Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences

TL;DR

This paper systematically studies standard $f$-divergences in von Neumann algebras, extending key variational formulas and exploring properties of quantum divergences, including the Rényi divergence, with improvements on previous results.

Contribution

It extends Kosaki's variational expression to arbitrary standard $f$-divergences and provides a comprehensive analysis of quantum divergences in von Neumann algebras.

Findings

Extended variational formulas for $f$-divergences.

Established properties of quantum $f$-divergences.

Improved results on relative Hamiltonians.

Abstract

We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the R\'enyi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.