Chun-Neng Chu | Searcharxiv

arXiv2026

Algorithms for Computing the Petz-Augustin Capacity

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-Rényi information and the Petz-Augustin information. To maximize the Petz-Rényi information, we show that it corresponds to a convex Hölder-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).

cs.IT↗

arXiv2025

A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean

We study the computation of the Petz-Augustin mean of order $α\in (0,1) \cup (1,\infty)$, defined as the minimizer of a weighted sum of $n$ Petz-Rényi divergences of order $α$ over the set of $d$-by-$d$ quantum states, where the Petz-Rényi divergence is a quantum generalization of the classical Rényi divergence. We propose the first algorithm with a non-asymptotic convergence guarantee for solving this optimization problem. The iterates are guaranteed to converge to the Petz-Augustin mean at a linear rate of $ O\left( \lvert 1 - 1/α\rvert^T \right) $ with respect to the Thompson metric for $α\in(1/2,1)\cup(1,\infty)$, where $ T $ denotes the number of iterations. The algorithm has an initialization time complexity of $O\left(nd^3\right)$ and a per-iteration time complexity of $O\left(nd^2 + d^3\right)$. Two applications follow. First, we propose the first iterative method with a non-asymptotic convergence guarantee for computing the Petz capacity of order $α\in(1/2,1)$, which generalizes the quantum channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. Second, we establish that the Petz-Augustin mean of order $α$, when all quantum states commute, is equivalent to the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities of common elasticity $ρ=1-1/α$, and our proposed algorithm can be interpreted as a tâtonnement dynamic. We then extend the proposed algorithm to inhomogeneous Fisher markets, where buyers have different elasticities, and prove that it achieves a faster convergence rate compared to existing tâtonnement-type algorithms.

quant-ph↗