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arXiv:quant-ph/0106024 · Physical Review Letters

Bell Inequalities for Arbitrarily High-Dimensional Systems

Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, Sandu Popescu

Published Jan 10, 2002• 983 citations
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Abstract

We develop a novel approach to Bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy. This is used to construct a family of Bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise. In particular, our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality.

DOI
10.1103/physrevlett.88.040404

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We develop a novel approach to Bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy. This is used to construct a family of Bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise. In particular, our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality.
We analyze the correlation structure of bipartite arbitrary-dimensional Bell inequalities via novel conditions of correlations in terms of differences of joint probabilities called correlators. The conditions of correlations are shown to be necessary for the multi-level Bell state. In particular, we find that the bipartite arbitrary-dimensional Bell-type inequalities introduced by Collins-Gisin-Linden-Massar-Popescu [Phys. Rev. Lett. 88, 040404 (2002)] and Son-Lee-Kim [Phys. Rev. Lett. 96, 060406 (2006)] are composed of correlators, and we reveal that the maximal violations by the Bell state just fit the conditions of quantum correlations. Correlators can be considered as essential elements of Bell inequalities.
We present a family of Bell inequalities for three parties and arbitrarily many outcomes, which can be seen as a natural generalization of the Mermin-Bell inequality. For a small number of outcomes, we verify that our inequalities define facets of the polytope of local correlations. We investigate the quantum violations of these inequalities, in particular with respect to the Hilbert space dimension. We provide strong evidence that the maximal quantum violation can be reached only using systems with local Hilbert space dimension exceeding the number of measurement outcomes. This suggests that our inequalities can be used as multipartite dimension witnesses.
Local measurements acting on entangled quantum states give rise to a rich correlation structure in the multipartite scenario. We introduce a versatile technique to build families of Bell inequalities witnessing different notions of multipartite nonlocality for any number of parties. The idea behind our method is simple: a known Bell inequality satisfying certain constraints, for example the Clauser–Horne–Shimony–Holt inequality, serves as the seed to build new families of inequalities for more parties. The constructed inequalities have a clear operational meaning, capturing an essential feature of multipartite correlations: their violation implies that numerous subgroups of parties violate the inequality chosen as seed. The more multipartite nonlocal the correlations, the more subgroups can violate the seed. We illustrate our construction using different seeds and designing Bell inequalities to detect k -way nonlocal multipartite correlations, in particular, genuine multipartite nonlocal correlations—the strongest notion of multipartite nonlocality. For one of our inequalities we prove analytically that a large class of pure states that are genuine multipartite entangled (GME) exhibit genuine multipartite nonlocality for any number of parties, even for some states that are almost product. We also provide numerical evidence that this family is violated by all GME pure states of three and four qubits. Our results make us conjecture that this family of Bell inequalities can be used to prove the equivalence between genuine multipartite pure-state entanglement and nonlocality for any number of parties.
Imagine two parties, Alice and Bob, who share an entangled quantum state. A well-established result that if Alice performs a two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt inequality and show that the maximum violation of these Bell inequalities allowed by quantum theory cannot be attained by a bipartite quantum system having support on a qubit at each site.
When three or more particles are considered, quantum correlations can be stronger than the correlations generated by so-called hybrid local hidden variable models, where some of the particles are considered as a single block inside which communication and signaling is allowed. We provide an exhaustive classification of Bell inequalities to characterize various hybrid scenarios in four- and five-particle systems. In quantum mechanics, these inequalities provide device-independent witnesses for the entanglement depth. In addition, we construct a family of inequalities to detect a nonlocality depth of $(n\ensuremath{-}1)$ in nnn-particle systems. Moreover, we present two generalizations of the original Svetlichny inequality, which was the first Bell inequality designed for hybrid models. Our results are based on the cone-projection technique, which can be used to completely characterize Bell inequalities under affine constraints; even for many parties, measurements, and outcomes.
Bell inequality tests where the detection efficiency is below a certain threshold ηcrit can be simulated with local hidden-variable models. Here, we introduce a method to identify Bell tests requiring low ηcrit and relatively low dimension d of the local quantum systems. The method has two steps. First, we show a family of bipartite Bell inequalities for which, for correlations produced by maximally entangled states, ηcrit can be upper bounded by a function of some invariants of graphs, and use it to identify correlations that require small ηcrit . We present examples in which, for maximally entangled states, ηcrit≤0.516 for d=16 , ηcrit≤0.407 for d=28 , and ηcrit≤0.326 for d=32 . We also show evidence that the upper bound for ηcrit can be lowered down to 0.415 for d=16 and present a method to make the upper bound of ηcrit arbitrarily small by increasing the dimension and the number of settings. All these upper bounds for ηcrit are valid (as it is the case in the literature) assuming no noise. The second step is based on the observation that, using the initial state and measurement settings identified in the first step, we can construct Bell inequalities with smaller ηcrit and better noise robustness. For that, we use a modified version of Gilbert's algorithm that takes advantage of the automorphisms of the graphs used in the first step. We illustrate its power by explicitly developing an example in which ηcrit is 12.38% lower and the required visibility is 14.62% lower than the upper bounds obtained in the first step. The tools presented here may allow for developing high-dimensional loophole-free Bell tests and loophole-free Bell nonlocality over long distances.