Nonlinear power-like and SVD-like iterative schemes with applications to entangled bipartite rank-1 approximation

Gauging the distance between a mixed state and the convex set of separable states in a bipartite quantum mechanical system over the complex field is an important but challenging task. As a first step toward this difficult problem, this paper investigates the rank-1 approximation of a bipartite system over the real field where the entanglement is characterized in terms of the Kronecker product of density matrices. The approximation is recast in the form of a nonlinear eigenvalue problem and a nonlinear singular value problem for which two iterative methods are proposed, respectively. This study offers insight into and might serve as the building block for the more complicated multipartite systems and higher-rank approximation problems. The main focus is on the convergence analysis. Numerical experiments seem to suggest that these easily constructed solvers have higher efficiency when comparing with some state-of-the-art optimization techniques.