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

TL;DR

This paper extends the concept of $S$-divergence and quantum Jensen divergence from matrices to broader algebraic structures, establishing their metric properties and answering open questions in quantum information theory.

Contribution

It introduces a generalized framework for trace-logarithmic divergences in tracial $C^*$-algebras and von Neumann algebras, proving metricity and addressing open questions about Hilbertianity.

Findings

The square root of the extended $S$-divergence defines a metric on the positive cone.

The quantum Jensen--Shannon divergence's square root is a metric on the positive cone in the tracial setting.

Symmetric quantum Jensen divergences from non-affine operator convex functions are metrics, with a generalized Nevanlinna--Stieltjes representation.

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.