Quantum reverse hypercontractivity: its tensorization and application to strong converses

TL;DR

This paper develops quantum reverse hypercontractivity inequalities, explores their tensorization, and applies these results to establish strong converse bounds in quantum information theory.

Contribution

It introduces quantum reverse hypercontractivity inequalities, generalizes key inequalities to the non-commutative setting, and applies them to quantum hypothesis testing and channel coding.

Findings

Derived quantum reverse hypercontractivity inequalities from log-Sobolev inequalities.

Proved tensorization results for quantum hypercontractivity.

Established strong converse bounds in quantum information theory.

Abstract

In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock-Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from $2$-log-Sobolev and $1$-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.