Multipartite Entanglement and Hyperdeterminants

Details Multipartite Entanglement and Hyperdeterminants Multipartite Entanglement and Hyperdeterminants

2002-12-27

arxiv.org/abs/quant-ph/0212146v1

Quant. Info. Comp. 2 (Special), 540-555 (2002)

Physics

Quantum Physics

12 pages, 5 eps figures. Contribution to the ERATO workshop on Quantum Information Science 2002 (September, 2002, Tokyo, Japan

Scientific paper

We classify multipartite entanglement in a unified manner, focusing on a duality between the set of separable states and that of entangled states. Hyperdeterminants, derived from the duality, are natural generalizations of entanglement measures, the concurrence, 3-tangle for 2, 3 qubits respectively. Our approach reveals how inequivalent multipartite entangled classes of pure states constitute a partially ordered structure under local actions, significantly different from a totally ordered one in the bipartite case. Moreover, the generic entangled class of the maximal dimension, given by the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the 4 or more 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 that of mixed states.

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