Diamond-smoothed max-relative-entropy AEP for quantum channels

Problem

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

\begin{equation} \begin{aligned} D_{\max}(\mathcal N\|\mathcal M) &:=\inf\{\lambda:J_{\mathcal N}\leq2^\lambda J_{\mathcal M}\},\\ D_{\max}^{\varepsilon}(\mathcal N\|\mathcal M) &:=\inf_{\substack{\widetilde{\mathcal N}\ {\rm CPTP}:\\ \frac12\|\widetilde{\mathcal N}-\mathcal N\|_\diamond \leq\varepsilon}} D_{\max}(\widetilde{\mathcal N}\|\mathcal M). \end{aligned} \tag{1} \end{equation}

The smoothing in Eq. (1) requires one channel \(\widetilde{\mathcal N}\) that approximates \(\mathcal N\) uniformly over all ancilla-assisted inputs. Define

\begin{equation} \begin{aligned} D_{\max}^{\varepsilon,\infty}(\mathcal N\|\mathcal M) &:=\limsup_{n\to\infty}\frac1n D_{\max}^{\varepsilon} (\mathcal N^{\otimes n}\|\mathcal M^{\otimes n}),\\ D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M) &:=\lim_{n\to\infty}\frac1n \sup_{\psi_{R A^n}} D\!\left((\operatorname{id}_R\otimes\mathcal N^{\otimes n})(\psi) \middle\| (\operatorname{id}_R\otimes\mathcal M^{\otimes n})(\psi)\right), \end{aligned} \tag{2} \end{equation}

where \(R\simeq A^{\otimes n}\) 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?

\begin{equation} \sup_{\varepsilon>0} D_{\max}^{\varepsilon,\infty}(\mathcal N\|\mathcal M) =D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M). \tag{3} \end{equation}

Source

Winter first proposed the identity in Eq. (3); Liu and Winter discussed it formally in the setting of channel-resource erasure [LW19]. Hirche restated it explicitly as the technical conjecture needed in quantum-network discrimination [Hir23].

Progress

Gour and Winter proved asymptotic equipartition statements for two channel-resource divergences using a more permissive “liberal” smoothing. That smoothing does not require the single uniformly diamond-close channel in Eq. (1), so it does not establish Eq. (3) [GW19].
Hirche proved the general bounds

\begin{equation} D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M) \leq D_{\max}^{\varepsilon,\infty}(\mathcal N\|\mathcal M) \leq D_{\max}(\mathcal N\|\mathcal M), \tag{4} \end{equation}

The bounds in Eq. (4) do not identify the asymptotic rate. Hirche also showed that Eq. (3) would collapse the amortized Umegaki relative entropy of superchannels to their regularized relative entropy [Hir23].
Fang, Gour, and Wang related an analogous AEP for unstabilized channel divergences without quantum-memory assistance to strong converses for channel discrimination. Their latest formulation leaves that general strong-converse problem unresolved and does not prove the stabilized, diamond-smoothed identity in Eq. (3) [FGW25].
Gour gives a counterexample in Theorems 11 and 15 of a 2026 preprint [Gou26]. For \(0<t<(\sqrt5-2)^2\) and \(0<\theta<2\arcsin(1/8)\), set

\begin{equation} \begin{aligned} \mathcal N&=\operatorname{id}_3,\qquad \mathcal M=t\operatorname{id}_3+(1-t)\mathcal U,\\ \mathcal U(X)&=UXU^\dagger,\qquad U=\operatorname{diag}(1,e^{i\theta},e^{-i\theta}). \end{aligned} \tag{5} \end{equation}

The channels in Eq. (5) have finite \(D_{\max}\). For every fixed \(0<\varepsilon<\sin^2\theta\),

\begin{equation} \begin{aligned} \lim_{n\to\infty}\frac1nD_{\max}^{\varepsilon} (\mathcal N^{\otimes n}\|\mathcal M^{\otimes n}) &=\log_2(1/t),\\ D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M) &\leq\tfrac12\log_2(1/t)+\log_2\frac4{1-t} <\log_2(1/t). \end{aligned} \tag{6} \end{equation}

Monotonicity and the unsmoothed upper bound make the supremum in Eq. (3) equal to \(\log_2(1/t)\). Equation (6) therefore refutes the proposed identity, including its vanishing-error version (Section VII, Eqs. (232)–(234)).

Comment

The universal identity in Eq. (3) is false by Gour’s preprint [Gou26]. The limit exists for this counterexample family; the separate question of its existence for arbitrary channel pairs is not settled by this result. The record therefore remains Unsolved while both requests are retained in the statement. The counterexample also does not decide the fixed-error parallel channel Stein problem or the conditional equality in the parallel versus nested-adaptive superchannel-discrimination problem.

References

[LW19]: Z.-W. Liu and A. Winter, “Resource Theories of Quantum Channels and the Universal Role of Resource Erasure,” arXiv preprint (2019).arXiv
[GW19]: G. Gour and A. Winter, “How to Quantify a Dynamical Quantum Resource,” Physical Review Letters 123, 150401 (2019).DOI arXiv
[Hir23]: C. Hirche, “Quantum Network Discrimination,” Quantum 7, 1064 (2023).DOI arXiv
[FGW25]: K. Fang, G. Gour, and X. Wang, “Towards the Ultimate Limits of Quantum Channel Discrimination and Quantum Communication,” Science China Information Sciences 68, 180509 (2025).DOI arXiv
[Gou26]: G. Gour, “Failure of the Asymptotic Equipartition Property for Quantum Channels,” arXiv preprint (2026).arXiv

Diamond-smoothed max-relative-entropy AEP for quantum channels

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