api/problems/op_ec184fc49232a9c0.json",
      "tex": "https://qiqc-op.com/problem/op_ec184fc49232a9c0/op_ec184fc49232a9c0.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "9cc81d7b56927be0be21604d037a60f1748ef71edd9ebe5a2c82dc23c3583e5d"
    },
    {
      "id": "op_f1e5a1cad168b38b",
      "ulid": "01M1HME780G89RD9W1SZ24KSRP",
      "aliases": [
        "op_f1e5a1cad168b38b",
        "01M1HME780G89RD9W1SZ24KSRP",
        "op-f1e5a1cad168b38b",
        "v2-honest-party-lockability-of-distillable-key",
        "open-problem-v2-problem-22"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-cryptography",
          "quantum-resource-theory"
        ],
        "topicIds": [
          "secret-key-distillation",
          "local-operations-and-classical-communication"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Honest-party lockability of distillable key",
      "status": "Unsolved",
      "fields": [
        "Quantum Cryptography",
        "Quantum Resource Theory"
      ],
      "topics": [
        "Secret-key distillation",
        "Local operations and classical communication"
      ],
      "tags": [
        "Quantum Cryptography",
        "Quantum Resource Theory",
        "Secret-key distillation",
        "Local operations and classical communication"
      ],
      "statement": "Can loss of one qubit held by an honest party reduce the two-way distillable secret key by an arbitrarily large amount? Let $K_D(A:B)_\\rho$ denote the asymptotic secret-key rate obtainable from $\\rho_{AB}$ by local operations and public two-way classical communication, with an adversary holding a purification. The question is whether there are finite-dimensional states $\\rho^{(r)}_{A_ra:B_r}$, with $\\dim a=2$, such that\n\n \\begin{equation}\n K_D(A_ra:B_r)_{\\rho^{(r)}}\n -K_D(A_r:B_r)_{\\operatorname{Tr}_a\\rho^{(r)}}\n \\xrightarrow[r\\to\\infty]{}\\infty.\n\\tag{1}\n\\end{equation} \nEquation (1) is the $AB$-locking question: the discarded qubit belongs to Alice rather than being transferred to the adversary.",
      "url": "https://qiqc-op.com/problem/op_f1e5a1cad168b38b/",
      "json": "https://qiqc-op.com/api/problems/op_f1e5a1cad168b38b.json",
      "tex": "https://qiqc-op.com/problem/op_f1e5a1cad168b38b/op_f1e5a1cad168b38b.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "38ae8492207dc7e9ecacf77d76ac43b27992399446fb6ad59228d5332ea07194"
    },
    {
      "id": "op_f60b9a99d7945e3b",
      "ulid": "01M1HME780ZVGZ03D56MC08Z5J",
      "aliases": [
        "op_f60b9a99d7945e3b",
        "01M1HME780ZVGZ03D56MC08Z5J",
        "op-f60b9a99d7945e3b",
        "v2-real-four-quhex-absolutely-maximally-entangled-state",
        "open-problem-v2-problem-41"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-resource-theory"
        ],
        "topicIds": [
          "absolutely-maximally-entangled-states"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Real four-quhex absolutely maximally entangled state",
      "status": "Unsolved",
      "fields": [
        "Quantum Resource Theory"
      ],
      "topics": [
        "Absolutely maximally entangled states"
      ],
      "tags": [
        "Quantum Resource Theory",
        "Absolutely maximally entangled states"
      ],
