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.Comment: 11 pages, 7 figures, submitted to PR