An alternative steepest direction method for the optimization in evaluating geometric discord

We address the evaluation of geometric discord of bipartite quantum states arising from quantum information theory. The problem corresponds to finding the best approximation of the orthogonal decomposition of a partially Hermite fourth-order tensor. By discussing the optimality condition of the problem, we reduce it to a homogenous polynomial optimization problem on the product of two unitary matrices. Based on the Riemannian manifold and Lie group theory, we propose an alternative steepest direction method for the problem. Numerical experiments show the efficiency of the method.