On the metric character of the quantum Jensen-Shannon divergence

TL;DR

This paper investigates whether the quantum Jensen-Shannon divergence functions as a metric, demonstrating that its square root satisfies the triangle inequality for pure states and through simulations for mixed states.

Contribution

It provides a formal proof for pure states and simulation evidence for mixed states that the square root of the quantum JSD is a metric.

Findings

Square root of quantum JSD satisfies triangle inequality for pure states.

Simulations suggest the same property holds for mixed states.

Quantum JSD is a reliable distinguishability measure.

Abstract

In a recent paper, the generalization of the Jensen Shannon divergence (JSD) in the context of quantum theory has been studied (Phys. Rev. A 72, 052310 (2005)). This distance between quantum states has shown to verify several of the properties required for a good distinguishability measure. Here we investigate the metric character of this distance. More precisely we show, formally for pure states and by means of simulations for mixed states, that its square root verifies the triangle inequality.