Metrics induced by Jensen-Shannon and related divergences on positive definite matrices

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 α∈[0,2]). When specialized to α=1, we obtain as a corollary a proof of the metric property of the Quantum Jensen-Shannon divergence that was conjectured by Lamberti et al. (2008) [13], and recently also proved by Virosztek (2019) [28]. A more intricate argument also establishes metric properties of Jensen-Rényi divergences (for α∈(0,1)); this argument develops a technique that may be of independent interest.