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Algorithms for Computing the Petz-Augustin Capacity

Chun-Neng Chu, Wei-Fu Tseng, Yen-Huan Li

TL;DR

This work addresses the non-asymptotic computation of the Petz-Augustin capacity $C_{\mathcal{W},\alpha}$ for classical-quantum channels by casting it as the maximization of two generalized mutual informations, $I_{\alpha}^{\text{R}}$ and $I_{\alpha}^{\text{A}}$. It introduces two algorithmic pipelines: (i) a Hölder-smooth optimization approach for $I_{\alpha}^{\text{R}}$ based on a convex reformulation $\tilde{g}_{\alpha}^{\text{R}}$ and Nesterov's universal fast gradient method, achieving a rate $O\left(\log(n)^{0.5/\alpha} T^{1-1.5/\alpha}\right)$ for $\alpha\in[1/2,1)$; and (ii) a Blahut-Arimoto-type nested scheme for $I_{\alpha}^{\text{A}}$ with an inner fixed-point solver that contracts in the Thompson metric and an outer entropic mirror-descent loop with rate $O\left(\log(n)/T\right)$ for $\alpha\in(1/2,1)$. The inner problem is analyzed for contractivity and efficiency, and the outer loop leverages relative smoothness to guarantee non-asymptotic convergence, with practical performance demonstrated on numerical tests. These results provide the first non-asymptotic guarantees for computing the Petz-Augustin capacity in the quantum setting and offer tools for constrained-input scenarios via PDHG, strengthening the operational relevance of quantum channel error exponents.

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-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).

Algorithms for Computing the Petz-Augustin Capacity

TL;DR

This work addresses the non-asymptotic computation of the Petz-Augustin capacity for classical-quantum channels by casting it as the maximization of two generalized mutual informations, and . It introduces two algorithmic pipelines: (i) a Hölder-smooth optimization approach for based on a convex reformulation and Nesterov's universal fast gradient method, achieving a rate for ; and (ii) a Blahut-Arimoto-type nested scheme for with an inner fixed-point solver that contracts in the Thompson metric and an outer entropic mirror-descent loop with rate for . The inner problem is analyzed for contractivity and efficiency, and the outer loop leverages relative smoothness to guarantee non-asymptotic convergence, with practical performance demonstrated on numerical tests. These results provide the first non-asymptotic guarantees for computing the Petz-Augustin capacity in the quantum setting and offer tools for constrained-input scenarios via PDHG, strengthening the operational relevance of quantum channel error exponents.

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-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).
Paper Structure (20 sections, 10 theorems, 23 equations, 2 figures, 1 algorithm)

This paper contains 20 sections, 10 theorems, 23 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Hölder-Smooth Optimization
  5. Blahut-Arimoto-Type Algorithm
  6. Related Work
  7. Preliminaries
  8. Notations
  9. Mirror Descent
  10. Relative Smoothness
  11. Hölder Smoothness
  12. Maximizing the Petz-Rényi Information
  13. Maximizing the Petz-Augustin Information
  14. Inner Subproblem: Computing the Petz-Augustin Information
  15. Outer Subproblem: Maximizing the Petz-Augustin Information
  16. ...and 5 more sections

Key Result

Lemma 3.1

(Lu2018) Consider the problem of minimizing a convex function $f$ over the set $\mathcal{C}$, where $f$ is $L$-smooth relative to a convex function $h$ on $\mathcal{C}$. Suppose that an iterative algorithm computes the next iterate by $x_{t+1} = T_{h,\mathcal{C}}(x_t, \nabla f(x_t), 1/L)$, with $x_1

Figures (2)

  • Figure 1: Computing Petz-Augustin capacity for order $\alpha = 0.6$.
  • Figure 2: Computing Petz-Augustin capacity for order $\alpha = 0.9$.

Theorems & Definitions (13)

  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2: Nesterov2015
  • Remark 4.3
  • Corollary 4.4
  • Remark 4.5
  • Lemma 5.1: Chu2025
  • Lemma 5.2: Chu2025
  • Lemma 5.3: Chu2025
  • Theorem 5.4: Chu2025
  • ...and 3 more