Correlation Detection and an Operational Interpretation of the Renyi Mutual Information

TL;DR

This paper provides an operational interpretation of quantum Renyi mutual information and conditional entropy through hypothesis testing, linking these measures to practical tasks in quantum channel coding and classical probability.

Contribution

It introduces an operational meaning for quantum Renyi measures in hypothesis testing, extending their relevance to quantum and classical information theory.

Findings

Renyi mutual information relates to composite hypothesis testing.

Operational interpretation of Renyi conditional entropy established.

Results apply to both quantum and classical distributions.

Abstract

A variety of new measures of quantum Renyi mutual information and quantum Renyi conditional entropy have recently been proposed, and some of their mathematical properties explored. Here, we show that the Renyi mutual information attains operational meaning 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 decoupled from the 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.