Degree of separability of bipartite quantum states

We investigate the problem of finding the optimal convex decomposition of a bipartite quantum state into a separable part and a positive remainder, in which the weight of the separable part is maximal. This weight is naturally identified with the degree of separability of the state. In a recent work, the problem was solved for two-qubit states using semidefinite programming. In this paper, we describe a procedure to obtain the optimal decomposition of a bipartite state of any finite dimension via a sequence of semidefinite relaxations. The sequence of decompositions thus obtained is shown to converge to the optimal one. This provides, for the first time, a systematic method to determine the so-called optimal Lewenstein-Sanpera decomposition of any bipartite state. Numerical results are provided to illustrate this procedure, and the special case of rank-2 states is also discussed.