Metric Properties: From $S$-Divergence to Quantum Jensen Divergence

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Bibliographic Details
Main Author:	Zhang, Teng
Format:	Preprint
Published:	2026
Subjects:	Operator Algebras 46L10, 46L30, 15A42, 94A17
Online Access:	Acceder al recurso
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_version_	1866918328584372224
author	Zhang, Teng
author_facet	Zhang, Teng
contents	We extend the trace-logarithmic $S$-divergence from matrices to tracial $C^*$-algebras and finite von Neumann algebras, and show that its square root defines a metric on the invertible positive cone. We also prove an integral representation of the quantum Jensen--Shannon divergence in terms of shifted trace-log distances, implying metricity of its square root on the full positive cone in the same tracial framework. In the matrix case, we answer two questions of Virosztek \cite{Vir21} on Hilbertianity. Finally, we show that symmetric quantum Jensen divergences generated by non-affine operator convex functions yield metrics in the tracial setting via a Nevanlinna--Stieltjes type representation of the derivative, which generalizes a result of Carlen, Lieb and Seiringer.
format	Preprint
id	arxiv_https___arxiv_org_abs_2602_08164
institution	arXiv
publishDate	2026
record_format	arxiv
spellingShingle	Metric Properties: From $S$-Divergence to Quantum Jensen Divergence Zhang, Teng Operator Algebras 46L10, 46L30, 15A42, 94A17 We extend the trace-logarithmic $S$-divergence from matrices to tracial $C^*$-algebras and finite von Neumann algebras, and show that its square root defines a metric on the invertible positive cone. We also prove an integral representation of the quantum Jensen--Shannon divergence in terms of shifted trace-log distances, implying metricity of its square root on the full positive cone in the same tracial framework. In the matrix case, we answer two questions of Virosztek \cite{Vir21} on Hilbertianity. Finally, we show that symmetric quantum Jensen divergences generated by non-affine operator convex functions yield metrics in the tracial setting via a Nevanlinna--Stieltjes type representation of the derivative, which generalizes a result of Carlen, Lieb and Seiringer.
title	Metric Properties: From $S$-Divergence to Quantum Jensen Divergence
topic	Operator Algebras 46L10, 46L30, 15A42, 94A17
url	https://arxiv.org/abs/2602.08164

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