Revisiting the equality conditions of the data processing inequality for the sandwiched R\'enyi divergence

TL;DR

This paper offers a clear and unified analysis of the conditions under which the data processing inequality for sandwiched quantum Re9nyi divergence becomes equality, including recoverability implications and new generalized conditions.

Contribution

It provides a simplified, comprehensive treatment of equality conditions for DPI in sandwiched Re9nyi divergence, extending previous results with new generalized criteria.

Findings

Equality in DPI implies recoverability via Petz map for all b5

New generalized equality conditions for sandwiched Re9nyi divergence

Unified framework for recent and new results on DPI equality

Abstract

We provide a transparent, simple and unified treatment of recent results on the equality conditions for the data processing inequality (DPI) of the sandwiched quantum R\'enyi divergence, including the statement that equality in the data processing implies recoverability via the Petz recovery map for the full range of $\alpha$ recently proven by Jen\v cov\'a. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al.