Chain rules for quantum channels

TL;DR

This paper derives quantum divergence chain rules for channels using matrix analysis, simplifying and generalizing previous results, and highlighting the data-processing inequality in quantum information theory.

Contribution

It introduces new Rénnyi divergence chain rules for quantum channels, improving upon prior derivations with simplified and more general results.

Findings

Derived several Rénnyi divergence chain rules for quantum channels

Simplified existing proofs and extended previous results

Reinforced the understanding of data-processing inequality in quantum context

Abstract

Divergence chain rules for channels relate the divergence of a pair of channel inputs to the divergence of the corresponding channel outputs. An important special case of such a rule is the data-processing inequality, which tells us that if the same channel is applied to both inputs then the divergence cannot increase. Based on direct matrix analysis methods, we derive several R\'enyi divergence chain rules for channels in the quantum setting. Our results simplify and in some cases generalise previous derivations in the literature.