  The doubly minimized Petz Renyi mutual information of order $\alpha$ is
defined as the minimization of the Petz divergence of order $\alpha$ of a fixed
bipartite quantum state relative to any product state. In this work, we
establish several properties of this type of Renyi mutual information,
including its additivity for $\alpha\in [1/2,2]$. As an application, we show
that the direct exponent of certain binary quantum state discrimination
problems is determined by the doubly minimized Petz Renyi mutual information of
order $\alpha\in (1/2,1)$. This provides an operational interpretation of this
type of Renyi mutual information, and generalizes a previous result for
classical probability distributions to the quantum setting.
