Correlation detection and an operational interpretation of the Rényi mutual information

Abstract

Recently, a variety of new measures of quantum Renyi mutual information and quantum Renyi conditional entropy have been proposed, and some of their mathematical properties explored. Here, we show that the Renyi mutual information attains operational significance in the context of composite hypothesis testing, when the null hypothesis is a fixed bipartite state and the alternate hypothesis consists of all product states that share one marginal with the null hypothesis. This hypothesis testing problem occurs naturally in channel coding, where it corresponds to testing whether a state is the output of a given quantum channel or of a “useless” channel whose output is independent of the channel input and environment. Similarly, we establish an operational interpretation of Renyi conditional entropy by choosing an alternative hypothesis that consists of product states that are maximally mixed on one system. Specialized to classical probability distributions, our results also establish an operational interpretation of Renyi mutual information and Renyi conditional entropy.