dorsal/arxiv
View SchemaClassification of multipartite entangled states by multidimensional determinants
| Authors | Akimasa Miyake |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0206111 |
| URL | https://arxiv.org/abs/quant-ph/0206111 |
| DOI | 10.1103/PhysRevA.67.012108 |
| Journal | Phys. Rev. A 67, 012108 (2003) |
Abstract
We find that multidimensional determinants "hyperdeterminants", related to entanglement measures (the so-called concurrence or 3-tangle for the 2 or 3 qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the $n \geq 4$ qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.
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"abstract": "We find that multidimensional determinants \"hyperdeterminants\", related to\nentanglement measures (the so-called concurrence or 3-tangle for the 2 or 3\nqubits, respectively), are derived from a duality between entangled states and\nseparable states. By means of the hyperdeterminant and its singularities, the\nsingle copy of multipartite pure entangled states is classified into an onion\nstructure of every closed subset, similar to that by the local rank in the\nbipartite case. This reveals how inequivalent multipartite entangled classes\nare partially ordered under local actions. In particular, the generic entangled\nclass of the maximal dimension, distinguished as the nonzero hyperdeterminant,\ndoes not include the maximally entangled states in Bell\u0027s inequalities in\ngeneral (e.g., in the $n \\geq 4$ qubits), contrary to the widely known\nbipartite or 3-qubit cases. It suggests that not only are they never locally\ninterconvertible with the majority of multipartite entangled states, but they\nwould have no grounds for the canonical n-partite entangled states. Our\nclassification is also useful for the mixed states.",
"arxiv_id": "quant-ph/0206111",
"authors": [
"Akimasa Miyake"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"doi": "10.1103/PhysRevA.67.012108",
"journal_ref": "Phys. Rev. A 67, 012108 (2003)",
"title": "Classification of multipartite entangled states by multidimensional determinants",
"url": "https://arxiv.org/abs/quant-ph/0206111"
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