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arXiv:2609.30268

Quantum Channel Stein Theorem beyond Definite Causal Order

Chengkai Zhu, Xin Wang

quant-phSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
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Abstract

Distinguishability is a central concept in information theory and extends naturally to quantum systems. This work shows that, for any two finite-dimensional memoryless quantum channels, parallel, adaptive and general testing strategies achieve the same Stein exponent at every fixed type-I error tolerance $ε\in(0,1)$, namely the regularized channel relative entropy. When this rate is finite, any larger type-II exponent forces the probability of correctly accepting the null hypothesis to decay exponentially. Thus, adaptivity provides no asymptotic advantage. The key step is proving the continuity of the regularized sandwiched Rényi channel divergence at order one. We also derive the exact strong-converse exponent for general testers.

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20 pages

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Distinguishability is a central concept in information theory and extends naturally to quantum systems. This work shows that, for any two finite-dimensional memoryless quantum channels, parallel, adaptive and general testing strategies achieve the same Stein exponent at every fixed type-I error tolerance ε∈(0,1)ε\in(0,1)ε∈(0,1), namely the regularized channel relative entropy. When this rate is finite, any larger type-II exponent forces the probability of correctly accepting the null hypothesis to decay exponentially. Thus, adaptivity provides no asymptotic advantage. The key step is proving the continuity of the regularized sandwiched Rényi channel divergence at order one. We also derive the exact strong-converse exponent for general testers.
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 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.
We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero--one testing law, and the subchannel asymptotic equipartition property.
This paper studies the difficulty of discriminating quantum channels under operational regimes, proves the quantum channel Stein's lemma (strong converse part), and provides a unified framework to show the operational meaning of quantum channel divergences. First, we establish the exponentially strong converse of quantum channel hypothesis testing under coherent strategies, meaning that any strategy to make the Type II error decays with an exponent larger than the regularized channel relative entropy will unavoidably result in the Type I error converging to one exponentially fast in the asymptotic limit. This result notably delivers the desirable quantum channel Stein's Lemma, enclosing a long-term open problem in quantum information theory. As a byproduct, we show the continuity of the regularized (amortized) Sandwiched R\'{e}nyi channel divergence at α=1\alpha=1α=1, resolving another open problem in the field. Second, we develop a framework to show the interplay between the strategies of channel discrimination, the operational regimes, and variants of channel divergences. This framework systematically underlies the operational meaning of quantum channel divergences in quantum channel discrimination. Our work establishes the ultimate limit of quantum channel discrimination, deepening our understanding of quantum channel discrimination and quantum channel divergences in the asymptotic regime. As quantum channel discrimination is strongly connected to many other fundamental tasks in quantum information theory, we expect plentiful applications on related topics such as quantum metrology and quantum communication.
We study binary discrimination of idempotent quantum channels. When the two channels share a common full-rank invariant state, we show that a simple image inclusion condition completely determines the asymptotic behavior: when it holds, a broad family of channel divergences collapse to a closed-form, single-letter expression, regularization is unnecessary, and all error exponents (Stein/Chernoff/Hoeffding/strong-converse) are explicitly computable with no adaptive advantage. Crucially, this yields the strong converse property for this channel family. When the inclusion fails, asymmetric exponents become infinite, implying perfect asymptotic discrimination. We apply the results to GNS-symmetric channels, showing discrimination rates for large number of self-iterations converge exponentially fast to those of the corresponding idempotent peripheral projections. Without a common invariant state, we show that the same image inclusion still guarantees additivity of the sandwiched Rényi cb-divergence for every α>1α>1α>1, which suffices to establish the strong converse property and single-letter expressions for the Stein exponents.
We prove a generalized quantum Stein lemma for i.i.d. normal states against convex, tensor-stable families on arbitrary von Neumann algebras. Assuming the existence of an alternative state with finite relative entropy from the null state, we show that, at every type-I error tolerance ε∈(0,1)\varepsilon\in (0,1)ε∈(0,1), the worst case type-II error exponent is achieved with the regularized relative entropy with a strong converse. The proof combines the integral representation of relative entropy by hockey-stick divergences, a modular testing bound, and convex minimax argument.