quant-ph

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Title: A complete family of separability criteria

Abstract: We introduce a new 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 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.
 Comments: 21 pages. Expanded introduction. References added, typos corrected. Accepted for publication in Physical Review A Subjects: Quantum Physics (quant-ph) Journal reference: Phys. Rev. A 69, 022308 (2004) DOI: 10.1103/PhysRevA.69.022308 Cite as: arXiv:quant-ph/0308032 (or arXiv:quant-ph/0308032v3 for this version)

Submission history

From: Federico M. Spedalieri [view email]
[v1] Tue, 5 Aug 2003 22:15:04 GMT (35kb)
[v2] Fri, 22 Aug 2003 22:34:28 GMT (37kb)
[v3] Wed, 5 Nov 2003 19:32:04 GMT (39kb)
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