Unique Games with Entangled Provers are Easy

TL;DR

This paper demonstrates that for unique games with entangled provers, the game value can be efficiently approximated using semidefinite programming, challenging the unique games conjecture in the entangled setting.

Contribution

It introduces a quantum rounding technique to approximate entangled game values via SDP and proves a parallel repetition theorem for these games.

Findings

Entangled unique games can be approximated by semidefinite programs.

The entangled variant of the unique games conjecture is false.

A parallel repetition theorem for entangled games is established.

Abstract

We consider one-round games between a classical verifier and two provers who share entanglement. We show that when the constraints enforced by the verifier are `unique' constraints (i.e., permutations), the value of the game can be well approximated by a semidefinite program. Essentially the only algorithm known previously was for the special case of binary answers, as follows from the work of Tsirelson in 1980. Among other things, our result implies that the variant of the unique games conjecture where we allow the provers to share entanglement is false. Our proof is based on a novel `quantum rounding technique', showing how to take a solution to an SDP and transform it to a strategy for entangled provers. Using our approximation by a semidefinite program we also show a parallel repetition theorem for unique entangled games.