A Linearly Convergent Algorithm for Petz-Augustin Means with Applications to Computing Petz-Augustin Capacity and Equilibrium Prices in Fisher Markets
The paper introduces a new iterative algorithm to find the Petz-Augustin mean Q⋆, which is crucial for defining quantum means. This method provides the first non-asymptotic convergence guarantee, showing linear convergence at a rate of O|1 - 1/α|^T under the Thompson metric for various orders α. It also applies this technique to compute Petz capacity and find equilibrium prices in specific economic models.