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

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].
Semidefinite programs return the same value, and hence no improvement, for

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

but the finite cases in Eq. (4) do not prove the statement for all \(n\) and all admissible \(d\) [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. A proof of Eq. (3) would therefore isolate dynamical non-stabilizer resources as necessary for purification rather than establish an unrestricted impossibility [CEM99].

Comment

Only the two-copy case is proved. The three- and four-copy evidence is numerical and cannot exclude a genuinely many-copy classically simulable protocol, and the completely positive-Wigner-preserving class in Eq. (2) is defined only for odd \(d\), so even dimensions above two are outside the present 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).DOI arXiv
[CEM99]: J. I. Cirac, A. K. Ekert, and C. Macchiavello, “Optimal Purification of Single Qubits,” Physical Review Letters 82, 4344–4347 (1999).DOI arXiv

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