Algorithms for Computing the Petz-Augustin Capacity

2026-01-10

arXiv.org

AI Summary

This work addresses the efficient computation of the Petz-Augustin capacity, a generalization of classical channel capacity that characterizes the optimal error exponent in classical-quantum channel coding. The authors propose the first algorithmic framework with non-asymptotic convergence guarantees for the associated maximization problems of Petz-Rényi and Petz-Augustin information. Their approach integrates an accelerated gradient method, mirror descent based on negative Shannon entropy, and a novel fixed-point iteration. By leveraging contractivity analysis under the Thompson metric and convex Hölder-smooth optimization theory, they establish a generalized connection between their algorithm and the mirror descent interpretation of the Blahut-Arimoto algorithm. The resulting method enables efficient numerical evaluation of the Petz-Augustin capacity, offering a theoretically grounded and practical computational tool for analyzing classical-quantum channel performance.

Technology Category

Machine Learning: Information TheorySearch and Optimization: Non-convex OptimizationReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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).

Problem

• Petz-Augustin capacity
• classical-quantum channel coding
• channel capacity
• error exponent
• mutual information

Innovation

• Petz-Augustin capacity
• entropic mirror descent
• fixed-point algorithm
• non-asymptotic convergence
• Thompson metric