On the asymmetry of the relative entropy

TL;DR

This paper investigates the asymmetry of quantum relative entropy, providing a sharp upper bound based on trace norm distance and eigenvalues, enhancing understanding of quantum state dissimilarity measures.

Contribution

It introduces a precise upper bound for the asymmetry of quantum relative entropy using trace norm and eigenvalues, clarifying its behavior in quantum information theory.

Findings

Derived a sharp upper bound for asymmetry

Bound depends on trace norm distance and eigenvalues

Enhances understanding of quantum state dissimilarity

Abstract

The quantum relative entropy $S(\rho||\sigma)$ is a widely used dissimilarity measure between quantum states, but it has the peculiarity of being asymmetric in its arguments. We quantify the amount of asymmetry by providing a sharp upper bound in terms of two parameters: the trace norm distance between the two states, and the smallest of the smallest eigenvalues of both states. The bound is essentially the asymmetry between two binary distributions governed by these two parameters.