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arXiv:2509.13280

Generalized Quantum Stein's Lemma and Reversibility of Quantum Resource Theories for Classical-Quantum Channels

Bjarne Bergh, Nilanjana Datta, Anirudh Khaitan

Published Sep 16, 2025• 0 citations
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Abstract

We extend the recent proof of the Generalized Quantum Stein's Lemma by Hayashi and Yamasaki [arXiv:2408.02722] to classical-quantum (c-q) channels. We analyze the composite hypothesis testing problem of testing a c-q channel $\mathcal{E}^{\otimes n}$ against a sequence of sets of c-q channels $(\mathcal{S}_n)_n$ (satisfying certain natural assumptions), under parallel strategies. We prove that the optimal asymptotic asymmetric error exponent is given by the regularization of Umegaki channel divergence, minimized over $\mathcal{S}_n$. This allows us to prove the reversibility of resource theories of classical-quantum channels in a natural framework, where the distance between channels (and hence also the notion of approximate interconvertibility of channels) is measured in diamond norm, and the set of free operations is the set of all asymptotically resource non-generating superchannels. The results we obtain are similar to the ones in the concurrent and independent work by Hayashi and Yamasaki [arXiv:2509.07271]. However the proof of the direct part of the GQSL uses different arguments and techniques to deal with the challenges that arise from dealing with c-q channels.

