Quantum Correlations: From Bell inequalities to Tsirelson’s theorem

The cut polytope and its relatives are good models of the correlations that can be obtained between events that can be well described by classical physics. Bell’s Theorem and subsequent experiments demonstrate that correlations obtainable between events at the quantum level cannot be modelled in this way. This raises the question of whether a “good” mathematical characterization of quantum correlation vectors can be obtained. An important special case was completely solved by Tsirelson, who showed that a projection of the elliptope provides the desired body. (This parallels the well know semi-definite programming approach to approximating max-cut.) I will survey this material and present some new joint work with Hiroshi Imai and Tsuyoshi Ito on a possible direction for extending Tsirelson’s theorem. 1 Classical Correlations Let A1, ..., An be a collection of n 0/1 valued random variables that belong to a common joint probability distibution. For 1 ≤ i < j ≤ n, we define new random variables Ai△Aj that are one when Ai = Aj and zero otherwise. Denote by 〈A〉 the expected value of a random variable A. The full correlation vector x based on A1, ..., An is the vector of length N = n+ n(n− 1)/2 given by the expected values: x = (〈Ai〉 , 〈Ai△Aj〉) ≡ (〈Ai〉1≤i≤n , 〈Ai△Aj〉1≤i<j≤n). Note that each element of the above vector lies between zero and one. Now consider any vector x = (x1, ..., xn, x12, ..., xn−1,n) ∈ [0, 1]N indexed as above, which we will call an outcome. We consider two related computational questions: Recognition. When is an outcome x a full correlation vector? Optimization. For any c ∈ RN what is the maximum value of cTx over all possible full correlation vectors x? It turns out that the recognition problem is NP-complete, and the optimization problem is NP-hard. This follows from the fact that the set of full correlation vectors is in fact the cut polytope CUTn+1 defined on the complete graph Kn+1. This polytope is defined as the convex hull of the 2N full correlation vectors obtained by deterministically setting each random variable Ai to either zero or one. For details of the above and other facts about cut polytopes, see the book by Deza and Laurent [8]. For a vector u = (u1, ..., ud) the L1-norm