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

By Pawel Horodecki, Maciej Lewenstein, Guifré Vidal, I. Cirac. 2000

143 citations recorded by the source corpus.

We consider low-rank density operators $\ensuremath{\varrho}$ supported on a $M\ifmmode\times\else\texttimes\fi{}N$ Hilbert space for arbitrary M and N $(Ml~N),$ and with a positive partial transpose (PPT) ${\ensuremath{\varrho}}^{{T}_{A}}g~0.$ For rank $r(\ensuremath{\varrho})l~N$ we prove that having a PPT is necessary and sufficient for $\ensuremath{\varrho}$ 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 $\ensuremath{\varrho}$ and ${\ensuremath{\varrho}}^{{T}_{A}}$ satisfies $r(\ensuremath{\varrho})+r({\ensuremath{\varrho}}^{{T}_{A}})l~2MN\ensuremath{-}M\ensuremath{-}N+2.$ 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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1b1016e005f6769cb339d0aa3a5b3ee27ce1aed0

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Citations in the corpus

Geometry of Quantum States: Index (2007, background): “It is known that there are no entangled PPT states of rank three or less [467].”
Entanglement Breaking Channels (2003, background): “This follows immediately from a (non-trivial) result in [10] which implies that a d(2) × d(2) density matrix of rank d is separable if and only if it has positive partial transpose.”
Entanglement detection (2008, background): “This state has also the minimal possible rank, since the minimal rank of a PPT entangled state in a dA × dB-system is max{dA, dB} + 1 [129].”