Generalized Tsirelson Inequalities, Commuting-Operator Provers, and Multi-Prover Interactive Proof Systems

TL;DR

This paper introduces a new method for bounding the power of nonlocal strategies in multi-prover quantum games, generalizes Tsirelson inequalities, and establishes complexity results for multi-prover interactive proof systems.

Contribution

It develops a novel approach using commuting-operator provers to derive tight bounds on nonlocal strategies and extends Tsirelson inequalities to multiple parties.

Findings

Derived tight n-party Tsirelson bounds for CHSH inequalities.

Proved NP-hardness of distinguishing entangled values in 3-prover games.

Established a new complexity-theoretic separation for multi-prover proof systems.

Abstract

A central question in quantum information theory and computational complexity is how powerful nonlocal strategies are in cooperative games with imperfect information, such as multi-prover interactive proof systems. This paper develops a new method for proving limits of nonlocal strategies that make use of prior entanglement among players (or, provers, in the terminology of multi-prover interactive proofs). Instead of proving the limits for usual isolated provers who initially share entanglement, this paper proves the limits for "commuting-operator provers", who share private space, but can apply only such operators that are commutative with any operator applied by other provers. Commuting-operator provers are at least as powerful as usual isolated but prior-entangled provers, and thus, limits for commuting-operator provers immediately give limits for usual entangled provers. Using this method, we obtain an n-party generalization of the Tsirelson bound for the Clauser-Horne- Shimony-Holt inequality for every n. Our bounds are tight in the sense that, in every n-party case, the equality is achievable by a usual nonlocal strategy with prior entanglement. We also apply our method to a 3-prover 1-round binary interactive proof for NEXP. Combined with the technique developed by Kempe, Kobayashi, Matsumoto, Toner and Vidick to analyze the soundness of the proof system, it is proved to be NP-hard to distinguish whether the entangled value of a 3-prover 1-round binary-answer game is equal to 1 or at most 1-1/p(n) for some polynomial p, where n is the number of questions. This is in contrast to the 2-prover 1-round binary-answer case, where the corresponding problem is efficiently decidable. Alternatively, NEXP has a 3-prover 1-round binary interactive proof system with perfect completeness and soundness 1-2^{-poly}.