The metric property of the quantum Jensen-Shannon divergence

TL;DR

This paper proves that the square root of the quantum Jensen-Shannon divergence functions as a true metric on the space of positive matrices and quantum states, establishing a fundamental property of this divergence measure.

Contribution

It demonstrates that the square root of the quantum Jensen-Shannon divergence is a valid metric, a previously unconfirmed property in quantum information theory.

Findings

The square root of the quantum Jensen-Shannon divergence is a metric.

This property holds on the cone of positive matrices and quantum state space.

The result solidifies the mathematical foundation of this divergence measure.

Abstract

In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.