Abstract
We consider low-rank density operators supported on a Hilbert space for arbitrary M and N and with a positive partial transpose (PPT) For rank 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 satisfies 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.
- Received 1 March 2000
DOI:https://doi.org/10.1103/PhysRevA.62.032310
©2000 American Physical Society