R\'enyi divergence inequalities via interpolation, with applications to generalised entropic uncertainty relations

Abstract

We investigate quantum R\'enyi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed R\'enyi generalisations of the quantum relative entropy. We may define R\'enyi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of R\'enyi order which replicate the chain and decomposition rules of Shannon and von Neumann entropies. Although weaker in general, these inequalities recover equivalence when the R\'enyi parameters tend to one. To this end we present R\'enyi mutual information decomposition rules, a new approach to the R\'enyi conditional entropy tripartite chain rules and a more general bipartite comparison. The derivation of these results relies on a novel complex interpolation approach for general spaces of linear operators. These new comparisons allow us to employ techniques that until now were only available for Shannon and von Neumann entropies. We can therefore directly apply them to the derivation of R\'enyi entropic uncertainty relations. Accordingly, we establish a family of R\'enyi information exclusion relations and provide further generalisations and improvements to this and other known relations, including the R\'enyi bipartite uncertainty relations.