R\'enyi divergences as weighted non-commutative vector valued $L_p$-spaces

TL;DR

This paper links weighted non-commutative vector valued $L_p$-spaces to sandwiched Rényi divergences, establishing their fundamental properties in von Neumann algebras and providing new proofs for finite-dimensional cases.

Contribution

It generalizes sandwiched Rényi divergences to von Neumann algebras using complex interpolation, and introduces new inequalities and theorems for these spaces.

Findings

Proves data-processing inequality and monotonicity in $\alpha$ for divergences.

Derives a Riesz-Thorin theorem for Araki-Masuda $L_p$-spaces.

Establishes an Araki-Lieb-Thirring inequality for states on von Neumann algebras.

Abstract

We show that Araki and Masuda's weighted non-commutative vector valued $L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched R\'enyi divergences with parameter $\alpha = \frac{p}{2}$. Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in $\alpha$. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $\alpha\to \{\frac{1}{2},1,\infty\}$ leading to minus the logarithm of Uhlmann's fidelity, Umegaki's relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz-Thorin theorem for Araki-Masuda $L_p$-spaces and an Araki-Lieb-Thirring inequality for states on von Neumann algebras.