Undecidability and incompleteness in quantum information theory and operator algebras

TL;DR

This paper surveys how recent undecidability results in quantum complexity theory impact operator algebras, leading to fundamental limitations and refutations of longstanding conjectures like the Connes Embedding Problem.

Contribution

It highlights the connection between quantum undecidability results and operator algebra incompleteness, including the refutation of the Connes Embedding Problem and the Aldous-Lyons conjecture.

Findings

Undecidability results imply incompleteness in operator algebras.

Refutation of the Connes Embedding Problem based on MIP* = RE.

Application of MIP* = RE to disprove the Aldous-Lyons conjecture.

Abstract

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