Open Access
February 2014 Second-order asymptotics for quantum hypothesis testing
Ke Li
Ann. Statist. 42(1): 171-189 (February 2014). DOI: 10.1214/13-AOS1185


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.


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Ke Li. "Second-order asymptotics for quantum hypothesis testing." Ann. Statist. 42 (1) 171 - 189, February 2014.


Published: February 2014
First available in Project Euclid: 18 February 2014

zbMATH: 1321.62155
MathSciNet: MR3178460
Digital Object Identifier: 10.1214/13-AOS1185

Primary: 62G10 , 62P35

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

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 1 • February 2014
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