# 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 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 Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win Gn if they win each instance of G.

We show that the entangled value of Gn decreases exponentially in n if the original game has entangled value strictly less than 1. We also extend this result to any distribution, but with worst parameters. 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.