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arXiv:quant-ph/0611144

Geometrical structure of entangled states and secant variety

Hoshang Heydari

Published Nov 14, 2006• 1 citation
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Abstract

We show that the secant variety of the Segre variety gives useful information about the geometrical structure of an arbitrary multipartite quantum system. In particular, we investigate the relation between arbitrary bipartite and three-partite entangled states and this secant variety. We also discuss the geometry of an arbitrary general multipartite state.

DOI
10.48550/arxiv.quant-ph/0611144

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We show that the secant variety of the Segre variety gives useful information about the geometrical structure of an arbitrary multipartite quantum system. In particular, we investigate the relation between arbitrary bipartite and three-partite entangled states and this secant variety. We also discuss the geometry of an arbitrary general multipartite state.
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.