A Mirror-Descent Algorithm for Computing the Petz-Rényi Capacity of Classical-Quantum Channels

Abstract

We study the computation of the $\hat{I}\pm$-Rényi capacity of a classical-quantum (c-q) channel for $\hat{I}\pm \in (0,1)$. 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.

This paper is independent and concurrent to arXiv:2601.06492