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 $\alpha \in [0,2]$), 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, $\mathbf{\text{79}}$, (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.