Mathematics Colloquium

The three problems referred to in the title originate in operator algebras, quantum information theory, and complexity theory respectively. Recently we established the complexity-theoretic equality MIP* = RE. This equality implies that the membership problem for certain quantum correlation sets is undecidable. Due to prior work by many others the result implies a negative answer to Tsirelson's problem (quantum information) as well as Connes' Embedding Problem (von Neumann algebras) and equivalent problems in operator algebras such as Kirchberg's QWEP (C* algebras). It leaves open the famous question about the existence of a non-hyperlinear group.

In the talk I will explain the characterization MIP* = RE and motivate it by describing its connection to the study of nonlocality in quantum information, Tsirelson's problem, and operator algebras. I will mention some proof ideas, which draw from the theory of probabilistic checking in complexity theory and approximate stability in group theory.

The main result is joint work with Ji, Natarajan, Wright and Yuen available as arXiv:2001.04383.