Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $N$ ($M\leq N$) and with a positive partial
transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ 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 $r(\varrho)+r(\varrho^{T_A}) \le
2MN-M-N+2$. 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 a
finite number of solutions, as expected in most cases.

31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess