Minimizing Quantum Renyi Divergences via Mirror Descent with Polyak Step Size

TL;DR

This paper introduces a novel mirror descent method with Polyak step size for efficiently computing quantum information quantities involving quantum Renyi divergences, overcoming limitations of existing convex optimization techniques.

Contribution

It proposes a new convex optimization approach with convergence guarantees for minimizing quantum Renyi divergences, applicable to quantum information theory problems.

Findings

The method converges faster than traditional algorithms.

It successfully computes various quantum information quantities.

The approach is theoretically proven to converge under weak conditions.

Abstract

Quantum information quantities play a substantial role in characterizing operational quantities in various quantum information-theoretic problems. We consider numerical computation of four quantum information quantities: Petz-Augustin information, sandwiched Augustin information, conditional sandwiched Renyi entropy and sandwiched Renyi information. To compute these quantities requires minimizing some order-$\alpha$ quantum Renyi divergences over the set of quantum states. Whereas the optimization problems are obviously convex, they violate standard bounded gradient/Hessian conditions in literature, so existing convex optimization methods and their convergence guarantees do not directly apply. In this paper, we propose a new class of convex optimization methods called mirror descent with the Polyak step size. We prove their convergence under a weak condition, showing that they provably converge for minimizing quantum Renyi divergences. Numerical experiment results show that entropic mirror descent with the Polyak step size converges fast in minimizing quantum Renyi divergences.