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110 records covering 110 distinct questions. Filter counts refer to records.
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Optimal precision dependence of low-energy Hamiltonian simulation
What is the tight precision dependence of worst-case query complexity for low-energy simulation in the regime (2)?
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Simultaneously optimal queries to both quantum linear-system oracles
Can one quantum linear-system algorithm attain both query bounds in (2) simultaneously?
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Quantum query complexity of Welded Tree path-finding
Does finding an explicit ENTRANCE-to-EXIT path in the standard Welded Tree oracle problem require exponentially many quantum queries?
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Two-way secret-key capacity of the qubit amplitude-damping channel
What is the exact secret-key capacity \(K(\mathcal A_p)\) of the qubit amplitude-damping channel for every damping probability \(p\in[0,1]\)?
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Qubit code distance bound
Does there exist a sequence of qubit quantum codes \(Q_j\subseteq(\mathbb C^2)^{\otimes n_j}\), with \(n_j\to\infty\), \(\dim Q_j\geq2\), and minimum distances \(d_j\), satisfying
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Polynomial-time quantum algorithm for approximate Shortest Vector Problem
Does the polynomial-factor approximate Shortest Vector Problem admit a polynomial-time quantum algorithm?
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Polynomial-time quantum algorithm for the Dihedral Hidden Subgroup Problem
Does the Dihedral Hidden Subgroup Problem admit a quantum algorithm whose running time is polynomial in the input length?
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Squashed entanglement of the qubit depolarizing channel
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
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Private capacity of the qubit depolarizing channel
What is the private classical capacity \(P(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by
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Additivity of the relative entropy of entanglement
Does the relative entropy of entanglement of every bipartite state equal its regularization, or is regularization genuinely necessary?
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QMA(2) versus QMA
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?
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Asymptotic growth of the stabilizer rank of T-state tensor powers
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?
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Additivity of the entanglement of purification
Is the entanglement of purification additive on tensor products?
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Smallest output dimension violating minimum output entropy additivity
What is the smallest output dimension in which the minimum output von Neumann entropy of quantum channels fails to be additive?
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Closed-form nonadditivity of the Holevo capacity and entanglement of formation
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.
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Computability of ordinary quantum capacity
Is the ordinary unassisted quantum capacity a computable function of a finite description of a finite-dimensional quantum channel?
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Ordinary-Petz recovery bound for conditional mutual information
Does the ordinary, unrotated Petz map universally recover a tripartite state with fidelity controlled by its conditional mutual information?
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Parallel versus nested-adaptive superchannel discrimination
Can nested-adaptive strategies improve the Stein exponent for discriminating two finite-dimensional quantum superchannels?
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Ordinary-Petz fidelity remainder for relative-entropy data processing
Does the ordinary Petz recovery map give a universal fidelity remainder for monotonicity of quantum relative entropy?
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All-code exponential strong converse for degradable channels
Does every finite-dimensional degradable quantum channel satisfy an all-code exponential strong converse for quantum communication at its quantum capacity?
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Diamond-smoothed max-relative-entropy AEP for quantum channels
Does the max-relative entropy of finite-dimensional quantum channels satisfy an asymptotic equipartition property under uniform diamond-norm smoothing?
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Entanglement of formation of generalized Bell-diagonal states
For every local dimension \(d\geq3\), determine the entanglement of formation of an arbitrary Weyl–Bell-diagonal state.
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Exponential strong converse for transpose-degradable channels
Does every finite-dimensional transpose-degradable channel satisfy an exponential strong converse for quantum communication at its single-letter quantum capacity?
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Minimal dimensions for strict transpose degradability
What are the componentwise-minimal dimension triples \((d_A,d_B,d_E)\) that admit a transpose-degradable but nondegradable channel?
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Positivity threshold for thermal-attenuator quantum capacity
For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be
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Unclassified existence parameters for homogeneous AME states
For which of the parameter pairs specified below does an absolutely maximally entangled state exist?
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Weyl–Heisenberg-covariant SICs in every dimension
Does every finite dimension admit a symmetric informationally complete measurement that is a single Weyl–Heisenberg orbit?
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Entanglement cost of an amplitude-damping-channel Choi state
What is the entanglement cost of the Choi state of the qubit amplitude-damping channel
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Statistical strength of CGLMP measurements
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?
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Entanglement cost of a qubit Bell-diagonal state
What is the entanglement cost of a qubit Bell-diagonal state for an arbitrary probability vector \(\mathbf p=(p_I,p_X,p_Y,p_Z)\)?
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Energy-constrained quantum capacity of a thermal attenuator
For \(0<\eta<1\) and \(0<\nu<\infty\), let the single-mode bosonic thermal attenuator be
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Two-way quantum capacity: amplitude-damping channel
What is the two-way quantum capacity \(\mathcal{Q}_2(\mathcal A_p)\) of the qubit amplitude-damping channel
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Zauner-symmetric Weyl–Heisenberg SIC fiducials
Does every finite dimension admit a Weyl–Heisenberg SIC fiducial that is an eigenvector of a Zauner Clifford unitary?
