Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences

TL;DR

This paper systematically studies maximal $f$-divergences in von Neumann algebras, providing new integral and variational expressions, and establishing key properties such as monotonicity, convexity, and continuity.

Contribution

It introduces general integral and variational formulas for maximal $f$-divergences in von Neumann algebras, extending previous work on standard $f$-divergences.

Findings

Established monotonicity inequality for maximal $f$-divergences

Proved joint convexity and lower semicontinuity

Derived the inequality between standard and maximal $f$-divergences

Abstract

As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerup's $L^1$-space, we present the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.