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arXiv:2601.10558 · IEEE International Symposium on Information Theory (ISIT)

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

Yu-Hong Lai, Hao-Chung Cheng

quant-phcs.ITmath.OCSubmitted Jan 15, 2026• Announced Jan 16, 2026• 1 citation
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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.

Comments
This paper is independent and concurrent to arXiv:2601.06492

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We study the computation of the ααα-Rényi capacity of a classical-quantum (c-q) channel for α∈(0,1)α\in(0,1)α∈(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.
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).
The Blahut-Arimoto algorithm is a well-known method to compute classical channel capacities and rate-distortion functions. Recent works have extended this algorithm to compute various quantum analogs of these quantities. In this paper, we show how these Blahut-Arimoto algorithms are special instances of mirror descent, which is a type of Bregman proximal method, and a well-studied generalization of gradient descent for constrained convex optimization. Using recently developed convex analysis tools, we show how analysis based on relative smoothness and strong convexity recovers known sublinear and linear convergence rates for Blahut-Arimoto algorithms. This Bregman proximal viewpoint allows us to derive related algorithms with similar convergence guarantees to solve problems in information theory for which Blahut-Arimoto-type algorithms are not directly applicable. We apply this framework to compute energy-constrained classical and quantum channel capacities, classical and quantum rate-distortion functions, and approximations of the relative entropy of entanglement, all with provable convergence guarantees.
Quantum information quantities play a substantial role in characterizing operational quantities in various quantum information-theoretic problems. We consider the numerical computation of four quantum information quantities: Petz-Augustin information, sandwiched Augustin information, conditional sandwiched Rényi entropy, and sandwiched Rényi information. To compute these quantities requires minimizing some order-α quantum Rényi divergences over the set of quantum states. Whereas the optimization problems are obviously convex, they violate standard bounded gradient/Hessian conditions in the literature, so existing convex optimization methods and their convergence guarantees do not directly apply. In this paper, we propose a new class of convex optimization methods called mirror descent with the Polyak step size. We prove their convergence under a weak condition, showing that they provably converge for minimizing quantum Rényi divergences. Numerical experiment results show that entropic mirror descent with the Polyak step size converges fast in minimizing quantum Rényi divergences.
In this work, we investigate the convergence rate of the quantum Blahut-Arimoto (BA) algorithm proposed for computing the coherent information of less noisy quantum channels. Previous work established exponential convergence under certain restrictive assumptions on the channel. Using a Taylor expansion around a positive definite optimizer, we show that, for an injective degradable channel whose degrading channel has a relative-entropy contraction coefficient strictly less than one, the algorithm converges at an exponential rate after sufficiently many iterations. We further verify this condition for qubit amplitude-damping (AD) channels in the relevant parameter regime. In addition, by analyzing the Bogoliubov-Kubo-Mori (BKM) metric on the Bloch ball, we derive an explicit characterization of the relative-entropy contraction coefficient of qubit AD channels. The proposed approach is further extended to calculate the relative contraction coefficient and the relative expansion coefficient of two qubit AD channels.
Quantum information quantities play a substantial role in characterizing operational quantities in various quantum information-theoretic problems. We consider numerical computation of four quantum information quantities: Petz-Augustin information, sandwiched Augustin information, conditional sandwiched Renyi entropy and sandwiched Renyi information. To compute these quantities requires minimizing some order-ααα quantum Renyi divergences over the set of quantum states. Whereas the optimization problems are obviously convex, they violate standard bounded gradient/Hessian conditions in literature, so existing convex optimization methods and their convergence guarantees do not directly apply. In this paper, we propose a new class of convex optimization methods called mirror descent with the Polyak step size. We prove their convergence under a weak condition, showing that they provably converge for minimizing quantum Renyi divergences. Numerical experiment results show that entropic mirror descent with the Polyak step size converges fast in minimizing quantum Renyi divergences.
We develop and analyze an entropic descent method for a nonconvex optimization problem on probability simplices arising in classical and quantum secure communication. The optimization variables describe a randomized encoder, allowing its input distribution and stochastic preprocessing to be optimized jointly. An exact relative-entropy identity yields a channel-independent majorization bound and an explicit mirror-descent iteration with a fixed unit step size. We establish quantitative descent, convergence of objective values, and stationarity of accumulation points on their support faces for arbitrary finite classical channels, without assuming degradedness or less noisiness. For strictly positive transition matrices, a square-root parametrization and a Kurdyka--Łojasiewicz argument establish finite length of the transformed trajectory and convergence of the entire sequence, including limits on the simplex boundary. For less-noisy channels, a restricted convex formulation admits global convergence with an explicit O(1/t)O(1/t)O(1/t) objective-error bound. We extend the relative-entropy construction to noncommuting quantum outputs and derive monotone alternating updates for covariant encoders, including correlated preprocessing over a fixed block. Numerical experiments compare the method with analytic benchmarks and constrained optimization, reveal multiple attracting values, and demonstrate strict improvements from stochastic preprocessing.