Recovery and the Data Processing Inequality for quasi-entropies

TL;DR

This paper establishes quantitative stability bounds for equality cases in Petz's monotonicity theorem related to quasi-relative entropies, with implications for quantum information theory.

Contribution

It provides new, elementary stability bounds for quasi-relative entropies and extends previous results with more general bounds involving the Petz recovery map.

Findings

Quantitative bounds for equality in Petz's theorem

Bounds involving the Petz recovery map

Elementary treatment in finite-dimensional von Neumann algebras

Abstract

We prove number of quantitative stability bounds for the cases of equality in Petz's monotonicity theorem for quasi-relative entropies defined in terms of an operator monotone decreasing functions. Included in our results is a bound in terms of the Petz recovery map, but we obtain more general results. The present treatment is entirely elementary and developed in the context of finite dimensional von Neumann algebras where the results are already non-trivial and of interest in quantum information theory.