Trace- and improved data processing inequalities for von Neumann algebras

TL;DR

This paper extends data-processing inequalities for relative entropy to general von Neumann algebras, providing bounds on quantum state recoverability and generalizing multi-trace inequalities, with applications in quantum field theory.

Contribution

It introduces a version of the data-processing inequality for von Neumann algebras with explicit bounds, generalizing finite-dimensional results to infinite-dimensional quantum systems.

Findings

Established a lower bound involving measured relative entropy.

Generalized multi-trace inequalities to all von Neumann algebras.

Applied results to quantum field theory contexts.

Abstract

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite dimensional density matrices, yields a bound how well a quantum state can be recovered after it has been passed through a channel. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.