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arXiv:quant-ph/0503164 · Physical Review Letters

Generic Bell Inequalities for Multipartite Arbitrary Dimensional Systems

Wonmin Son, Jinhyoung Lee, M. S. Kim

Published Feb 17, 2006• 110 citations
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Abstract

We present generic Bell inequalities for multipartite arbitrary dimensional systems. The inequalities that any local realistic theory must obey are violated by quantum mechanics for even dimensional systems. A large set of variants are shown to naturally emerge from the generic Bell inequalities. We discuss particular variants of Bell inequalities that are violated for all the systems including odd dimensional systems.

DOI
10.1103/physrevlett.96.060406

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The structure of Bell-type inequalities detecting genuine multipartite nonlocality, and hence detecting genuine multipartite entanglement, is investigated. We first present a simple and intuitive approach to Svetlichny's original inequality, which provides a clear understanding of its structure and of its violation in quantum mechanics. Based on this approach, we then derive a family of Bell-type inequalities for detecting genuine multipartite nonlocality in scenarios involving an arbitrary number of parties and systems of arbitrary dimension. Finally, we discuss the tightness and quantum mechanical violations of these inequalities.
Quantum correlations between spatially separated parts of a d-dimensional bipartite system (d ≥ 2) have no classical analog. Such correlations, also called entanglements, are not only conceptually important, but also have a profound impact on information science. In theory the violation of Bell inequalities based on local realistic theories for d-dimensional systems provides evidence of quantum nonlocality. Experimental verification is required to confirm whether a quantum system of extremely large dimension can possess this feature, however it has never been performed for large dimension. Here, we report that Bell inequalities are experimentally violated for bipartite quantum systems of dimensionality d = 16 with the usual ensembles of polarization-entangled photon pairs. We also estimate that our entanglement source violates Bell inequalities for extremely high dimensionality of d > 4000. The designed scenario offers a possible new method to investigate the entanglement of multipartite systems of large dimensionality and their application in quantum information processing.
A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.
Quantum correlations between spatially separated parts of a ddd-dimensional bipartite system (d≥2d\geq 2d≥2) have no classical analog. Such correlations, also called entanglements, are not only conceptually important, but also have a profound impact on information science. In theory the violation of Bell inequalities based on local realistic theories for ddd-dimensional systems provides evidence of quantum nonlocality. Experimental verification is required to confirm whether a quantum system of extremely large dimension can possess this feature, however it has never been performed for large dimension (e.g., d≥1000d\geq 1000d≥1000). Here, we report that Bell inequalities are experimentally violated for bipartite quantum systems of extremely high dimensionality with the usual ensembles of polarization-entangled photon pairs. Our entanglement source violates Bell inequalities for extremely high dimensionality of d>4000d>4000d>4000. The designed scenario offers a possible new method to investigate the entanglement of multipartite systems of large dimensionality and their application in quantum information processing.
We propose a hierarchy of Bell-type inequalities for arbitrary nnn-partite systems that identifies the different degrees of nonlocality ranging from standard to genuine multipartite nonlocality. After introducing the definition of nonsignaling mmm locality, we show that the observed joint probabilities in any nonsignaling mmm-local realistic models should satisfy the $(m\ensuremath{-}1)\mathrm{th}$ Bell-type inequality. When m=2m=2m=2, the corresponding inequality reduces to the one shown earlier [Q. Chen et al., Phys. Rev. Lett. 112, 140404 (2014)] whose violation indicates genuine multipartite nonlocality, and when m=nm=nm=n, the corresponding inequality is just Hardy's inequality whose violation indicates standard multipartite nonlocality. Furthermore, several examples are provided to demonstrate their hierarchy of multipartite nonlocality.