DOI
10.48550/arxiv.2509.13280

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Channel conversion constitutes a pivotal paradigm in information theory and its applications to quantum physics, providing a unified problem setting that encompasses celebrated results such as Shannon's noisy-channel coding theorem. Quantum resource theories (QRTs) offer a general framework to study such problems under a prescribed class of operations, such as those for encoding and decoding. In QRTs, quantum states serve as static resources, while quantum channels give rise to dynamical resources. A recent major advance in QRTs is the generalized quantum Stein's lemma, which characterizes the optimal error exponent in hypothesis testing to discriminate resource states from free states, enabling a reversible QRT framework for static resources where asymptotic conversion rates are fully determined by the regularized relative entropy of resource. However, applications of QRTs to channel conversion require a framework for dynamical resources. The earlier extension of the reversible framework to a fundamental class of dynamical resources, represented by classical-quantum (CQ) channels, relied on state-based techniques and imposed an asymptotic continuity assumption on operations, which prevented its applicability to conventional channel coding scenarios. To overcome this problem, we formulate and prove a generalized quantum Stein's lemma directly for CQ channels, by developing CQ-channel counterparts of the core proof techniques used in the state setting. Building on this result, we construct a reversible QRT framework for CQ channel conversion that does not require the asymptotic continuity assumption, and show that this framework applies to the analysis of channel coding scenarios. These results establish a fully general toolkit for CQ channel discrimination and conversion, enabling their broad application to core conversion problems for this fundamental class of channels.
We prove a generalized quantum Stein lemma for parallel discrimination of an arbitrary finite-dimensional channel against families of alternative channels. The alternatives are compact and convex, closed under tensor products, and contain a faithful replacer. At every fixed type-I error tolerance, the optimal type-II error exponent equals the regularized Umegaki channel relative entropy minimized over the alternatives, and both defining limits exist. Inputs may be entangled across channel uses and with a reference system. The main technical result constructs exactly trace-preserving approximations with exponentially small diamond error, dominated in completely positive order by free channels at every rate above the common exponent. The proof combines uniform auxiliary map approximation, iterative reduction of the domination rate, and tensor amplification. It yields asymptotic equipartition with trace-preserving smoothing at every subexponentially vanishing error, an exponential strong converse, and stability under vanishing diamond-norm perturbations. Under additional permutation invariance and closure under insertion of a fixed faithful input state and discarding of its output, we also prove an explicit finite-block completion bound. This quantitative construction uses weighted discarding, local operator corrections, and comparison with auxiliary extensions. Applications include state-preserving, coherence-restricted, covariant, entanglement-breaking, and positive-partial-transpose channels. For isometric targets against entanglement-breaking or positive-partial-transpose alternatives, we obtain exact finite-block testing and smoothing formulas.
The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras. After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.
We establish the quantum channel Stein lemma at every fixed type-I error tolerance in (0,1)(0,1)(0,1) and prove an exponential strong converse under arbitrary adaptive strategies. For any finite-dimensional channel pair, the optimal type-II exponent equals the regularized channel relative entropy and is attained by parallel strategies. When this divergence is finite, every strictly larger type-II rate forces the probability of accepting the null hypothesis to decay exponentially, uniformly over strategies and quantum memories. The proof rests on a uniform Stinespring approximation: tensor powers of the null dilation are approximated by tensor powers of the alternative dilation after applying an auxiliary linear map to the environment. The operator-norm error is exponentially small under a squared-norm bound at any rate above the regularized relative entropy. We construct the approximation from a weak testing bound by iterative rate reduction and a fixed-block tensor expansion with an exact residual. It implies right continuity at order one of the regularized sandwiched Rényi channel divergence, which yields the adaptive converse through the channel chain rule. We also obtain a sharp hockey-stick threshold and a subchannel smoothing asymptotic equipartition property for fixed and subexponentially vanishing diamond-norm errors.
We solve the generalised quantum Stein’s lemma, proving that the Stein exponent associated with entanglement testing, namely, the quantum hypothesis testing task of distinguishing betweenn\boldsymbol {n}ncopies of an entangled stateρAB\boldsymbol {\rho _{AB}}ρAB​and a generic separable stateσAn:Bn\boldsymbol {\sigma _{A^{n}:B^{n}}}σAn:Bn​, equals the regularised relative entropy of entanglement. Not only does this determine the ultimate performance of entanglement testing, but it also establishes the reversibility of all quantum resource theories under asymptotically resource non-generating operations, with the regularised relative entropy of resource governing the asymptotic transformation rate between any two quantum states. As a by-product, we prove that the same Stein exponent can also be achieved when the null hypothesis is only approximately i.i.d., in the sense that it can be modelled by an ‘almost power state’. To solve the problem we introduce two techniques. The first is a procedure that we call ‘blurring’, which, informally, transforms a permutationally symmetric state by making it more evenly spread across nearby type classes. Blurring alone suffices to prove the generalised Stein’s lemma in the fully classical case, but not in the quantum case. Our second technical innovation, therefore, is to perform a second quantisation step to lift the problem to an infinite-dimensional bosonic quantum system; we then solve it there by using techniques from continuous-variable quantum information. Rather remarkably, the second-quantised action of the blurring map corresponds to a pure loss channel. A careful examination of this second quantisation step is the core of our quantum solution.
The generalized quantum Stein's lemma characterizes the optimal asymptotic exponent of the type-II error in quantum hypothesis testing for an independent and identically distributed (IID) null hypothesis against a composite alternative hypothesis. Classically, a probabilistic mixture of IID sources arises as a natural generalization of IID sources, and, in the non-composite setting, the optimal type-II error exponent in hypothesis testing for such classical mixed sources is known to be characterized concisely by the worst-case component of the mixture. In this work, we extend these foundational results to composite quantum hypothesis testing where the null hypothesis is a mixed source, i.e., a probabilistic mixture of IID quantum states, and the alternative hypothesis is composite as in the generalized quantum Stein's lemma. When the type-I error vanishes asymptotically, we characterize the optimal type-II error exponent of this composite quantum hypothesis testing problem in terms of the worst-case component of the mixture, by developing techniques for the non-commutative quantum setting inspired by the classical information-spectrum analysis. We also show that the analogous characterization does not hold in general for a fixed nonzero type-I error threshold, by providing a counterexample beyond the vanishing type-I error regime. These results clarify the applicability of the generalized quantum Stein's lemma to highly non-IID null hypotheses arising from arbitrary finite probabilistic mixtures of IID quantum states.