Algorithms for Computing the Petz-Augustin Capacity

TL;DR

This paper introduces the first algorithms with provable convergence guarantees for computing the Petz-Augustin capacity, a key quantity in quantum information theory related to channel capacity and error exponents.

Contribution

It develops novel algorithms for Petz-Augustin capacity computation, including a convex optimization approach for Petz-Rényi information and a fixed-point method for Petz-Augustin information.

Findings

Algorithms have non-asymptotic convergence guarantees.

The Petz-Augustin information maximization uses a novel fixed-point algorithm.

The approach generalizes the mirror-descent interpretation of the Blahut-Arimoto algorithm.

Abstract

We propose the first algorithms with non-asymptotic convergence guarantees for computing the Petz-Augustin capacity, which generalizes the channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. This capacity can be equivalently expressed as the maximization of two generalizations of mutual information: the Petz-R\'{e}nyi information and the Petz-Augustin information. To maximize the Petz-R\'{e}nyi information, we show that it corresponds to a convex H\"{o}lder-smooth optimization problem, and hence the universal fast gradient method of Nesterov (2015), along with its convergence guarantees, readily applies. Regarding the maximization of the Petz-Augustin information, we adopt a two-layered approach: we show that the objective function is smooth relative to the negative Shannon entropy and can be efficiently optimized by entropic mirror descent; each iteration of entropic mirror descent requires computing the Petz-Augustin information, for which we propose a novel fixed-point algorithm and establish its contractivity with respect to the Thompson metric. Notably, this two-layered approach can be viewed as a generalization of the mirror-descent interpretation of the Blahut-Arimoto algorithm due to He et al. (2024).