# Winter school on Connes' embedding problem and quantum information theory

Connes' embedding problem is one of the long standing open problems in Operator Algebras. Since it was raised in 1976, several equivalent formulations, seemingly unrelated to it, have been found, also in other areas of mathematics. In particular, it is equivalent to the (weak) Tsirelson problem in Quantum Information Theory.

The aim of the winter school is to upgrade the knowledge of all operator algebraists at NTNU and UiO, in particular master and PhD students, on these important topics and to review some of the recent developments.

This course is a part of the project "Pure Mathematics in Norway, 2018-2022", supported by the Trond Mohn Foundation (earlier called the Bergen Research Foundation) and the Tromsø Research Foundation.

**Practical information**: How to get to the city center and to the main campus of UiO.

The main programme will consist of four series of lectures given by some of the experts on these topics (see below). There will also be three additional lectures, by Mikael Rørdam, Alexander Müller-Hermes and Jitendra Prakash, all from the University of Copenhagen. The detailed programme is now available.

- Benoît Collins (Kyoto University, Japan):
**A non-commutative probability point of view on the Connes embedding problem****.**

**Abstract:** We will review the micro state approach to the Connes problem developed in free probability, as well as some relations between this problem and free entropy / free dimension questions. We will also consider some moment problems in matrix algebras and finite von Neumann algebras, and mention some reformulations of the Connes problem, some positive results, and some no-go results. This second part will be inspired from works from/with Dykema and Brannan (among others). Time allowing, we will discuss non commutative real algebraic reformulations of the Connes problem, after Klep and Schweighofer and related topics.

- Magdalena E. Musat (University of Copenhagen, Denmark):
**Von Neumann algebras meet Quantum Information Theory**.

**Abstract**: Available here.

**Lecture notes (day 1)**: Available here.

- Narutaka Ozawa (RIMS Kyoto, Japan):
**Connes's Embedding Problem and its equivalents.**

**Abstract**: I will survey the operator algebraic aspects of Connes’ Embedding Conjecture and Tsirelson's problem. I will also cover some of the recent works of W. Slofstra.

- Vern Paulsen (University of Waterloo, Canada):
**C*-algebras and non-local games**.

**Abstract**: There are currently several different mathematical models that attempt to describe the conditional probabiity densities that can occur when two labs in an entangled state conduct a finite set of quantum experiments. The Tsirelson conjectures are concerned with whether or not these various models give rise to the same sets of conditional probability densities. Thanks to the work of a number of researchers we now know that one of these conjectures is equivalent to Connes' embedding conjecture.

Many of the best results on these conjectures have come from the study of certain families of games, called non-local games. In these talks we will introduce these ideas and show that for each synchronous non-local game, there is an affiliated C*-algebra whose representation theory tells us if the game has a perfect strategy in each of the possible models. In the case of the graph isomorphism game, this C*-algebra is related to the quantum permutation group and the game theory perspective gives new information about this C*-algebra.

The number of participants is strictly limited and participation is only possible by personal invitation from the organizing committee.