Second-order asymptotics for quantum hypothesis testing

TL;DR

This paper derives second-order asymptotics for quantum hypothesis testing, revealing how the error of the first kind grows smoothly and providing tight finite-sample bounds, with implications for quantum information theory.

Contribution

It introduces the second-order asymptotics for quantum hypothesis testing, extending quantum Stein's lemma and offering a unified, elementary approach with finite-sample bounds.

Findings

Smooth growth of first-kind error probability analyzed

Tight bounds for finite sample sizes established

Extension of quantum Stein's lemma to second-order asymptotics

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.