[Paper Review] A multi-prover interactive proof for NEXP sound against entangled provers

Tsuyoshi Ito, Thomas Vidick|arXiv (Cornell University)|Jul 2, 2012

Complexity and Algorithms in Graphs29 references50 citations

TL;DR

This paper establishes that the class of languages with multi-prover interactive proofs using entangled provers, denoted MIP*, contains NEXP—the class of problems decidable in non-deterministic exponential time. The authors prove that the multilinearity test, a key component in classical interactive proof systems, remains sound even when provers share entanglement, thereby demonstrating that entanglement does not weaken the computational power of multi-prover interactive proofs.

ABSTRACT

We prove a strong limitation on the ability of entangled provers to collude in a multiplayer game. Our main result is the first nontrivial lower bound on the class MIP* of languages having multi-prover interactive proofs with entangled provers; namely MIP* contains NEXP, the class of languages decidable in non-deterministic exponential time. While Babai, Fortnow, and Lund (Computational Complexity 1991) proved the celebrated equality MIP = NEXP in the absence of entanglement, ever since the introduction of the class MIP* it was open whether shared entanglement between the provers could weaken or strengthen the computational power of multi-prover interactive proofs. Our result shows that it does not weaken their computational power: MIP* contains MIP. At the heart of our result is a proof that Babai, Fortnow, and Lund's multilinearity test is sound even in the presence of entanglement between the provers, and our analysis of this test could be of independent interest. As a byproduct we show that the correlations produced by any entangled strategy which succeeds in the multilinearity test with high probability can always be closely approximated using shared randomness alone.

Motivation & Objective

• To resolve the fundamental open question of whether shared entanglement between provers strengthens or weakens the computational power of multi-prover interactive proof systems.
• To establish a nontrivial lower bound on the complexity class MIP*, showing it contains NEXP, thus extending the classical MIP = NEXP result to the quantum setting with entanglement.
• To analyze the robustness of the multilinearity test under quantum entanglement, proving it remains sound even when provers use entangled strategies.
• To show that any entangled strategy succeeding in the multilinearity test with high probability can be closely approximated using only shared randomness, implying a form of classical simulatability.
• To provide a foundational result for quantum complexity theory by demonstrating that entanglement does not reduce the soundness of classical proof systems in this context.

Proposed method

• Adapt the classical multilinearity test from Babai, Fortnow, and Lund (1991) to the quantum setting with entangled provers, proving its soundness under entanglement.
• Use a novel analysis of the multilinearity test that bounds the probability of success under entangled strategies, relying on the structure of linear functionals over finite fields.
• Apply a bias-approximation technique to show that any entangled strategy succeeding with high probability must produce correlations close to those achievable with shared randomness alone.
• Construct a protocol where the verifier uses a polynomial-time algorithm to generate a biased probability space over functions, enabling soundness analysis under entanglement.
• Leverage the arithmetization of Boolean circuits and the construction of low-bias probability spaces to simulate the multilinearity test with quantum provers.
• Combine the soundness of the multilinearity test under entanglement with the completeness of a related protocol to achieve perfect completeness and bounded soundness.

Experimental results

Research questions

• RQ1 Does shared entanglement between provers in a multi-prover interactive proof system weaken or strengthen the computational power of the system?
• RQ2 Is the multilinearity test, a cornerstone of classical interactive proof systems, still sound when provers are allowed to share entanglement?
• RQ3 Can the correlations produced by entangled provers in a multilinearity test be approximated by classical shared randomness?
• RQ4 What is the minimum lower bound on the complexity class MIP* under entanglement, and does it contain NEXP?
• RQ5 Can the soundness of classical interactive proof systems be preserved when the provers are quantumly entangled?

Key findings

• The class MIP* contains NEXP, meaning that multi-prover interactive proofs with entangled provers are at least as powerful as classical multi-prover interactive proofs.
• The multilinearity test remains sound even when provers use entangled strategies, which is a key technical contribution showing that entanglement does not enable cheating in this test.
• Any entangled strategy that succeeds in the multilinearity test with high probability can be closely approximated using only shared classical randomness, indicating that quantum nonlocality does not provide an advantage in this setting.
• The soundness error in the protocol is bounded by 5/8 plus a term that decreases polynomially with the field size, ensuring that the protocol is sound for any non-zero input.
• The construction of a low-bias probability space over functions via arithmetic expressions enables the verification of multilinearity under entanglement.
• The result implies that MIP ⊆ MIP*, showing that entanglement does not reduce the computational power of multi-prover interactive proof systems.