Operational criterion and constructive checks for the separability of low-rank density matrices

We consider low-rank density operators supported on a M×N Hilbert space for arbitrary M and N $(M<~N),$ and with a positive partial transpose (PPT) ^TA>~0. For rank r()<~N we prove that having a PPT is necessary and sufficient for to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of and ^TA satisfies r()+r(^TA)<~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 system of couple polynomial equations has a finite number of solutions, as expected in most cases.