dorsal/arxiv
View SchemaA complete family of separability criteria
| Authors | Andrew C. Doherty, Pablo A. Parrilo, Federico M. Spedalieri |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0308032 |
| URL | https://arxiv.org/abs/quant-ph/0308032 |
| DOI | 10.1103/PhysRevA.69.022308 |
| Journal | Phys. Rev. A 69, 022308 (2004) |
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.
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"abstract": "We introduce a new family of separability criteria that are based on the\nexistence of extensions of a bipartite quantum state $\\rho$ to a larger number\nof parties satisfying certain symmetry properties. It can be easily shown that\nall separable states have the required extensions, so the non-existence of such\nan extension for a particular state implies that the state is entangled. One of\nthe main advantages of this approach is that searching for the extension can be\ncast as a convex optimization problem known as a semidefinite program (SDP).\nWhenever an extension does not exist, the dual optimization constructs an\nexplicit entanglement witness for the particular state. These separability\ntests can be ordered in a hierarchical structure whose first step corresponds\nto the well-known Positive Partial Transpose (Peres-Horodecki) criterion, and\neach test in the hierarchy is at least as powerful as the preceding one. This\nhierarchy is complete, in the sense that any entangled state is guaranteed to\nfail a test at some finite point in the hierarchy, thus showing it is\nentangled. The entanglement witnesses corresponding to each step of the\nhierarchy have well-defined and very interesting algebraic properties that in\nturn allow for a characterization of the interior of the set of positive maps.\nCoupled with some recent results on the computational complexity of the\nseparability problem, which has been shown to be NP-hard, this hierarchy of\ntests gives a complete and also computationally and theoretically appealing\ncharacterization of mixed bipartite entangled states.",
"arxiv_id": "quant-ph/0308032",
"authors": [
"Andrew C. Doherty",
"Pablo A. Parrilo",
"Federico M. Spedalieri"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.69.022308",
"journal_ref": "Phys. Rev. A 69, 022308 (2004)",
"title": "A complete family of separability criteria",
"url": "https://arxiv.org/abs/quant-ph/0308032"
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