Complete family of separability criteria

By Andrew C. Doherty, Pablo A. Parrilo, F. Spedalieri. 2003

407 citations recorded by the source corpus.

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.

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

View Paper: https://www.semanticscholar.org/paper/947b899d33b42f1ff7239e2584af18bbdb9f561c

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Citations in the corpus

Faithful Squashed Entanglement (2010, methodology): “The algorithm, which we analyse in more detail in Section IV, is very simple and in fact is the basis of a well-known hierarchy of efficient tests for separability [51].”
The Church of the Symmetric Subspace (2013, background): “Thus, the set of k -extendible states comprises a hierarchy of relaxations of the set of separable states [13].”
Geometry of Quantum States: Index (2007, methodology): “A technique of extending the system (and the map) a certain number of times gives a constructive test for positivity for a large class of maps [270].”
Random matrix techniques in quantum information theory (2015, background): “rmutations of the B-systems and has ˆ AB as a marginal: [id nk Tr k r1](˙ AB ) = ˆ AB: Obviously, any separable state ˆ AB is r-extendible, for all r1. Doherty, Parrilo, and Spedalieri have shown in [DPS04] that these conditions are also sucient. Theorem 4.23. A bipartite quantum state ˆ AB 2D nk is”