Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost

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 - varepsilon < 1, ω^*\(Gⁿ\) decreases exponentially in n. First, for any game G on the uniform distribution, we show that ω^*\(Gⁿ\) = \(1 - varepsilon²\)^{Ωleft\(frac{n}{log(|I||O|\)} - |log\(varepsilon\)|right)}, 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^*\(Gⁿ\) le \(1 - varepsilon²\)^{Ω\(frac{n}{Qlog(|I||O|\)} - frac{|log\(varepsilon\)|}{Q})} where p is the input distribution of G and Q= frac{|I|² max_xy p_xy² }{min_xy p_xy }. This implies parallel repetition with exponential decay as long as min_xy \{p_xy\} neq 0 for general games. To prove this parallel repetition, we introduce the concept of Superposed Information Cost for entangled games which is inspired from the information cost used in communication complexity.