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Published Online: 03 October 2012
Accepted: August 2012

Geometric descriptions of entangled states by auxiliary varieties

Journal of Mathematical Physics 53, 102203 (2012); https://doi.org/10.1063/1.4753989
Frédéric Holweck1, a), Jean-Gabriel Luque2, b), and Jean-Yves Thibon3, c)
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Abstract
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 × 2 × (n + 1), for n ⩾ 1, quantum systems and a new description with the 2 × 3 × 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.

ACKNOWLEDGMENTS

This paper is partially supported by the ANR project PhysComb, ANR-08-BLAN-0243-04.

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