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Tripartite probability distributions and communication complexity

We show that every tripartite quantum correlation generated with a Schmidt state (in particular every correlation generated with the GHZ state) can be simulated with the sending of two bits of classical communication from Alice to Bob and Charlie plus the sending of two bits of classical communication from Bob to Charlie. This extends recent results which showed that the maximal violation of Bell inequalities attainable by these correlations is uniformly bounded. For simplicity, we state and prove the result for three parties, but the generalization to the case of $n$ parties follows easily. We also show that every $n$-partite probability distribution generated with local resources plus $c$-bits of local communication can violate a Bell inequality by at most a factor of $2^c$.