  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.
