Doubly minimized Petz and sandwiched Renyi mutual information: Properties

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. The doubly minimized sandwiched Renyi mutual information is defined analogously using the sandwiched divergence in place of the Petz divergence. In this work, we establish several properties of these two types of Renyi mutual information. In particular, for the Petz case, we prove additivity for $\alpha\in [1/2,2]$. For the sandwiched case, we establish a novel duality relation for $\alpha\in [2/3,\infty]$ via Sion's minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of $\alpha$. Previously, additivity for the sandwiched case was known only for $\alpha\in [1,\infty]$, but it had been conjectured to hold for $\alpha\in [1/2,\infty]$.