Undecidability and incompleteness in quantum information theory and operator algebras

TL;DR

This survey exposes a web of undecidability phenomena in operator algebras anchored by the landmark quantum-complexity result $ ext{MIP}^*= ext{RE}$. By translating this into a continuous-logic framework, the authors derive Gödelian refutations of the Connes Embedding Problem and related axiomatizability barriers for both von Neumann and C*-algebras, including QWEP, MF, and nuclearity classes. The work also connects these logical undecidability results to Tsirelson’s problem and shows that certain probabilistic conjectures (notably Aldous–Lyons) can be refuted using the same foundational machinery, revealing deep links between quantum information, operator algebras, and stochastic group theory. The overarching theme is that computability limitations of quantum correlations propagate to fundamental structural questions about operator algebras, with wide-ranging implications for model theory, algebra, and probability.

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 Gödelian 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.

Theorems & Definitions (46)

• Theorem 2.1: $\operatorname{MIP}^*=\operatorname{RE}$
• Corollary 2.2
• Remark 2.3
• Proposition 2.4
• Theorem 4.1
• Theorem 4.2: Completeness theorem for continuous logic
• Corollary 4.3
• Theorem 4.4
• Lemma 4.5
• Lemma 4.6