Metrics Induced by Quantum Jensen-Shannon-Renyí and Related Divergences

11/06/2019

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by   Suvrit Sra, et al.

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We study symmetric divergences on Hermitian positive definite matrices generated by functions closely related to Pick-Nevanlinna functions. Our main result proves that the square root of these divergences is a distance metric. As a corollary we obtain the metric property for Quantum Jensen-Shannon-Tsallis divergences (parameterized by α∈ [0,2]), which in turn (for α=1) yields a proof of the metric property conjectured by Briët and Harremoës a decade ago (Properties of classical and quantum Jensen-Shannon divergence, Phy. Rev. A, 79, 052311 (2009)). A somewhat more intricate argument also establishes metric properties of Jensen-Renyí divergences (for α∈ (0,1)), and outlines a techniques for proving this property for symmetric divergences involving completely monotonic functions.