Classification of multipartite entangled states by multidimensional determinants

Journal Article · Wed Jan 01 04:00:00 EST 2003 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.67.012108· OSTI ID:20634076

Miyake, Akimasa ^[1]

---

1. Department of Physics, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033 (Japan)

We find that multidimensional determinants 'hyperdeterminants', related to entanglement measures (the so-called concurrence, or 3-tangle for two or three 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{>=}4 qubits), contrary to the widely known bipartite or three-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.

• OSTI ID: 20634076
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 1 Vol. 67; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
BELL THEOREM
CORRELATIONS
DUALITY
ELECTRIC GROUNDS
ENERGY LEVELS
INFORMATION THEORY
MIXED STATE
QUANTUM MECHANICS
SINGULARITY