[Paper Review] MIP*=RE

Zhengfeng Ji, Anand Natarajan+3 more|arXiv (Cornell University)|Jan 13, 2020

Quantum Computing Algorithms and ArchitectureComputer Science33 citations

TL;DR

The paper proves MIP* (entangled-prover interactive proofs) equals RE (recursively enumerable languages), by constructing an interactive proof for the Halting problem and using a compression framework; as a consequence, entangled-value decision problems become undecidable and Tsirelson’s problem is resolved negatively.

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.

Motivation & Objective

• Motivate and formalize the MIP* framework and its significance in quantum interactive proofs.
• Show that MIP*(2,1) captures RE by reducing Halting to entangled nonlocal games.
• Leverage compression, introspection, and low-degree tests to construct undecidable entangled-value problems.
• Derive consequences for Tsirelson’s problem and Connes’ Embedding Conjecture.

Proposed method

• Define and analyze quantum and classical correlation sets Cqs, Cqa, and Cqc.
• Introduce a compression procedure for a family of normal-form nonlocal games to preserve entangled value across scales.
• Construct an infinite family of nonlocal games G_M,n such that val*(G_M,n) distinguishes halting vs non-halting M.
• Use introspection and quantum low-degree tests to robustly self-test distributions and enable the compression framework.
• Iteratively apply the compression to obtain a fixed-point game GM with val*(GM)=1 iff M halts.
• Show that undecidability of the entangled value implies a negative answer to Tsirelson’s problem and Connes’ Embedding Conjecture.

Experimental results

Research questions

• RQ1 Can RE be captured by a one-round two-prover interactive proof system with entangled provers?
• RQ2 Does there exist an efficient reduction from the Halting problem to deciding the entangled value of a two-prover nonlocal game?
• RQ3 What are the implications of MIP* = RE for Tsirelson’s problem and Connes’ Embedding Conjecture?
• RQ4 Can a compression/introspection framework overcome the gap limitations in prior MIP* constructions?

Key findings

• MIP* equals RE, giving MIP* = RE as a complete characterization of entangled-prover interactive proofs.
• There exists an efficient reduction from the Halting problem to deciding whether a two-player nonlocal game has entangled value 1 or at most 1/2.
• An explicit separation shows the closure of quantum tensor-product correlations is strictly contained in quantum commuting correlations, refuting Tsirelson’s problem.
• The results imply Connes’ Embedding Conjecture is false via established connections to von Neumann algebras.
• A synchronous/ PCC strategy witnesses val*(GM)=1 in halting cases, underpinning the compression-based Halting proof.