  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, \textbf{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.
