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

We consider low-rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary M and N $(M<~N),$ and with a positive partial transpose (PPT) ${\varrho }^{T_A}>~0.$ For rank $r\left(\varrho \right)<~N$ we prove that having a PPT is necessary and sufficient for $\varrho$ 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 $\varrho$ and ${\varrho }^{T_A}$ satisfies $r\left(\varrho \right)+r\left({\varrho }^{T_A}\right)<~2MN-M-N+2.\mathrm{}$ 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.