The doubly minimized Petz Rényi mutual information of order α is defined as the minimization of the Petz divergence of order α of a fixed bipartite quantum state relative to any product state. The doubly minimized sandwiched Rényi 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 Rényi mutual information. In particular, for the Petz case, we prove additivity for α ∈ [1/2, 2]. For the sandwiched case, we establish a novel duality relation for α ∈ [2/3,∞] via Sion’s minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of α. Previously, additivity for the sandwiched case was known only for α ∈ [1,∞], but it had been conjectured to hold for α ∈ [1/2,∞].

The doubly minimized Petz Rényi mutual information of order α is defined as the minimum of the Petz divergence of order α of a given bipartite quantum state relative to all product states. The doubly minimized sandwiched Rényi mutual information is defined analogously, with the Petz divergence replaced by the sandwiched divergence. In this work, we study certain binary quantum state discrimination problems related to correlation detection. We show that the corresponding direct exponent is determined by the doubly minimized Petz Rényi mutual information of order α ∈ (1/2, 1), and that the strong converse exponent is determined by the doubly minimized sandwiched Rényi mutual information of order α ∈ (1,∞). This provides an operational interpretation of these types of Rényi mutual information and generalizes previous results for classical probability distributions to the quantum setting. For completeness, we also study the corresponding moderate deviation regime both below and above the threshold, and determine the Stein exponent and the second-order asymptotics.