Abstract
We study the computation of the Petz-Augustin mean of order α ∈ (0,1) ∪ (1,∞), defined as the minimizer of a weighted sum of n Petz-R\'enyi divergences of order α over the set of d-by-d quantum states, where the Petz-R\'enyi divergence is a quantum generalization of the classical R\'enyi 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 ) with respect to the Thompson metric for α∈(1/2,1)∪(1,∞), where T denotes the number of iterations. The algorithm has an initialization time complexity of O(nd³) and a per-iteration time complexity of O(nd² + d³). Two applications follow. First, we propose the first iterative method with a non-asymptotic convergence guarantee for computing the Petz capacity of order α∈(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 t\^{a}tonnement 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 t\^{a}tonnement-type algorithms.