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IEEE International Symposium on Information Theory (ISIT)

On error exponents in quantum hypothesis testing

T. Ogawa, Masahito Hayashi

Published Jan 1, 2003• 3 citations
Publisher

Abstract

On error exponents in the simple quantum hypothesis testing problem, a noncommutative analogue of the Hoeffding bound (W. Hoeffding, 1991) is presented. If the hypotheses, namely two density operators, are mutually commutative, the bound is identical with the classical counter part and thus is optimal. As a by-product, a simple proof of the direct part of the quantum Stein's lemma [F. Hiai (1991)] [T. Ogawa (2000)] is also given.

DOI
10.1109/isit.2003.1228496

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On error exponents in the simple quantum hypothesis testing problem, a noncommutative analogue of the Hoeffding bound (W. Hoeffding, 1991) is presented. If the hypotheses, namely two density operators, are mutually commutative, the bound is identical with the classical counter part and thus is optimal. As a by-product, a simple proof of the direct part of the quantum Stein's lemma [F. Hiai (1991)] [T. Ogawa (2000)] is also given.
In the simple quantum hypothesis testing problem, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and its pinching. Concerning the error exponents, the upper bounds lead to a noncommutative analogue of the Hoeffding bound, which is identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. The upper bounds also provide a simple proof of the direct part of the quantum Stein's lemma.
In the simple quantum hypothesis testing problem for two density operators, upper bounds on the error probabilities are shown based on a key operator inequality between a density operator and a conditional expectation of it. Concerning the error exponents, the upper bounds lead to a noncommutative analog of the Hoeffding bound, which is identical with the classical counterpart if two density operators commute. The upper bounds also provide a simple proof of the direct part of the quantum Stein's lemma.
An upper bound on simple quantum hypothesis testing in the asymmetric setting is shown using a useful inequality by Audenaert et al. [Phys. Rev. Lett. 98, 160501 (2007)] which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which is identical with the classical counterpart if the hypotheses, composed of two density operators, are mutually commutative. Its attainability has been a long-standing open problem. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding.
We study the error exponents in quantum hypothesis testing between two sets of quantum states, extending the analysis beyond the independent and identically distributed case to encompass composite correlated hypotheses. In particular, we introduce and compare two natural extensions of the quantum Hoeffding divergence and anti-divergence to sets of quantum states, establishing their equivalence or quantitative relations. In the error exponent regime, we generalize the quantum Hoeffding bound to stable sequences of convex, compact sets of quantum states, demonstrating that the optimal Type-I error exponent, under an exponential constraint on the Type-II error, is precisely characterized by the regularized quantum Hoeffding divergence between the sets. In the strong converse exponent regime, we establish a general lower bound on the exponent in terms of the regularized quantum Hoeffding anti-divergence, and we prove a matching upper bound when the null hypothesis is a singleton, under additional assumptions. The generality of these results enables applications in various contexts, including (i) refining the generalized quantum Stein’s lemma by [Fang, Fawzi & Fawzi, 2024]; (ii) exhibiting counterexamples to the continuity of the regularized Petz R´enyi divergence and Hoeffding divergence; (iii) obtaining error exponents for adversarial channel discrimination and resource detection problems.
The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result of Hiai and Petz (1991) to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy, which yields the strong converse in quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi (1989) in classical hypothesis testing.
The hypothesis testing problem of two quantum states is treated. We show a new inequality between the error of the first kind and the second kind, which complements the result of Hiai and Petz to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the first kind error when the power exponent for the second kind error exceeds the quantum relative entropy, and the bound yields the strong converse in the quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi in the classical hypothesis testing.