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We study metric properties of symmetric divergences on Hermitian positivedefinite matrices. In particular, we prove that the square root of thesedivergences is a distance metric. As a corollary we obtain a proof of themetric property for Quantum Jensen-Shannon-(Tsallis) divergences (parameterizedby $\alpha\in [0,2]$), which in turn (for $\alpha=1$) yields a proof of themetric property of the Quantum Jensen-Shannon divergence that was conjecturedby Lamberti \emph{et al.} a decade ago (\emph{Metric character of the quantumJensen-Shannon divergence}, Phy.\ Rev.\ A, \textbf{79}, (2008).) A somewhatmore intricate argument also establishes metric properties of Jensen-R\'enyidivergences (for $\alpha \in (0,1)$), and outlines a technique that may be ofindependent interest.

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