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Aaronson–Ambainis conjecture
Must every bounded low-degree real polynomial on the Boolean cube with non-negligible variance have a variable of inverse-polynomial influence?
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Strict inclusion of degradable channels in the less-noisy class
Do there exist finite-dimensional quantum channels that are less noisy in Watanabe’s regularized sense but are not degradable?
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Polynomial-time quantum algorithm for Learning With Errors
Does the Learning With Errors problem in its standard worst-case-hard parameter regime admit a polynomial-time quantum algorithm?
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Parity is not in QAC0
Can polynomial-size constant-depth \(\mathsf{QAC}^0\) circuits compute the parity function, or equivalently, is \(\mathrm{PARITY}\notin\mathsf{QAC}^0\)?
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The PPT-squared conjecture
Must the composition of any two compatible PPT completely positive maps be entanglement breaking?
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Is bipartite Quantum Max-Cut in BPP?
Is the following bipartite Quantum Max-Cut promise problem in \(\mathrm{BPP}\)?
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Average-case approximation hardness of random Ising partition functions
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?
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Average-case approximation hardness of squared normalized gaps of random cubic polynomials
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 \(\mathbb{F}_2\)?
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Average-case approximation hardness of random-circuit output probabilities
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?
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Permanent-of-Gaussians Conjecture
Is the following estimation task \(\#\mathrm{P}\)-hard under randomized polynomial-time Turing reductions?
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Quantum query complexity of Triangle Finding
What is the bounded-error quantum query complexity of finding a triangle in an \(n\)-vertex graph given oracle access to its adjacency matrix?
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Polynomial-time quantum algorithm for Graph Isomorphism
Does the Graph Isomorphism problem admit a polynomial-time quantum algorithm?
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Multi-slot overhead of virtual channel conjugation
What is the optimal quasiprobability overhead of implementing the complex conjugate of an unknown quantum channel from \(n\) queries?
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Bidirectional classical-communication cost of bipartite channel simulation
What is the asymptotic classical-communication cost of simulating a bipartite quantum channel with bidirectional classical communication and non-signalling assistance?
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Closed-form exact PPT distillable entanglement
What computable expression, if any, equals the regularized exact PPT distillable entanglement of a bipartite state?
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Universal purification with classically simulable operations
Can classically simulable operations purify an unknown depolarized pure state from any number of copies?
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Advantage of fully general superchannel-discrimination strategies
Can a fully general adaptive strategy attain a larger Stein exponent than every nested-adaptive strategy for discriminating two quantum superchannels?
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Amortization collapse for superchannel divergences
Does amortization collapse for the max-relative entropy and geometric Rényi divergence of arbitrary finite-dimensional quantum superchannels?
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Uniformly efficient HSW pretty-good decoding
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…
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Multimode constrained output entropy of a pure-loss channel
Is Guha’s multimode Strong Conjecture 2 true for every correlated input state?
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Universal finite truncation of quantum and private capacities
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?
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Candidate pure-loss second-order converse
Does the pure-loss bosonic channel admit the following candidate second-order classical converse under a maximum-photon-number occupation constraint?
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Exactly solvable nondegradable quantum channels
Does there exist a finite-dimensional quantum channel that is neither degradable nor antidegradable and whose unassisted quantum capacity is known exactly?
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Structural criterion for classical–entanglement trade-off advantage
Characterize the finite-dimensional quantum channels for which joint classical–entanglement coding strictly outperforms time sharing between unassisted and unlimited-entanglement classical communication.
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Strong converse for general mixed-state quantum compression
Does general finite-dimensional i.i.d.
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Quantum capacity of the Gaussian random-displacement channel
What is the unconstrained, unassisted quantum capacity of the single-mode Gaussian random-displacement channel?
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Convergence of the JRF iteration for mixed-state discrimination
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?
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Minimum LU–LC counterexample for graph states
What is the least number of qubits for which two graph states can be locally unitary equivalent without being locally Clifford equivalent?
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Asymptotic metrology with quantum-controlled causal order
Does quantum control of causal order yield a persistent asymptotic metrological advantage over parallel access for some smooth finite-dimensional channel family?
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Equal-weight low-Choi-rank decompositions of quantum channels
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?
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Nontrivial mutually degradable channel pairs
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?
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Umegaki relative entropy of local recovery
Does the conditional mutual information of every finite-dimensional tripartite state dominate its Umegaki relative entropy of local recovery?
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Multiplicativity for polarized Werner–Holevo channels
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?
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The quantum PCP conjecture
Is the constant-relative-gap local Hamiltonian problem QMA-hard?
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Delayed-onset additivity violation for minimum output Rényi entropy
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?
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Transpose degradability beyond degradability
Does there exist a finite-dimensional transpose-degradable quantum channel that is not degradable?
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Fixed-error parallel Stein lemma for quantum channels
Let \(\mathcal N,\mathcal M:\mathcal L(A)\to\mathcal L(B)\) be quantum channels on finite-dimensional systems.
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Finite nontrivial LU moduli of AME states
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?
