Alternating minimization for computing doubly minimized Petz Renyi mutual information

TL;DR

This paper introduces an alternating minimization algorithm to compute the doubly minimized Petz Renyi mutual information for quantum states, proving its convergence for a broad range of parameters and extending previous classical results to general quantum states.

Contribution

It develops and proves the convergence of an alternating minimization method for calculating doubly minimized PRMI for any quantum state, filling a gap in numerical approaches.

Findings

Convergence of the algorithm is linear for α in (1,2].

Convergence is sublinear for α in (1/2,1).

Results extend previous classical state findings to all quantum states.

Abstract

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 $\rho_{AB}$.