      "statement": "Does there exist an $\\operatorname{AME}(4,6)$ state whose coefficients are real in a product basis? Fix $[6]=\\{0,\\ldots,5\\}$ and ask for a normalized state\n\n \\begin{equation}\n |\\psi\\rangle\n =\\sum_{i,j,k,l\\in[6]}T_{ijkl}|i\\rangle_A|j\\rangle_B\n |k\\rangle_C|l\\rangle_D,\n \\qquad\n T_{ijkl}\\in\\mathbb{R},\n \\qquad\n \\sum_{i,j,k,l\\in[6]}T_{ijkl}^{2}=1 .\n\\tag{1}\n\\end{equation} \nThe state in (1) must have maximally mixed reductions on every pair of parties: for each $S\\subseteq\\{A,B,C,D\\}$ with $|S|=2$,\n\n \\begin{equation}\n \\rho_S:=\\operatorname{Tr}_{S^{c}}|\\psi\\rangle\\!\\langle\\psi|\n =\\frac{I_{36}}{36} .\n\\tag{2}\n\\end{equation} \nEquivalently, define the $36\\times36$ flattening $U$ and its reshuffling $U^R$ and second-factor partial transpose $U^\\Gamma$ by\n\n \\begin{equation}\n U_{(i,j),(k,l)}:=6T_{ijkl},\n \\qquad\n (U^R)_{(i,k),(j,l)}:=U_{(i,j),(k,l)},\n \\qquad\n (U^\\Gamma)_{(i,j),(k,l)}:=U_{(i,l),(k,j)} .\n\\tag{3}\n\\end{equation} \nHere the three matrices in (3) are the coefficient flattenings for the bipartitions $AB|CD$, $AC|BD$, and $AD|CB$, up to the displayed ordering of tensor factors. Consequently, (2) is equivalent to the orthogonal $2$-unitarity conditions\n\n \\begin{equation}\n U\\in O(36),\n \\qquad\n U^R\\in O(36),\n \\qquad\n U^\\Gamma\\in O(36) .\n\\tag{4}\n\\end{equation} \nThus the problem is also to construct an orthogonal $2$-unitary matrix of order $36$, or prove that none exists.",
      "url": "https://qiqc-op.com/problem/op_f60b9a99d7945e3b/",
      "json": "https://qiqc-op.com/api/problems/op_f60b9a99d7945e3b.json",
      "tex": "https://qiqc-op.com/problem/op_f60b9a99d7945e3b/op_f60b9a99d7945e3b.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "3f95665e401452db524da8541117dfae70fa0609c65a9528a9edae810528849b"
    },
    {
      "id": "op_fcd21a1a5021e464",
      "ulid": "01M1HME780H9TAVH85TF8KJDS5",
      "aliases": [
        "op_fcd21a1a5021e464",
        "01M1HME780H9TAVH85TF8KJDS5",
        "op-fcd21a1a5021e464",
        "v2-capacity-achieving-codes-for-amplitude-damping",
        "open-problem-v2-problem-2"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-error-correction",
          "quantum-communication"
        ],
        "topicIds": [
          "quantum-capacity",
          "quantum-coding-theory",
          "decoding-algorithms"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Capacity-achieving codes for amplitude damping",
      "status": "Solved",
      "fields": [
        "Quantum Error Correction",
        "Quantum Communication"
      ],
      "topics": [
        "Quantum capacity",
        "Quantum coding theory",
        "Decoding algorithms"
      ],
      "tags": [
        "Quantum Error Correction",
        "Quantum Communication",
        "Quantum capacity",
        "Quantum coding theory",
        "Decoding algorithms"
      ],
      "statement": "What is the constructive quantum code for achieving the quantum capacity $\\mathcal{Q}(\\mathcal A_p)$ of the qubit amplitude-damping channel\n\n \\begin{equation}\n \\mathcal A_p(\\rho)=A_0\\rho A_0^\\dagger+A_1\\rho A_1^\\dagger,\n \\qquad 0\\le p\\le1?\n\\tag{1}\n\\end{equation} \nThe operators appearing in Eq. (1) are\n\n \\begin{equation}\n \\begin{aligned}\n A_0&=\\lvert0\\rangle\\!\\langle0\\rvert\n +\\sqrt{1-p}\\,\\lvert1\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}1&0\\\\0&\\sqrt{1-p}\\end{pmatrix},\\\\\n A_1&=\\sqrt p\\,\\lvert0\\rangle\\!\\langle1\\rvert\n =\\begin{pmatrix}0&\\sqrt p\\\\0&0\\end{pmatrix}.\n \\end{aligned}\n\\tag{2}\n\\end{equation} \nEquation (2) uses $p$ as the decay probability of the excited state.",
      "url": "https://qiqc-op.com/problem/op_fcd21a1a5021e464/",
      "json": "https://qiqc-op.com/api/problems/op_fcd21a1a5021e464.json",
