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Quantum Correlations: From Bell inequalities to Tsirelson's April 24, 2007

Summary: Quantum Correlations: From Bell inequalities to Tsirelson's
David Avis
April 24, 2007
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 AiAj
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


Source: Avis, David - School of Computer Science, McGill University


Collections: Computer Technologies and Information Sciences