Convexity properties of the quantum R\'enyi divergences, with applications to the quantum Stein's lemma

TL;DR

This paper establishes finite-size bounds on the deviation of the optimal type II error in quantum hypothesis testing, leveraging new convexity properties of quantum Re9nyi divergences to improve understanding of quantum Stein's lemma.

Contribution

It introduces and applies convexity properties of a new quantum Re9nyi divergence to derive finite-size bounds in quantum hypothesis testing.

Findings

Finite-size bounds on type II error deviations.

Utilization of new quantum Re9nyi divergence properties.

Enhanced understanding of quantum Stein's lemma.

Abstract

We show finite-size bounds on the deviation of the optimal type II error from its asymptotic value in the quantum hypothesis testing problem of Stein's lemma with composite null-hypothesis. The proof is based on some simple properties of a new notion of quantum R\'enyi divergence, recently introduced in [M\"uller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013)], and [Wilde, Winter, Yang, arXiv:1306.1586].