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

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 is an element of [0, 2]). When specialized to alpha = 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-Renyi divergences (for alpha is an element of (0, 1)); this argument develops a technique that may be of independent interest. (C) 2020 Elsevier Inc. All rights reserved.