Decomposition Rules for Quantum R\'enyi Mutual Information with an Application to Information Exclusion Relations

TL;DR

This paper develops decomposition rules for quantum Re9nyi mutual information, extending classical relations to inequalities between different Re9nyi entropies, and applies these to establish generalized information exclusion relations.

Contribution

It introduces new decomposition rules for quantum Re9nyi mutual information and applies them to derive generalized information exclusion relations.

Findings

Decomposition rules generalize classical mutual information relations.

Established an information exclusion relation for Re9nyi mutual information.

Utilized advanced interpolation techniques in the proof.

Abstract

We prove decomposition rules for quantum R\'enyi mutual information, generalising the relation $I(A:B) = H(A) - H(A|B)$ to inequalities between R\'enyi mutual information and R\'enyi entropy of different orders. The proof uses Beigi's generalisation of Reisz-Thorin interpolation to operator norms, and a variation of the argument employed by Dupuis which was used to show chain rules for conditional R\'enyi entropies. The resulting decomposition rule is then applied to establish an information exclusion relation for R\'enyi mutual information, generalising the original relation by Hall.