Alternating minimization for computing doubly minimized Petz Renyi mutual information

The doubly minimized Petz Renyi mutual information (PRMI) of order $α$ is defined as the minimization of the Petz divergence of order $α$ of a fixed bipartite quantum state $ρ_{AB}$ relative to any product state $σ_A\otimes τ_B$. 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 $σ_A$ and $τ_B$ asymptotically converges to the doubly minimized PRMI for any $α\in (\frac{1}{2},1)\cup (1,2]$, by proving linear convergence of the objective function values with respect to the number of iterations for $α\in (1,2]$ and sublinear convergence for $α\in (\frac{1}{2},1)$. Previous studies have only addressed the specific case where $ρ_{AB}$ is a classical-classical state, while our results hold for any quantum state $ρ_{AB}$.