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

Yu-Hong Lai, Hao-Chung Cheng

TL;DR

This work tackles the computation of the Petz–Rényi capacity $C_\alpha(W)$ for classical-quantum channels with $\alpha\in(0,1)$ by recasting it as minimizing a trace-power objective $S(p)=\mathrm{Tr}[M(p)^{\beta}]$ over the simplex, where $M(p)=\sum_x p_x W_x^{\alpha}$ and $\beta=1/\alpha>1$. The authors develop an exponentiated-gradient mirror-descent algorithm using the negative-entropy map, yielding a simple constant-stepsize update and a truncation safeguard to keep iterates in the interior. They establish sharp Hessian-based bounds that yield entropy-relative smoothness and global sublinear convergence, and, under a tangent-space nondegeneracy condition plus a mild spectral lower bound, prove local linear convergence in KL divergence on a truncated simplex. Numerical experiments on random noncommuting c-q channels confirm the capacity trends across $\alpha$, provide a convergence certificate via a duality gap, and illustrate scaling with problem size. Overall, the paper delivers both theoretical guarantees and practical algorithms for accurate Petz–Rényi capacity computation in finite-dimensional c-q settings.

Abstract

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

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

TL;DR

This work tackles the computation of the Petz–Rényi capacity for classical-quantum channels with by recasting it as minimizing a trace-power objective over the simplex, where and . The authors develop an exponentiated-gradient mirror-descent algorithm using the negative-entropy map, yielding a simple constant-stepsize update and a truncation safeguard to keep iterates in the interior. They establish sharp Hessian-based bounds that yield entropy-relative smoothness and global sublinear convergence, and, under a tangent-space nondegeneracy condition plus a mild spectral lower bound, prove local linear convergence in KL divergence on a truncated simplex. Numerical experiments on random noncommuting c-q channels confirm the capacity trends across , provide a convergence certificate via a duality gap, and illustrate scaling with problem size. Overall, the paper delivers both theoretical guarantees and practical algorithms for accurate Petz–Rényi capacity computation in finite-dimensional c-q settings.

Abstract

We study the computation of the -Rényi capacity of a classical-quantum (c-q) channel for . 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.
Paper Structure (18 sections, 12 theorems, 46 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 12 theorems, 46 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and Convexity
  3. Convex Reformulation
  4. The Algorithm
  5. Mirror-descent update and closed form
  6. Truncated-simplex safeguard (iterates stay in $\Delta_\delta$)
  7. Constant stepsize selection
  8. Analysis: Hessian and Kernel Bounds
  9. Exact Directional Hessian
  10. Kernel Bounds
  11. Relative Smoothness and Global Convergence
  12. Relative Strong Convexity and Linear Rate
  13. Numerical Experiments
  14. Experimental setup
  15. Capacity sweep and runtime profile
  16. ...and 3 more sections

Key Result

Lemma 1

For $\beta\ge 1$, the map $p\mapsto S(p)=\operatorname{Tr}\left[M(p)^\beta\right]$ is convex on $\Delta$.

Figures (2)

  • Figure 1: Evaluation on a noncommuting channel ($|\mathcal{X}|=10$, $d=6$). (Left) Estimated Petz--Rényi capacity $C_\alpha(W)$ versus $\alpha$. (Center) Wall-clock runtime versus $\alpha$. (Right) Iterations-to-stop versus $\alpha$.
  • Figure 2: Convergence traces (gap) for $\alpha\in\{0.2,0.5,0.8\}$. The gap $g(p^t)$ provides a first-order stationarity certificate on the simplex.

Theorems & Definitions (24)

  • Remark 1: Regularity and the role of $\delta$
  • Lemma 1: Convexity Car09
  • Remark 2: Capacity via $S$
  • Lemma 2: Directional Hessian
  • proof
  • Lemma 3: Integral Representation
  • proof
  • Lemma 4: Uniform Upper Bound
  • proof
  • Lemma 5: Lower Bound
  • ...and 14 more