Anchored parallel repetition for nonlocal games

Abstract

We introduce a simple transformation on two-player nonlocal games, called "anchoring", and prove an exponential-decay parallel repetition theorem for all anchored games in the setting of quantum entangled players. This transformation is inspired in part by the Feige-Kilian transformation (SICOMP 2000), and has the property that if the quantum value of the original game $G$ is $v$ then the quantum value of the anchored game $G_{\perp }$ is $1-(1-\alpha {)}^2\cdot (1-v)$ where $\alpha$ is a parameter of the transformation. In particular the anchored game has quantum value $1$ if and only if the original game $G$ has quantum value $1$. This provides the first gap amplification technique for general two-player nonlocal games that achieves exponential decay of the quantum value.

42 pages. Original version was published as "Hardness amplification for entangled games via anchoring" in the proceedings of Symposium on Theory of Computing 2017. This version is a revision to give more details on the proof of the quantum parallel repetition result. Classical multiplayer parallel repetition results no longer included, but can still be found in arXiv:1509.07466v1