MIP*=RE

TL;DR

This paper proves that the class MIP* equals RE, showing that quantum interactive proof systems with entangled provers can decide all recursively enumerable languages, leading to implications for the Halting Problem and Connes' embedding conjecture.

Contribution

It establishes the equality MIP* = RE using advanced quantum low-degree tests and recursive compression, connecting quantum complexity with fundamental problems in mathematics.

Findings

Decidability of the Halting Problem reduces to entangled value in nonlocal games.

The set of quantum tensor product correlations is strictly contained in quantum commuting correlations.

Refutes Connes' embedding conjecture by demonstrating the strict inclusion of correlation sets.

Abstract

We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. Our proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS 2018) and the classical low-individual degree test of (Ji, et al., 2020) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019). An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a two-player nonlocal game has entangled value $1$ or at most $1/2$. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure $C_{qa}$ of the set of quantum tensor product correlations is strictly included in the set $C_{qc}$ of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of Connes' embedding conjecture from the theory of von Neumann algebras.