Sam Harris, Department of Pure Mathematics, University of Waterloo
"Connes' embedding problem and quantum XOR games"
One of the most significant outstanding problems in operator algebras is Connes' embedding problem. In recent years, a deep connection has been exhibited between this problem and the theory of quantum bipartite correlations (in particular, via the weak Tsirelson problem). Motivated by this connection, we will briefly outline the theory of two-player quantum XOR games, and how unitary correlations can be thought of as strategies for these games. These correlations naturally arise from states on various tensor products of operator systems, where the operator system in question is the universal one generated by the entries of an $n \times n$ contraction. We will show that Connes' embedding problem holds if and only if every quantum XOR game with a winning strategy in the quantum commuting model also has a winning strategy in the approximate finite-dimesional (quantum approximate) model.