A fixed-point algorithm for matrix projections with applications in quantum information

TL;DR

This paper introduces a fast fixed-point iterative algorithm for matrix projection under the Bures distance, with applications in quantum information, demonstrating exponential convergence and efficiency over standard solvers.

Contribution

It presents a novel fixed-point algorithm for matrix projections in quantum information, with proven exponential convergence and practical advantages over existing methods.

Findings

Algorithm converges exponentially fast.

Numerical results show superior speed compared to SDP solvers.

Applicable to quantum resource theories and Shannon theory.

Abstract

We develop a fixed-point iterative algorithm that computes the matrix projection with respect to the Bures distance on the set of positive definite matrices that are invariant under some symmetry. We prove that the fixed-point iteration algorithm converges exponentially fast to the optimal solution in the number of iterations. Moreover, it numerically shows fast convergence compared to the off-the-shelf semidefinite program solvers. Our algorithm, for the specific case of Bures-Wasserstein barycenter, recovers the fixed-point iterative algorithm originally introduced in (\'Alvarez-Esteban et al., 2016). Our proof is concise and relies solely on matrix inequalities. Finally, we discuss several applications of our algorithm in quantum resource theories and quantum Shannon theory.