Interactive proofs with approximately commuting provers

TL;DR

This paper introduces a new approach to understanding the power of the class MIP* with entangled provers by establishing bounds based on approximately commuting strategies, connecting complexity theory, quantum physics, and operator algebras.

Contribution

It presents a rounding scheme for the hierarchy of semidefinite programs, linking approximate commutation to the class MIP* and its variants, and explores implications for computability and cryptography.

Findings

Bound on commutator norm in approximately commuting strategies

Upper bound on MIP*_δ in terms of exponential time

NEXP contained in MIP*_{2^{-poly}} with high completeness and soundness

Abstract

The class $\MIP^*$ of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Little is known about the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. This hierarchy converges to a value that is only known to coincide with the provers' maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes' embedding conjecture. No bounds on the rate of convergence are known. We introduce a rounding scheme for the hierarchy, establishing that any solution to its $N$-th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by $O(\ell^2/\sqrt{N})$ in operator norm, where $\ell$ is the number of possible answers per prover. Our rounding scheme motivates the introduction of a variant of $\MIP^*$, called $\MIP_\delta^*$, in which the soundness property is required to hold as long as the commutator of operations performed by distinct provers has norm at most $\delta$. Our rounding scheme implies the upper bound $\MIP_\delta^* \subseteq \DTIME(\exp(\exp(\poly)/\delta^2))$. In terms of lower bounds we establish that $\MIP^*_{2^{-\poly}}$, with completeness $1$ and soundness $1-2^{-\poly}$, contains $\NEXP$. The relationship of $\MIP_\delta^*$ to $\MIPstar$ has connections with the mathematical literature on approximate commutation. Our rounding scheme gives an elementary proof that the Strong Kirchberg Conjecture implies that $\MIPstar$ is computable. We discuss applications to device-independent cryptography.