Geometric measures of entanglement in multipartite pure states via complex-valued neural networks

The geometric measure of entanglement of a multipartite pure state is defined it terms of its geometric distance from the set of separable pure states. The quantum eigenvalue problem is derived to compute the separable pure state nearest to the given multipartite pure state. Computing the modulus largest quantum eigenvalue for a multipartite pure state is equivalent to finding the best complex rank-one approximation of the complex unit tensors, associated with the multipartite pure states. This paper is devoted to present a complex-valued neural networks approach for the computation of the quantum eigenvalue problem for multipartite pure states. We design the neural networks for computing the best rank-one tensor approximation of complex tensors, and prove that the solution of the networks is locally asymptotically stable in the sense of Lyapunov stability theory. This solution also converges to the local optimal solutions of the best complex rank-one tensor approximation. We illustrate our theoretical results via numerical simulations.