Quantum Skew Divergence

TL;DR

This paper explores the quantum extension of the skew divergence, analyzing its properties, relationships with other measures, and applications in quantum information theory, including new inequalities and a proof of a longstanding conjecture.

Contribution

It introduces the quantum skew divergence, studies its properties, and applies it to derive new inequalities and prove Bravyi's Small Incremental Mixing conjecture.

Findings

Derived new continuity inequalities for quantum Jensen-Shannon divergence

Established relationships between quantum skew divergence and other measures

Provided a short proof of Bravyi's Small Incremental Mixing conjecture

Abstract

In this paper we study the quantum generalisation of the skew divergence, which is a dissimilarity measure between distributions introduced by L. Lee in the context of natural language processing. We provide an in-depth study of the quantum skew divergence, including its relation to other state distinguishability measures. Finally, we present a number of important applications: new continuity inequalities for the quantum Jensen-Shannon divergence and the Holevo information, and a new and short proof of Bravyi's Small Incremental Mixing conjecture.