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Upper and lower bounds on the quantum violation of tripartite Bell correlation inequalities

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Two-party Bell correlation inequalities (that is, inequalities involving only correlations between dichotomic observables at each site, such as the CHSH inequality) are well-understood: Grothendieck's inequality stipulates that the quantum bias can only be a constant factor larger than the classical bias, and the maximally entangled state is always the most nonlocal resource. In part due to the complex nature of multipartite entanglement, tripartite inequalities are much more unwieldy. In a recent breakthrough result, Perez-Garcia et. al. (quant-ph/0702189) showed using tools originating from the study of operator algebras that in this setting the quantum-classical violation could be arbitrarily large. Moreover, they showed that GHZ states could only lead to bounded violations, so that they were not the most non-local states. We extend and simplify their results in a number of ways: - We show that large families of states, including generalizations of GHZ states and stabilizer states, can only lead to bounded violations. - We prove bounds on the maximal quantum-classical violation as a function both of the local dimension (this was already shown in Perez-Garcia et. al., but we give a much simpler proof), and of the number of settings per site. - We provide a simple probabilistic construction of an inequality for which there is an unbounded quantum-classical gap. Our construction is simpler, and has better parameters, than the one in Perez-Garcia et.al. It is essentially optimal in terms of the local dimension of one of the parties, and off by a quadratic factor in terms of the number of settings. In this talk I will survey some of these results, focusing on the tools that have so far been useful for their analysis (and do not involve operator algebras!). Based on joint works with Jop Briet, Harry Buhrman, and Troy Lee. Some of this work is available at arXiv:0911.4007.