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arXiv:quant-ph/0110125 · IEEE International Symposium on Information Theory (ISIT)

A New Proof of the Direct Part of Stein's Lemma in Quantum Hypothesis Testing

Tomohiro Ogawa, Masahito Hayashi

Published Oct 21, 2001• 5 citations
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Abstract

The direct part of Stein's lemma in quantum hypothesis testing is revisited based on a key operator inequality between a density operator and its pinching. The operator inequality is used to show a simple proof of the direct part of Stein's lemma without using Hiai-Petz's theorem, along with an operator monotone function, and in addition it is also used to show a new proof of Hiai-Petz's theorem.

DOI
10.48550/arxiv.quant-ph/0110125

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The direct part of Stein's lemma in quantum hypothesis testing is revisited based on a key operator inequality between a density operator and its pinching. The operator inequality is used to show a simple proof of the direct part of Stein's lemma without using Hiai-Petz's theorem, along with an operator monotone function, and in addition it is also used to show a new proof of Hiai-Petz's theorem.
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.
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.
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.
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.
The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.