Geometric descriptions of entangled states by auxiliary varieties
- Laboratoire M3M, Universite de Technologie de Belfort-Montbeliard, 90010 Belfort Cedex (France)
- Universite de Rouen, Laboratoire d'Informatique, du Traitement de l'Information et des Systemes (LITIS), Avenue de l'Universite - BP 8 6801 Saint-etienne-du-Rouvray Cedex (France)
- Universite Paris-Est Marne-la-Vallee, Laboratoire d'Informatique Gaspard-Monge, 77454 Marne-la-Vallee Cedex 2 (France)
The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting, we describe well-known classifications of multipartite entanglement such as 2 Multiplication-Sign 2 Multiplication-Sign (n+ 1), for n Greater-Than-Or-Slanted-Equal-To 1, quantum systems and a new description with the 2 Multiplication-Sign 3 Multiplication-Sign 3 quantum system. Our results complete the approach of Miyake and make stronger connections with recent work of algebraic geometers. Moreover, for the quantum systems detailed in this paper, we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.
- OSTI ID:
- 22093757
- Journal Information:
- Journal of Mathematical Physics, Vol. 53, Issue 10; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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