Generalized Stein lemma for fully quantum channel resources

Problem

Let \(\mathcal N:\mathcal L(A)\to\mathcal L(B)\) be a finite-dimensional quantum channel. For each \(n\), let \(\mathfrak F_n\) be a nonempty compact, convex, permutation-invariant set of channels from \(A^{\otimes n}\) to \(B^{\otimes n}\), closed under tensor products and containing \(\mathcal R_\omega^{\otimes n}\) for one full-rank state \(\omega\), where \(\mathcal R_\omega(X):=\operatorname{Tr}(X)\omega\). Define

\begin{equation} D_{\rm ch}(\mathcal N^{\otimes n}\|\mathcal M_n) :=\sup_{\psi_{R_nA^n}} D\!\left( (\operatorname{id}_{R_n}\otimes\mathcal N^{\otimes n})(\psi) \middle\| (\operatorname{id}_{R_n}\otimes\mathcal M_n)(\psi) \right), \qquad R_n\simeq A^{\otimes n}, \tag{1} \end{equation}

where the supremum in Eq. (1) is over density operators and \(D\) is the quantum relative entropy. The distance to the free set is

\begin{equation} E_n^{\rm QQ}(\mathcal N\|\mathfrak F_n) :=\inf_{\mathcal M_n\in\mathfrak F_n} D_{\rm ch}(\mathcal N^{\otimes n}\|\mathcal M_n). \tag{2} \end{equation}

Equation (2) is the \(n\)-use relative-entropy distance to the free channel set. For \(\varepsilon\in(0,1)\), define the optimal worst-case type-II error of a parallel quantum-input/quantum-output test by

\begin{equation} \begin{aligned} \beta_{\varepsilon,n}^{\rm QQ}(\mathcal N\|\mathfrak F_n) :=\inf_{\substack{\psi_{R_nA^n},\ 0\leq Q\leq I\\ \operatorname{Tr}[Q(\operatorname{id}_{R_n}\otimes \mathcal N^{\otimes n})(\psi)]\geq1-\varepsilon}} \ \sup_{\mathcal M_n\in\mathfrak F_n} \operatorname{Tr}\!\left[ Q(\operatorname{id}_{R_n}\otimes\mathcal M_n)(\psi) \right]. \end{aligned} \tag{3} \end{equation}

Under what additional structural assumptions on \((\mathfrak F_n)_{n\geq1}\), if any, do both limits exist and obey the fully quantum generalized Stein identity

\begin{equation} \lim_{n\to\infty}-\frac1n\log_2 \beta_{\varepsilon,n}^{\rm QQ}(\mathcal N\|\mathfrak F_n) =\lim_{n\to\infty}\frac1n E_n^{\rm QQ}(\mathcal N\|\mathfrak F_n) \qquad\text{for every }\varepsilon\in(0,1)? \tag{4} \end{equation}

Source

The fully quantum formulation is implicit in the two generalized Stein theorems for classical-input channels, both of which isolate their classical-input structure from the unresolved quantum-input setting [HY25b], [BDK25].

Progress

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For an i.i.d. resource state tested against admissible composite sets of free states, the generalized quantum Stein lemma identifies the fixed-error exponent with the regularized relative entropy of resource [HY25a].
Two independent works prove the channel analogue for classical–quantum channels. In that setting the channel relative entropy reduces to

\begin{equation} D_{\rm CQ}(\Phi\|\Psi)=\max_x D(\rho_x\|\sigma_x), \qquad \Phi:x\mapsto\rho_x,\quad\Psi:x\mapsto\sigma_x, \tag{5} \end{equation}

and its pointwise structure supplies the additivity and minimax steps needed for a fixed-error theorem. These steps apply to Eq. (5) but are unavailable in this form for QQ channels [HY25b], [BDK25].
The CQ proof explicitly states that analogous properties for fully quantum channels remain unclear. Entangled quantum inputs introduce a reference system and a nontrivial order between input optimization and the worst-case free-channel optimization in Eq. (3) [HY25b].

Comment

The state theorem and the CQ-channel theorems do not imply Eq. (4) for genuinely quantum inputs. An adaptive quantum-comb version would be a further problem and is not included in the present statement.

References

[HY25a]: M. Hayashi and H. Yamasaki, “The Generalized Quantum Stein’s Lemma and the Second Law of Quantum Resource Theories,” Nature Physics 21, 1988–1993 (2025).DOI arXiv
[HY25b]: M. Hayashi and H. Yamasaki, “Generalized Quantum Stein’s Lemma for Classical-Quantum Dynamical Resources,” arXiv preprint (2025).arXiv
[BDK25]: B. Bergh, N. Datta, and A. Khaitan, “Generalized Quantum Stein’s Lemma and Reversibility of Quantum Resource Theories for Classical-Quantum Channels,” arXiv preprint (2025).arXiv

Generalized Stein lemma for fully quantum channel resources

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