Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies

TL;DR

This paper establishes an operational interpretation for the quantum Renyi relative entropies in quantum hypothesis testing, clarifying their relevance depending on the parameter lpha, and demonstrates their attainability and monotonicity.

Contribution

It provides the first operational interpretation of the recently introduced quantum Renyi relative entropies in the context of quantum hypothesis testing.

Findings

Operational interpretation for lpha>1 in hypothesis testing

Traditional and new definitions of quantum Renyi entropies are optimal for different lpha ranges

New proof of monotonicity of quantum Renyi entropies under CPTP maps

Abstract

We show that the new quantum extension of Renyi's \alpha-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter \alpha: for \alpha<1, the right choice seems to be the traditional definition, whereas for \alpha>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi \alpha-relative entropies are asymptotically attainable by measurements for \alpha>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.