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Information-theoretic aspects of quantum channel discrimination

Mark M. Wilde

Published Jul 17, 2026• 0 citations
Publisher

Abstract

I will give an overview of quantum channel discrimination, reviewing the task, its operational significance, connections to other information-theoretic tasks like communication and estimation, and progress in establishing characterizations and bounds on error exponents. Inevitably, quantum effects and additivity issues arise in quantum channel discrimination, revealing fundamental differences between classical and quantum information. Delivered as a talk at "Additivity Problems in Quantum and Classical Information Theory", held at Banff International Research Station during July 12-17, 2026

DOI
10.5281/zenodo.21404893

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I will give an overview of quantum channel discrimination, reviewing the task, its operational significance, connections to other information-theoretic tasks like communication and estimation, and progress in establishing characterizations and bounds on error exponents. Inevitably, quantum effects and additivity issues arise in quantum channel discrimination, revealing fundamental differences between classical and quantum information. Delivered as a talk at "Additivity Problems in Quantum and Classical Information Theory", held at Banff International Research Station during July 12-17, 2026
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.
Quantum channel estimation and discrimination are fundamentally related information processing tasks of interest in quantum information science. In this paper, we analyze these tasks by employing the right logarithmic derivative Fisher information and the geometric Rényi relative entropy, respectively, and we also identify connections between these distinguishability measures. A key result of our paper is that a chain-rule property holds for the right logarithmic derivative Fisher information and the geometric Rényi relative entropy for the interval α∈ (0 , 1) of the Rényi parameter α. In channel estimation, these results imply a condition for the unattainability of Heisenberg scaling, while in channel discrimination, they lead to improved bounds on error rates in the Chernoff and Hoeffding error exponent settings. More generally, we introduce the amortized quantum Fisher information as a conceptual framework for analyzing general sequential protocols that estimate a parameter encoded in a quantum channel. We then use this framework, beyond the aforementioned application, to show that Heisenberg scaling is not possible when a parameter is encoded in a classical–quantum channel. We then identify a number of other conceptual and technical connections between the tasks of estimation and discrimination and the distinguishability measures involved in analyzing each. As part of this work, we present a detailed overview of the geometric Rényi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.
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