MIP*=RE (2001.04383v3)

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.

• The paper establishes that entangled-prover interactive proofs (MIP*) are computationally equivalent to recursively enumerable languages (RE).
• The authors introduce quantum low-degree tests and game-theoretic reductions to encode complex problems, including the Halting Problem, within MIP* systems.
• The work redefines computational boundaries in quantum complexity theory, bridging classical enumeration methods with quantum interactive proofs.

An Expert Overview of the Result $\MIP^* = \RE$

The paper "$\MIP^* = \RE$" establishes a notable result in the field of quantum computation and complexity theory, addressing a long-standing question about the class of languages decidable by interactive proofs with entangled quantum provers, denoted as $\MIP^*$. This document's central theorem asserts that the complexity class $\MIP^*$ is equivalent to the class $\RE$ of recursively enumerable languages, demonstrating the profound computational capabilities of entangled-prover systems. This overview will succinctly discuss the key concepts, findings, and implications presented in the paper.

Key Concepts

Entangled-Prover Interactive Proofs

The focus of the paper is multiprover interactive proof systems where the provers may utilize entanglement, yielding the class $\MIP^*$. In contrast to classical multiprover interactive proofs ($\MIP$), where the communicative constraints are strict yet derive from classical information theory, $\MIP^*$ systems leverage quantum entanglement to enable a richer and more complex interaction framework.

Characterizing $\MIP^*$

The equivalence $\MIP^* = \RE$ establishes that the class of problems decidable by a verifier interacting with multiple entangled provers encompasses the entirety of recursively enumerable sets, which are characterized by the ability to be listed (or enumerated) by a Turing machine, albeit not necessarily decidable.

Technical Merits

1. Quantum Low-Degree Test: Integral to the arguments is the rigorous expansion of quantum low-degree testing, ensuring that interactive protocols with quantum verifiers can enforce constraints that assert cooperative consistency among participating provers at a quantum level.
2. Game-Theoretic Reductions: The authors utilize nonlocal games to translate problems in $\RE$ into questioning frameworks for $\MIP^*$ systems. Establishing that every recursively enumerable language can be encoded and decided through an appropriately constructed nonlocal game is core to the thesis.
3. Entanglement Complexity: By focusing on entangled-prover tactics, the paper delineates how dimensionality in hilbert space (entanglement size) influences computational power, showcasing that uncomputable problems can be mapped to decision protocols for entangled provers.

Strong Numerical Impacts

The authors demonstrate that the Halting Problem, a prototypical undecidable problem, can be reduced into the framework of interactive proof games under this setting. Through intricate reduction techniques and recursive compression frameworks, they prove that robust bounds on decision processes can be manipulated to simulate enumerative procedures equivalent to $\RE$.

Theoretical and Practical Implications

The result $\MIP^* = \RE$ knots together profound theoretical insights with practical computational scenarios, suggesting several implications:

• Theoretical Depth: It refines our understanding of quantum computational complexity, drawing explicit lines between classical enumeration processes and novel quantum-verifier paradigms.
• Quantum Information Foundations: Addressing questions linked to quantum correlation sets, notably Tsirelson's problem and Connes' embedding conjecture, and demonstrating undecidability through quantum protocols offers insights into the mathematical and physical underpinnings of quantum mechanics.
• Computational Boundaries: This result transforms the perceived boundaries of computationally hard problems, hinting at the expansive nature of quantum computation capabilities.

Future Directions

This development propels further research in quantum complexity theory. The extent of computational systems utilizing quantum entanglement opens doors for examining undecidability in new contexts, exploring deeper residues in computational logic tied to infinite-dimensional Hilbert spaces, and potentially yielding architectures for realizing quantum-enhanced computing devices.

In conclusion, by bridging $\MIP^*$ to $\RE$, the paper not only breaks new ground in understanding quantum interactive proofs but also enriches the discourse surrounding undecidability, computational limits, and the role of entanglement—a synthesis of complexity theory and quantum mechanics with profound implications.