The Parallel-Repeated Magic Square Game is Rigid

TL;DR

This paper proves that the n-round parallel repetition of the Magic Square game exhibits rigidity, meaning near-perfect success implies players' strategies are close to the ideal entangled state and measurements, with bounds depending polynomially on n and the error.

Contribution

It establishes the rigidity of the parallel-repeated Magic Square game, providing quantitative bounds on the closeness to ideal strategies based on success probability.

Findings

Players' shared state is close to 2n EPR pairs when success probability is high.

Players' measurements are close to the ideal measurements under local isometry.

The bounds on closeness depend polynomially on the number of rounds and error.

Abstract

We show that the $n$-round parallel repetition of the Magic Square game of Mermin and Peres is rigid, in the sense that for any entangled strategy succeeding with probability $1 -\varepsilon$, the players' shared state is $O(\mathrm{poly}(n\varepsilon))$-close to $2n$ EPR pairs under a local isometry. Furthermore, we show that, under local isometry, the players' measurements in said entangled strategy must be $O(\mathrm{poly}(n\varepsilon))$ close to the "ideal" strategy when acting on the shared state.