A Mirror-Descent Algorithm for Computing the Petz-R\'enyi Capacity of Classical-Quantum Channels

TL;DR

This paper introduces a mirror descent algorithm to compute the Petz-Rényi capacity of classical-quantum channels, extending the Blahut-Arimoto method with proven convergence guarantees.

Contribution

It develops a generalized exponentiated-gradient algorithm for Petz-Rényi capacity calculation, providing convergence analysis and conditions for linear convergence.

Findings

Global sublinear convergence of the proposed algorithm.

Local linear convergence under certain conditions.

Explicit contraction factor in the convergence analysis.

Abstract

We study the computation of the $\alpha$-R\'enyi capacity of a classical-quantum (c-q) channel for $\alpha\in(0,1)$. We propose an exponentiated-gradient (mirror descent) iteration that generalizes the Blahut-Arimoto algorithm. Our analysis establishes relative smoothness with respect to the entropy geometry, guaranteeing a global sublinear convergence of the objective values. Furthermore, under a natural tangent-space nondegeneracy condition (and a mild spectral lower bound in one regime), we prove local linear (geometric) convergence in Kullback-Leibler divergence on a truncated probability simplex, with an explicit contraction factor once the local curvature constants are bounded.