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