Universal purification with classically simulable operations

Problem

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

\begin{equation} \mathcal D_\delta(\psi):=(1-\delta)\psi+\delta\,\frac{\mathbb 1_d}{d}, \qquad \operatorname{Tr}\bigl[\psi\,\mathcal D_\delta(\psi)\bigr] =1-\frac{d-1}{d}\,\delta. \tag{1} \end{equation}

Let \(\mathcal A_2\) 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

\begin{equation} F^{\mathcal A_d}_{\delta}(n,s) :=\sup\Bigl\{\frac1s\int d\psi\, \operatorname{Tr}\bigl[\psi\, \mathcal E\bigl(\mathcal D_\delta(\psi)^{\otimes n}\bigr)\bigr] :\ \mathcal E\in\mathcal A_d,\ \int d\psi\,\operatorname{Tr}\, \mathcal E\bigl(\mathcal D_\delta(\psi)^{\otimes n}\bigr)=s \Bigr\}, \tag{2} \end{equation}

where \(\mathcal E\) maps the \(n\) copies to one \(d\)-dimensional system and \(d\psi\) is the Haar measure on pure states. Is

\begin{equation} F^{\mathcal A_d}_{\delta}(n,s)=1-\frac{d-1}{d}\,\delta \tag{3} \end{equation}

for 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)?

Source

He, Zhu, Yao, Liu, Li, and Wang prove Eq. (3) for two copies and explicitly conjecture that universal purification is impossible without non-stabilizer resources for every dimension and every number of copies [HZY+26].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

Yuxuan Zhang gives a counterexample to Eq. (3) for qubits using five copies at \(\delta=2/3\). Choose one of the six signed Pauli axes uniformly and measure four inputs in that basis. If all four outcomes match the chosen sign, prepare that Pauli eigenstate; otherwise retain the untouched fifth input. Pauli measurements, stabilizer preparation, retention, discarding, and classical randomization make this a trace-preserving, completely stabilizer-preserving map, including in the presence of reference systems. Its fidelity is at least \(11693/17496=2/3+29/17496\) for every pure target. Its Haar fidelity gives

\begin{equation} F^{\mathcal A_2}_{2/3}(5,1)\geq\frac{325}{486} =\frac23+\frac1{486}>\frac23. \tag{4} \end{equation}

Thus Eq. (4) refutes the archived universal equality with success probability one. Discarding extra inputs and independently accepting the output extend the same conditional fidelity bound to every \(n\geq5\) and \(0<s\leq1\) at this noise strength [Zha26], pp. 2–3.
For two copies the no-go statement is a theorem: for qubits under completely stabilizer-preserving maps and for every odd \(d\) under completely positive-Wigner-preserving maps, \(F^{\mathcal A_d}_{\delta}(2,s)=1-\frac{d-1}{d}\delta\) for every \(0<s\leq1\), so neither deterministic nor postselected classically simulable processing improves the Haar-averaged fidelity [HZY+26].
Zhang also gives the four-copy postselection map with Kraus operator \(K=|0\rangle\langle0000|\). At \(\delta=1/2\), its Haar-averaged success probability is \(s=121/1280\) and its conditional Haar fidelity is \(547/726=3/4+5/1452>3/4\). This uses the ratio of integrals in Eq. (2): it improves the success-weighted average, not every individual target. The deterministic five-copy construction above has a strict fidelity gain for every target without postselection [Zha26], p. 3.
The arXiv version of He et al. reports numerical no-improvement results for

\begin{equation} (d,n)\in\{(2,3),(2,4),(3,3),(3,4)\}, \tag{5} \end{equation}

in arXiv:2504.10516v2, Table 1 [HZY+26]. The four-copy qubit assertion in Eq. (5), under that version’s displayed success-weighted fidelity definition, is contradicted by Zhang’s four-copy example; the manuscript does not establish the cause of the numerical discrepancy [Zha26], pp. 3–4. These numerical claims do not affect the analytical two-copy theorem or the five-copy counterexample.

The final journal supplement, Appendix F, restricts its qubit scans to \(0<\delta<1-1/\sqrt3\), with noise and success-probability steps of \(0.01\). Zhang’s four-copy example at \(\delta=1/2\) and five-copy example at \(\delta=2/3\) lie outside that sampled noise range [HZY+26].
Without the restriction to classically simulable maps, projecting the \(n\) copies of a depolarized qubit onto their symmetric subspace is the optimal universal purifier and raises the fidelity above the single-copy value. This unrestricted result does not determine the optimal fidelity attainable with classically simulable maps [CEM99].
He, Xiong, and Wang report two-copy resource laws for a feasible Haar-averaged fidelity and success-probability pair \((f,s)\) in the purification regime \(f\geq\lambda_0=1-(d-1)\delta/d\). Theorem 1 gives an exact linear law in \(f-\lambda_0\) for exponentiated purification mana in odd dimensions; the logarithmic mana includes the normalization \(-\log_2 s\). Theorem 2 bounds the minimum of \(s^{-1}\mathcal R(J^{\mathcal E}/d^2)\) over feasible successful trace-nonincreasing maps by two linear functions of \(f-\lambda_0\) for \(d=2^m\). Here \(J^{\mathcal E}\) is the unnormalized Choi operator and \(\mathcal R\) is stabilizer robustness. The bounds coincide for \(d=2\) (Appendix D, Eqs. (S79)–(S81)). These preprint results concern depolarizing noise and two input copies; they do not determine the attainable fidelity for arbitrary \(n\) [HXW26].
Reported progress: PR #83 · Issue #108.

Comment

Solved by counterexample to the universally quantified equality in Eq. (3): the admissible qubit instance in Eq. (4) suffices to refute it [Zha26]. The cited resolving result is a working manuscript archived on Zenodo; it is not presented as a peer-reviewed journal publication. The analytical two-copy no-go theorem remains unaffected. Zhang’s result does not determine optimal fidelities, the minimum copy number for a deterministic gain for every target, or the odd-dimensional cases. The completely positive-Wigner-preserving class in Eq. (2) is defined only for odd \(d\), so even dimensions above two remain outside the archived formulation.

References

[HZY+26]: K. He, C. Zhu, H. Yao, J. Liu, Y. Li, and X. Wang, “No-Go Theorems for Universal Quantum State Purification via Classically Simulable Operations,” Physical Review Letters 136, 090204 (2026). Final Supplemental Material, Appendix F: numerical parameter ranges.DOI arXiv link
[CEM99]: J. I. Cirac, A. K. Ekert, and C. Macchiavello, “Optimal Purification of Single Qubits,” Physical Review Letters 82, 4344–4347 (1999).DOI arXiv
[Zha26]: Y. Zhang, “Purifying an unknown qubit with stabilizer operations,” working manuscript dated 24 September 2026, pp. 2–4, preserved in Agentic Proofs for QIQC: Collected Manuscripts, version 1.0 (26 September 2026). archived purification manuscript; collection doi:10.5281/zenodo.22969546.link DOI
[HXW26]: K. He, E. Xiong, and X. Wang, “A Nonstabilizerness Resource Law for Universal Quantum State Purification,” arXiv:2607.08626v1 (2026), Theorems 1–2 and Appendices C–D.DOI arXiv

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