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A parallel repetition theorem for all entangled games

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Published on Jan 31, 2017

"The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known: in fact, it was an open question of whether the entangled value of every repeated game goes to $0$ as the number of repetitions goes to infinity (provided the base game is non-trivial, i.e., doesn't have entangled value $1$).

We prove that the entangled value of a two-player non-trivial game $G$ repeated $n$ times in parallel is at most $c_G n^{-1/4} \log n$ for a constant $c_G$ depending on $G$. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to $0$ for \emph{all} non-trivial games. Central to our proof is a combination of both classical and quantum correlated sampling.

Furthermore, we prove a \emph{concentration bound} for parallel repetition, which shows that the probability the fraction of games won in the parallel repetition of $G$ significantly deviates from the entangled value of $G$ goes to $0$ polynomially fast. We then give an application of this concentration bound to the so-called Quantum PCP Conjecture."

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