Workshop

We briefly recall the original statement of the Connes Embedding Problem(CEP), which was in terms of traces, and Kirchberg's equivalent problem about tensor norms on group algebras.

We then introduce synchronous games, synchronous correlations and their connections with traces. This leads to the concept of the fundamental orthogonality relations(FOR) of a synchronous game.

From this viewpoint, MIP*=RE gives a negative answer to the CEP by showing the existence of a FOR that can be realized in a tracial C*-algebra, but not in the algebra considered by Connes.

In this talk, we will explore the interaction between operator algebras and quantum information theory through a discussion on quantum graphs. Quantum graphs are an operator generalization of classical graphs that have appeared in different branches of mathematics including operator algebras, non-commutative topology, operator systems theory and quantum information theory. I will present an overview of the theory of quantum graphs and discuss the connections between different perspectives using operator algebraic methods. We will then investigate a coloring problem for quantum graphs using a quantum-input classical-output nonlocal game. Using this framework, we show that every quantum graph has a finite quantum coloring, but not necessarily finite classical coloring. We will develop a combinatorial characterization of quantum graph coloring using the winning strategies of this game and obtain various lower bounds for the chromatic numbers of quantum graphs. This is based on joint work with Michael Brannan and Samuel Harris.

In this talk I will give a light survey on some recent works on bisychronous correlations and bisynchronous games. Bisynchronous correlations form a special class of synchronous correlations, where there is the additional constraint that the probability of Alice and Bob giving the same outputs whenever they receive distinct inputs is always zero.

I will explain how bisynchronous correlations are connected to traces on quantum permutation groups, and how bisynchronous correlations and bisynchronous games fit within the context of MIP*=RE. Time permitting I'll also explain how for certain classes of bisynchronous games (e.g., graph isomorphism games), one can identify perfect strategies using purely combinatorial-algebraic methods by studying representation categories of certain quantum automorphism groups.

Certain spin models are universal, meaning that they contain all other models when seen in the appropriate light. What is the relation between universal spin models, universal Turing machines and notions of universality in neural networks? And what is their relation to forms of unreachability such as uncomputability or undecidability? I will explore these questions in the light of our recent framework for universality (arxiv:2307.06851). I will also share some philosophical perspectives on these structures.