Quantum $f$-divergences via Nussbaum-Szko{\l}a Distributions in Semifinite von Neumann Algebras

TL;DR

This paper extends the equivalence of quantum and classical $f$-divergences, via Nussbaum-Szko{ }a distributions, from bounded operator algebras to all semifinite von Neumann algebras.

Contribution

It generalizes previous results by proving the equivalence for normal states on any semifinite von Neumann algebra.

Findings

Quantum $f$-divergence equals classical $f$-divergence for these states.

Extension from $ ext{B}( ext{H})$ to all semifinite von Neumann algebras.

Provides a broader framework for quantum-classical divergence correspondence.

Abstract

In this article, we prove that the quantum $f$-divergence between two normal states on a semifinite von~Neumann algebra is equal to the classical $f$-divergence between two corresponding classical states, which are called Nussbaum-Szko{\l}a distributions. This result has been proved by the second named author and T.C.~John for normal states on the von~Neumann algebra $\mathbb{B}(\mathscr{H})$ of all bounded operators on a Hilbert space $\mathscr{H}$. We extend their result for normal states on any semifinite von~Neumann algebra, not only $\mathbb{B}(\mathscr{H})$.