      "tex": "https://qiqc-op.com/problem/op_fcd21a1a5021e464/op_fcd21a1a5021e464.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "24c66f232043e284fa8f1e7b3290786307002d8494171145d7652fae884b784a"
    },
    {
      "id": "op_ff2e80b425aebb86",
      "ulid": "01M1HME78033RK9X53VANQQXBM",
      "aliases": [
        "op_ff2e80b425aebb86",
        "01M1HME78033RK9X53VANQQXBM",
        "op-ff2e80b425aebb86",
        "v2-zauner-symmetric-weyl-heisenberg-sic-fiducials",
        "open-problem-v2-problem-19"
      ],
      "metadata": {
        "type": "Problem",
        "schemaVersion": "1.0",
        "revision": 2,
        "createdBy": "01M1Q787QRVXGPCXG6KEQTF7N1",
        "createdAt": "2026-09-04T22:04:59Z",
        "role": "primary",
        "parentProblemId": null,
        "parentClauseId": null,
        "origin": "source-stated",
        "posed": null,
        "areaIds": [
          "quantum-metrology"
        ],
        "topicIds": [
          "symmetric-informationally-complete-measurements"
        ],
        "keywords": [],
        "difficulty": "unrated",
        "verificationCost": "unrated",
        "relatedProblemIds": []
      },
      "title": "Zauner-symmetric Weyl–Heisenberg SIC fiducials",
      "status": "Unsolved",
      "fields": [
        "Quantum metrology"
      ],
      "topics": [
        "Symmetric informationally complete measurements"
      ],
      "tags": [
        "Quantum metrology",
        "Symmetric informationally complete measurements"
      ],
      "statement": "Does every finite dimension admit a Weyl–Heisenberg SIC fiducial that is an eigenvector of a Zauner Clifford unitary? For $d\\geq2$, define the displacement operators $D_{\\mathbf p}:=D_{p,q}:=X_d^pZ_d^q$ for $\\mathbf p=(p,q)^{\\mathsf T}\\in\\mathbb Z_d^2$, using\n\n \\begin{equation}\n X_d\\lvert j\\rangle=\\lvert j+1\\!\\!\\pmod d\\rangle,\n \\qquad\n Z_d\\lvert j\\rangle=e^{2\\pi i j/d}\\lvert j\\rangle,\n \\qquad \\mathbf p=(p,q)^{\\mathsf T}\\in\\mathbb Z_d^2.\n\\tag{1}\n\\end{equation} \nEquation (1) fixes the Weyl–Heisenberg orbit up to irrelevant phases. Let $U_Z$ be a Clifford unitary whose action on displacement operators is\n\n \\begin{equation}\n U_ZD_{\\mathbf p}U_Z^\\dagger\\doteq D_{F_Z\\mathbf p},\n \\qquad\n F_Z=\n \\begin{pmatrix}\n 0&-1\\\\\n 1&-1\n \\end{pmatrix},\n\\tag{2}\n\\end{equation} \nwhere indices are reduced modulo $d$ and $\\doteq$ denotes equality up to phase. Equation (2) fixes the distinguished order-three Clifford symmetry. The question is whether, for every $d\\geq2$, there are a unit vector $\\lvert\\phi\\rangle$ and a phase $e^{i\\theta}$ such that\n\n \\begin{equation}\n U_Z\\lvert\\phi\\rangle=e^{i\\theta}\\lvert\\phi\\rangle,\n \\qquad\n \\bigl|\\langle\\phi\\rvert D_{\\mathbf p}\\lvert\\phi\\rangle\\bigr|^2\n =\\frac{1}{d+1}\n \\quad\\text{for every }\\mathbf p\\in\\mathbb Z_d^2\\setminus\\{\\mathbf0\\}.\n\\tag{3}\n\\end{equation} \nEquation (3) simultaneously imposes Zauner symmetry and the SIC overlap equations.",
      "url": "https://qiqc-op.com/problem/op_ff2e80b425aebb86/",
      "json": "https://qiqc-op.com/api/problems/op_ff2e80b425aebb86.json",
      "tex": "https://qiqc-op.com/problem/op_ff2e80b425aebb86/op_ff2e80b425aebb86.tex",
      "created": "2026-09-01",
      "updated": "2026-09-04",
      "createdAt": "2026-09-01T05:13:32.000Z",
      "updatedAt": "2026-09-04T23:38:14.000Z",
      "sha256": "35c1f839b846ae10dee182ba8ca4627efb2228c36928bf70216764c7e55d1e72"
    }
  ]
}