YWC*A Mini-Course I

Mini-course title: Quantized Functional Analysis, Tensor Norms and the Grothendieck Program

Speaker: Magdalena Musat (University of Copenhagen)

Lecture I: Classical and quantum correlations, tensor norms, and an asymptotic behaviour of quantum channels

Abstract: In 1980, Tsirelson showed that Bell's inequalities---that have played an important role in distinguishing classical  correlations from quantum ones, and that were used to test, and ultimately disprove the Einstein-Podolski-Rosen postulate of "hidden variables", coincide with Grothendieck's famous inequalities from functional analysis. Tsirelson further studied sets of quantum correlations arising under two different assumptions of commutativity of observables. While he showed that they are the same in the finite dimensional case, the equality of these sets was later proven to be equivalent to the most famous still open question in operator algebras theory: the Connes embedding problem.

I will survey several reformulations of the Connes embedding problem, starting with Kirchberg's deep work from 1993, continuing with the results of Fritz (2009) and Ozawa (2012) connecting it to Tsirelson's conjecture, and the more recent work, joint with Haagerup (2015), concerning an asymptotic property of quantum channels posessing a certain factorizability property (that originates in operator algebras), introduced by Anantharaman-Delaroche. I will also discuss the remarkable recent breaktrough of Slofstra (June, 2016).

This is part of the Young Women in C*-Algebras, August 5-6.