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

We prove decomposition rules for quantum R'enyi mutual information, generalising the relation $I(A:B)=H(A)-H(A\mathrm{\mid }B)$ 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.