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arXiv:2502.06399

A Linearly Convergent Algorithm for Petz-Augustin Means with Applications to Computing Petz-Augustin Capacity and Equilibrium Prices in Fisher Markets

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

Published Feb 10, 2025• 0 citations
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Abstract

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. The first application concerns the Petz-Augustin 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. To compute this quantity, we propose the first iterative method with a non-asymptotic convergence guarantee, which uses our algorithm for computing the Petz-Augustin mean as a subroutine. The second application concerns the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities. We establish that when all buyers have CES utilities of common elasticity, the equilibrium prices are equivalent to the Petz-Augustin mean when all quantum states commute. In this case, our proposed algorithm is a tâtonnement process à la Walras. We further extend the algorithm to inhomogeneous Fisher markets, where buyers have CES utilities with different elasticities, and prove a faster convergence rate than existing tâtonnement-type algorithms.

DOI
10.48550/arxiv.2502.06399

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We study the computation of the Petz-Augustin mean of order α∈(0,1)∪(1,∞)α\in (0,1) \cup (1,\infty)α∈(0,1)∪(1,∞), defined as the minimizer of a weighted sum of nnn Petz-Rényi divergences of order ααα over the set of ddd-by-ddd 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(∣1−1/α∣T)O\left( \lvert 1 - 1/α\rvert^T \right)O(∣1−1/α∣T) with respect to the Thompson metric for α∈(1/2,1)∪(1,∞)α\in(1/2,1)\cup(1,\infty)α∈(1/2,1)∪(1,∞), where TTT denotes the number of iterations. The algorithm has an initialization time complexity of O(nd3)O\left(nd^3\right)O(nd3) and a per-iteration time complexity of O(nd2+d3)O\left(nd^2 + d^3\right)O(nd2+d3). Two applications follow. The first application concerns the Petz-Augustin capacity of order α∈(1/2,1)α\in(1/2,1)α∈(1/2,1), which generalizes the quantum channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. To compute this quantity, we propose the first iterative method with a non-asymptotic convergence guarantee, which uses our algorithm for computing the Petz-Augustin mean as a subroutine. The second application concerns the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities. We establish that when all buyers have CES utilities of common elasticity, the equilibrium prices are equivalent to the Petz-Augustin mean when all quantum states commute. In this case, our proposed algorithm is a tâtonnement process à la Walras. We further extend the algorithm to inhomogeneous Fisher markets, where buyers have CES utilities with different elasticities, and prove a faster convergence rate than existing tâtonnement-type algorithms.
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).
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.
The doubly minimized Petz Renyi mutual information (PRMI) of order ααα is defined as the minimization of the Petz divergence of order ααα of a fixed bipartite quantum state ρABρ_{AB}ρAB​ relative to any product state σA⊗τBσ_A\otimes τ_BσA​⊗τB​. To date, no closed-form expression for this measure has been found, necessitating the development of numerical methods for its computation. In this work, we show that alternating minimization over σAσ_AσA​ and τBτ_BτB​ asymptotically converges to the doubly minimized PRMI for any α∈(12,1)∪(1,2]α\in (\frac{1}{2},1)\cup (1,2]α∈(21​,1)∪(1,2], by proving linear convergence of the objective function values with respect to the number of iterations for α∈(1,2]α\in (1,2]α∈(1,2] and sublinear convergence for α∈(12,1)α\in (\frac{1}{2},1)α∈(21​,1). Previous studies have only addressed the specific case where ρABρ_{AB}ρAB​ is a classical-classical state, while our results hold for any quantum state ρABρ_{AB}ρAB​.
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.
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.
We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.