Alternating minimization for computing doubly minimized Petz Renyi mutual information

The doubly minimized Petz Renyi mutual information (PRMI) of order $\alpha$ is defined as the minimization of the Petz divergence of order $\alpha$ of a fixed bipartite quantum state $\rho_{AB}$ relative to any product state $\sigma_A\otimes \tau_B$. To date, no closed-form expression for this measure has been found, necessitating the development of numerical methods for its computation. In this work, we show that alternating minimization over $\sigma_A$ and $\tau_B$ asymptotically converges to the doubly minimized PRMI for any $\alpha\in (\frac{1}{2},1)\cup (1,2]$, by proving linear convergence of the objective function values with respect to the number of iterations for $\alpha\in (1,2]$ and sublinear convergence for $\alpha\in (\frac{1}{2},1)$. Previous studies have only addressed the specific case where $\rho_{AB}$ is a classical-classical state, while our results hold for any quantum state $\rho_{AB}$.