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

DOI10.1016/J.LAA.2020.12.023zbMATH Open1464.15031arXiv1911.02643OpenAlexW3119350730MaRDI QIDQ2656633

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Publication date: 16 March 2021

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Abstract: 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 $alphain[0,2]$), 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, extbf{79}, (2008).) A somewhat more intricate argument also establishes metric properties of Jensen-R'enyi divergences (for $alphain(0,1)$), and outlines a technique that may be of independent interest.

Full work available at URL: https://arxiv.org/abs/1911.02643

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