Operational criterion and constructive checks for the separability of low-rank density matrices - art. no. 032310

We consider low-rank density operators rho supported on a MXN Hilbert space for arbitrary M and N (M less than or equal to N), and with a positive par tial transpose (PPT) rho(TA) greater than or equal to 0. For rank r(rho)les s than or equal to N we prove that having a PPT is necessary and sufficient for rho to be separable; in this case we also provide its minimal decompos ition in terms of pure product states. It follows from this result that the re is no rank-3 bound entangled states having a PPT. We also present a nece ssary and sufficient condition for the separability of generic density matr ices for which the sum of the ranks of rho and rho(TA) satisfies r(rho) + r (rho(TA)) less than or equal to 2MN-M-N+2. This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a sys tem of couple polynomial equations has a finite number of solutions, as exp ected in most cases.