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A complete family of separability criteria


A. Doherty, P.A. Parrilo, F. Spedalieri

Phys. Rev. A, vol. 69, 022308

We introduce a new family of separability criteria that are based on the existence of extensions of a bipartite quantum state $ ho$ 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 non-existence 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 (SDP). 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.

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% Autogenerated BibTeX entry
@Article { DohPar:2004:IFA_1751,
    author={A. Doherty and P.A. Parrilo and F. Spedalieri},
    title={{A complete family of separability criteria}},
    journal={Phys. Rev. A},
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