Fixed-error parallel Stein lemma for quantum channels

Problem

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

\begin{equation} \begin{aligned} D(\rho\|\sigma) &:={\rm Tr}\!\left[\rho(\log_2\rho-\log_2\sigma)\right],\\ D_H^\varepsilon(\rho\|\sigma) &:=-\log_2\inf_{\substack{0\leq Q\leq I\\ {\rm Tr}(Q\rho)\geq1-\varepsilon}} {\rm Tr}(Q\sigma), \end{aligned} \tag{1} \end{equation}

where \(D(\rho\|\sigma)=+\infty\) 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

\begin{equation} \mathbf D_{\rm ch}(\mathcal N\|\mathcal M) :=\sup_{\psi_{RA}\in\mathcal D(R\otimes A)} \mathbf D\!\left( (\operatorname{id}_R\otimes\mathcal N)(\psi) \middle\| (\operatorname{id}_R\otimes\mathcal M)(\psi) \right), \qquad R\simeq A. \tag{2} \end{equation}

Using Eq. (2), define the regularized channel relative entropy by

\begin{equation} D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M) :=\lim_{n\to\infty}\frac1n D_{\rm ch}(\mathcal N^{\otimes n}\|\mathcal M^{\otimes n}). \tag{3} \end{equation}

Whenever Eq. (3) is finite, does the fixed-error parallel Stein limit exist and satisfy

\begin{equation} \lim_{n\to\infty}\frac1n D_{H,{\rm ch}}^\varepsilon (\mathcal N^{\otimes n}\|\mathcal M^{\otimes n}) =D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M) \qquad\text{for every }\varepsilon\in(0,1)? \tag{4} \end{equation}

Source

Fang, Gour, and Wang prove the weak channel Stein lemma and identify the fixed-error, equivalently strong-converse-threshold, extension as unresolved for general channel pairs [FGW25].

Progress

The weak channel Stein lemma proves

\begin{equation} \lim_{\delta\downarrow0}\lim_{n\to\infty}\frac1n D_{H,{\rm ch}}^\delta (\mathcal N^{\otimes n}\|\mathcal M^{\otimes n}) =D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M). \tag{5} \end{equation}

Equation (5) yields the required lower bound at every fixed error, but not the converse inequality [FGW25].
Sandwiched-Rényi converses bound the fixed-error limsup by

\begin{equation} \limsup_{n\to\infty}\frac1nD_{H,{\rm ch}}^\varepsilon (\mathcal N^{\otimes n}\|\mathcal M^{\otimes n}) \leq\inf_{\alpha>1} \widetilde D_{\alpha,{\rm ch}}^\infty(\mathcal N\|\mathcal M). \tag{6} \end{equation}

Closing Eq. (6) requires the still-unproved continuity identity

\begin{equation} \inf_{\alpha>1}\widetilde D_{\alpha,{\rm ch}}^\infty (\mathcal N\|\mathcal M) \stackrel{?}{=}D_{\rm ch}^\infty(\mathcal N\|\mathcal M). \tag{7} \end{equation}

Equation (7) is the general strong-converse threshold problem [FGW25].
The equality in Eq. (4) is known when \(\mathcal M\) is a replacer channel [CMW16]. It is also known for suitable pairs of idempotent channels that share a full-rank invariant state; this structured result explicitly leaves the general channel case open [SB26].
For finite \(D_{\max}(\mathcal N\|\mathcal M)\), Gour’s Theorem 14, statements (1), (3), and (4), equates Eq. (4) with a subchannel-smoothed max-relative-entropy AEP at rate \(D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M)\) for every fixed \(\delta\in(0,1)\), and with \(E_{2^{nr}}(\mathcal N^{\otimes n}\|\mathcal M^{\otimes n})\to0\) for every \(r>D_{\rm ch}^{\infty}(\mathcal N\|\mathcal M)\) [Gou26]. Here subchannels are completely positive trace-nonincreasing maps; smoothing is uniform over reference-assisted inputs in generalized trace distance \(\tfrac12(\|X\|_1+|\operatorname{Tr}X|)\), with \(X\) the output difference. The channel hockey-stick divergence \(E_\gamma\) maximizes \(\operatorname{Tr}(\rho-\gamma\sigma)_+\) over common reference-assisted inputs. These equivalent assertions remain open in general.

Comment

The question concerns parallel tests with an arbitrary entangled input across the channel uses. The weak-error result and the known structured channel families do not establish Eq. (4) for an arbitrary finite-dimensional pair \(\mathcal N,\mathcal M\). The failure of CPTP-smoothed AEP does not refute this Stein limit: trace-preserving completion is an additional constraint [Gou26].

References

[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
[CMW16]: T. Cooney, M. Mosonyi, and M. M. Wilde, “Strong Converse Exponents for a Quantum Channel Discrimination Problem and Quantum-Feedback-Assisted Communication,” Communications in Mathematical Physics 344, 797–829 (2016).DOI arXiv
[SB26]: S. Singh and B. Bergh, “Discriminating Idempotent Quantum Channels,” arXiv preprint (2026).arXiv
[Gou26]: G. Gour, “Failure of the Asymptotic Equipartition Property for Quantum Channels,” arXiv preprint (2026).arXiv

Fixed-error parallel Stein lemma for quantum channels

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