A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean

We study the computation of the Petz-Augustin mean of order $\alpha \in (0,1) \cup (1,\infty)$, defined as the minimizer of a weighted sum of $n$ Petz-R\'enyi divergences of order $\alpha$ 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 $O\left( \lvert 1 - 1/\alpha \rvert^T \right)$ with respect to the Thompson metric for $\alpha\in(1/2,1)\cup(1,\infty)$, 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. First, we propose the first iterative method with a non-asymptotic convergence guarantee for computing the Petz capacity of order $\alpha\in(1/2,1)$, 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 $\alpha$, when all quantum states commute, is equivalent to the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities of common elasticity $\rho=1-1/\alpha$, 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.