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

Teng Zhang

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arXiv · 2602.08164

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

Teng Zhang

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Abstract

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.

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Teng Zhang. 2026-02-08. Metric Properties: From $S$-Divergence to Quantum Jensen Divergence. https://arxiv.org/abs/2602.08164

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