The Annals of Statistics

Second-order asymptotics for quantum hypothesis testing

Ke Li

Full-text: Open access

Abstract

In the asymptotic theory of quantum hypothesis testing, the minimal error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states in an increasing way. This is well known as the direct part and strong converse of quantum Stein’s lemma.

Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows smoothly according to a lower order of the error exponent of the second kind, and hence we obtain the second-order asymptotics for quantum hypothesis testing. This actually implies quantum Stein’s lemma as a special case. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have potential applications in quantum information theory.

Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the optimality part in a unified fashion.

Article information

Source
Ann. Statist., Volume 42, Number 1 (2014), 171-189.

Dates
First available in Project Euclid: 18 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1392733184

Digital Object Identifier
doi:10.1214/13-AOS1185

Mathematical Reviews number (MathSciNet)
MR3178460

Zentralblatt MATH identifier
1321.62155

Subjects
Primary: 62P35: Applications to physics 62G10: Hypothesis testing

Keywords
Quantum hypothesis testing quantum Stein’s lemma second-order asymptotics finite sample size

Citation

Li, Ke. Second-order asymptotics for quantum hypothesis testing. Ann. Statist. 42 (2014), no. 1, 171--189. doi:10.1214/13-AOS1185. https://projecteuclid.org/euclid.aos/1392733184


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