# The Connes Embedding Problem and connections to quantum information theory

Mikael Rørdam (University of Copenhagen)
Analysis Seminar
Thursday, 13 February, 2020 | 14:15–15:00 | Aud. D4 (1531-219)
Contact: Steen Thorbjørnsen
The Einstein-Podolsky-Rosen paradox, leading to the notion of entanglement, can mathematically be understood in terms of analysing certain convex sets of correlation matrices (of probabilities). The correlations matrices arising from the classical model, for example, corresponds to the assumption of “hidden variable”, while “quantum models” are realized by operators on Hilbert spaces. Tsirelson’s conjecture asks if the two quantum models (based on tensor products of operators, respectively, based on commuting operators) coincide. By the work of Ozawa, Fritz and others it is known that Tsirelson’s conjecture is equivalent to the famous Connes Embedding Problem, that, very recently, may have found a solution in the negative. After reviewing some background, I will move on to describe some more recent results obtained in collaboration with Musat on non-closure of certain sets of correlation matrices, and explain how that leads to the conclusion that one must employ infinite dimensional von Neumann algebras to understand certain classes of unital quantum channels between matrix algebras, the so-called factorizable ones.