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Generalized Stein lemma for fully quantum channel resources
Let \(\mathcal N:\mathcal L(A)\to\mathcal L(B)\) be a finite-dimensional quantum channel.
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POVM steering threshold of higher-dimensional Werner states
What is the exact steering threshold for arbitrary POVMs on a higher-dimensional Werner state?
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Secret key from every Bell-nonlocal behavior
Does every finite-alphabet Bell-nonlocal behavior have a strictly positive asymptotic secret-key rate against arbitrary individual nonsignalling attacks?
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SIC-POVM existence in every dimension
Does a symmetric informationally complete positive-operator-valued measure exist in every finite dimension \(d\ge2\)?
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Unconditional classical verification with one quantum prover
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?
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Finite-alphabet nonsignalling simulation of entangled qubits
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?
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Quantum capacity of a qubit Pauli channel
What is the quantum capacity \(\mathcal{Q}(\Lambda_{\mathbf p})\) of the qubit Pauli channel defined by
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Universal simulation with two PR boxes
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?
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Quantum LDPC codes at the Pauli hashing bound
For a probability vector \(\mathbf p=(p_I,p_X,p_Y,p_Z)\), the qubit Pauli channel is
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Gaussian entanglement of formation beyond bisymmetry
Does Gaussian entanglement of formation equal unrestricted entanglement of formation for every non-bisymmetric multimode Gaussian state?
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Achievability of the Rains bound under PPT-preserving channels
Consider distilling the bipartite Bell-diagonal state
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Resources for implementing Gibbs-preserving channels
What are the exact one-shot coherence and work costs of implementing an arbitrary Gibbs-preserving channel by thermal operations?
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Collective cost of tensor-power state preparation
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.
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One-bit simulation of partially entangled qubits
Can shared randomness and one classical bit exactly simulate every pair of local projective measurements on every pure entangled two-qubit state?
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Long-range vacuum CHSH violation
Does the vacuum of a massive free scalar Bose field violate the CHSH inequality between bounded localization regions at arbitrarily large spacelike separation?
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Trace-exponential lower bound for matrix-word averages
Is the normalized trace average of all words in two positive-definite matrices always bounded below by the corresponding trace exponential?
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NPT bound entanglement and the rank-five frontier
Does there exist a finite-dimensional bipartite state with non-positive partial transpose (NPT) that is undistillable by local operations and classical communication (LOCC)?
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Minimal-support frontier for absolutely maximally entangled states
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.
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Lockability of two-way distillable entanglement
Can discarding one local qubit reduce two-way distillable entanglement by an arbitrarily large amount?
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Quantum capacity of a bosonic thermal attenuator
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
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Polynomial shared-resource lower bounds for routing
Does an explicit total Boolean family require polynomial shared-state cost for bounded-error one-round \(f\)-routing?
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Extensible causality and process purification
Is every finite-dimensional extensibly causal process matrix purifiable?
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Absolute separability from spectra
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.
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Existence of an eight-ququart perfect tensor
Does an absolutely maximally entangled state of eight ququarts exist?
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Tough error models
Determine \(c(e,n)\) and construct tough error models in the sense of [KW05].
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Semialgebraicity of closed quantum correlations
Is the closure of the finite-dimensional tensor-product correlation set semialgebraic in every fixed bipartite Bell scenario?
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Distillable entanglement of Bell-diagonal states
What is the distillable entanglement \(D(\rho_{\mathbf p})\) under local operations and classical communication (LOCC) for the Bell-diagonal state
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Secret key from every entangled state
Does every finite-dimensional entangled bipartite state have positive asymptotic distillable secret key?
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Maximum number of mutually unbiased bases
For each integer \(d\geq2\), determine the maximum number \(\mu(d)\) of pairwise mutually unbiased orthonormal bases of \(\mathbb C^d\).
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Optimal CGLMP measurements for a maximally entangled state
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?
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Quantum violations of bipartite Bell facets
Must every nontrivial facet Bell inequality in a finite bipartite scenario have a quantum violation?
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Relative entropy of entanglement for two qubits
Find a closed formula for the relative entropy of entanglement of every two-qubit density operator \(\rho\), including an explicit closest separable state.
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Three-dimensional self-correcting quantum memory
Does there exist a passive self-correcting quantum memory in three spatial dimensions?
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Square-root remainder in generalized quantum equipartition
Let \(H\) be a \(d\)-dimensional Hilbert space.
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Complete facet descriptions for Bell polytopes
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.
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Constant trace-distance separability testing
What is the computational complexity of testing bipartite separability with a constant trace-distance promise gap?
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Honest-party lockability of distillable key
Can loss of one qubit held by an honest party reduce the two-way distillable secret key by an arbitrarily large amount?
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Real four-quhex absolutely maximally entangled state
Does there exist an \(\operatorname{AME}(4,6)\) state whose coefficients are real in a product basis?
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Capacity-achieving codes for amplitude damping
What is the constructive quantum code for achieving the quantum capacity \(\mathcal{Q}(\mathcal A_p)\) of the qubit amplitude-damping channel
Solved
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