Complete family of separability criteria

Doherty, Andrew C., Parrilo, Pablo A. and Spedalieri, Federico M. (2004) Complete family of separability criteria. Physical Review A, 69 2: 022308-1-022308-20. doi:10.1103/PhysRevA.69.022308

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Author Doherty, Andrew C.
Parrilo, Pablo A.
Spedalieri, Federico M.
Title Complete family of separability criteria
Journal name Physical Review A   Check publisher's open access policy
ISSN 1050-2947
Publication date 2004-02-01
Sub-type Article (original research)
DOI 10.1103/PhysRevA.69.022308
Open Access Status File (Publisher version)
Volume 69
Issue 2
Start page 022308-1
End page 022308-20
Total pages 20
Place of publication College Park, Maryland, United States
Publisher American Physical Society
Language eng
Abstract We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state rho to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.
Keyword Optics
Physics, Atomic, Molecular & Chemical
Mixed-state Entanglement
Reduction Criterion
Bound Entanglement
Quantum States
Key Agreement
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collections: Excellence in Research Australia (ERA) - Collection
School of Physical Sciences Publications
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Created: Wed, 19 Sep 2007, 17:22:41 EST