Undecidability and incompleteness in quantum information theory and operator algebras

Abstract

We survey a number of incompleteness results in operator algebras stemming from the recent undecidability result in quantum complexity theory known as ${\mathrm{MIP}}^{\ast }=\mathrm{RE}$, the most prominent of which is the Gödelian refutation of the Connes Embedding Problem. We also discuss the very recent use of ${\mathrm{MIP}}^{\ast }=\mathrm{RE}$ in refuting the Aldous-Lyons conjecture in probability theory.

38 pages. To appear in a special issue of Monatshefte für Mathematik celebrating the 100th anniversary of Gödel's matriculation at the University of Vienna