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We consider low rank density operators $\varrho$ supported on a $M\times N$Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partialtranspose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we provethat having a PPT is necessary and sufficient for $\varrho$ to be separable; inthis case we also provide its minimal decomposition in terms of pure productstates. It follows from this result that there is no rank 3 bound entangledstates having a PPT. We also present a necessary and sufficient condition forthe separability of generic density matrices for which the sum of the ranks of$\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le2MN-M-N+2$. This separability condition has the form of a constructive check,providing thus also a pure product state decomposition for separable states,and it works in those cases where a system of couple polynomial equations has afinite number of solutions, as expected in most cases.Comment: RevTex, 10 pages, no figure

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