The Essentially Entangled Component of Multipartite Mixed Quantum States, its Properties and an Efficient Algorithm for its Extraction

We introduce with geometric means a density matrix decomposition of a multipartite quantum system of a finite dimension into two density matrices: a separable one, also known as the best separable approximation, and an essentially entangled one, which contains no product states components. We show that this convex decomposition can be achieved in practice with the help of an algorithm based on linear programming, which in the general case scales polynomially with the dimension of the multipartite system. Furthermore, we suggest methods for analyzing the multipartite entanglement content of the essentially entangled component and derive analytically an upper bound for its rank. We illustrate the algorithm at an example of a composed system of total dimension 12 undergoing loss of coherence due to classical noise and we trace the time evolution of its essentially entangled component. We suggest a "geometric" description of entanglement dynamics and show how it explains the well-known phenomena of sudden death and revival of multipartite entanglement.