In a two-player game, two cooperating but non communicating players, Alice
and Bob, receive inputs taken from a probability distribution. Each of them
produces an output and they win the game if they satisfy some predicate on
their inputs/outputs. The entangled value ω∗(G) of a game G is the
maximum probability that Alice and Bob can win the game if they are allowed to
share an entangled state prior to receiving their inputs.
The n-fold parallel repetition Gn of G consists of n instances of
G where the players receive all the inputs at the same time and produce all
the outputs at the same time. They win Gn if they win each instance of G.
In this paper we show that for any game G such that ω∗(G)=1−ε<1, ω∗(Gn) decreases exponentially in n. First, for
any game G on the uniform distribution, we show that ω∗(Gn)=(1−ε2)Ω(log(∣I∣∣O∣)n−∣log(ε)∣), where ∣I∣ and ∣O∣ are the sizes of the input
and output sets. From this result, we show that for any entangled game G,
ω∗(Gn)≤(1−ε2)Ω(Qlog(∣I∣∣O∣)n−Q∣log(ε)∣) where p is the input distribution of G and
Q=minxypxy∣I∣2maxxypxy2. This implies parallel
repetition with exponential decay as long as minxy{pxy}=0 for
general games. To prove this parallel repetition, we introduce the concept of
\emph{Superposed Information Cost} for entangled games which is inspired from
the information cost used in communication complexity.Comment: In the first version of this paper we presented a different, stronger
Corollary 1 but due to an error in the proof we had to modify it in the
second version. This third version is a minor update. We correct some typos
and re-introduce a proof accidentally commented out in the second versio