Metrics Induced by Jensen-Shannon and Related Divergences on Positive Definite Matrices

TL;DR

This paper proves that certain symmetric divergences on positive definite matrices, including quantum Jensen-Shannon divergences, satisfy metric properties, providing rigorous foundations for their use in quantum information theory.

Contribution

It establishes the metric property of Jensen-Shannon and related divergences on positive definite matrices, including quantum cases, resolving longstanding conjectures.

Findings

Square roots of these divergences are distances.

Quantum Jensen-Shannon divergence is proven to be a metric.

Metric properties of Jensen-Rényi divergences are also established.

Abstract

We study metric properties of symmetric divergences on Hermitian positive definite matrices. In particular, we prove that the square root of these divergences is a distance metric. As a corollary we obtain a proof of the metric property for Quantum Jensen-Shannon-(Tsallis) divergences (parameterized by $\alpha\in [0,2]$), which in turn (for $\alpha=1$) yields a proof of the metric property of the Quantum Jensen-Shannon divergence that was conjectured by Lamberti \emph{et al.} a decade ago (\emph{Metric character of the quantum Jensen-Shannon divergence}, Phy.\ Rev.\ A, \textbf{79}, (2008).) A somewhat more intricate argument also establishes metric properties of Jensen-R\'enyi divergences (for $\alpha \in (0,1)$), and outlines a technique that may be of independent interest.