Extremal Geometric Measure of Entanglement and Riemannian Optimization Methods

In this paper, we study the extremal geometric measure of quantum entanglement of superposition states of several symmetric pure states with parameters. This problem is equivalent to calculate their minimum entanglement value or maximum entanglement value. We propose the block coordinate descent (BCD) method to calculate the minimum entanglement value and establish its convergence analysis. We propose a gradient projection ascent with min-oracle(GPAM) method to calculate the maximum entanglement value. The subproblems of these two problems are optimization problems with a complex unit spherical constraint. The complex unit sphere is a Riemannian manifold. We propose an inexact Riemannian Newton-CG method to solve this Riemannian optimization problem. The numerical examples are presented to demonstrate the effectiveness of the proposed methods.