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