Entanglement Polytopes: Multiparticle Entanglement from Single-Particle Information

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Science  07 Jun 2013:
Vol. 340, Issue 6137, pp. 1205-1208
DOI: 10.1126/science.1232957

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Unraveling Entanglement

Entanglement is a curious property of some quantum mechanical systems, exploited in applications such as quantum information processing. Walter et al. (p. 1205) used an algebraic geometry approach to represent the entanglement of a multiparticle system in a pure state in the geometric space whose axes are associated with the properties of the individual particles. In that space, entanglement classes—collections of entangled states that can be transformed into each other—correspond to different convex polytopes, making it possible to distinguish between the classes.


Entangled many-body states are an essential resource for quantum computing and interferometry. Determining the type of entanglement present in a system usually requires access to an exponential number of parameters. We show that in the case of pure, multiparticle quantum states, features of the global entanglement can already be extracted from local information alone. This is achieved by associating any given class of entanglement with an entanglement polytope—a geometric object that characterizes the single-particle states compatible with that class. Our results, applicable to systems of arbitrary size and statistics, give rise to local witnesses for global pure-state entanglement and can be generalized to states affected by low levels of noise.

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