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

Abstract

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 G ⁿ 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 G ⁿ 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 ^*(G ⁿ ) decreases exponentially in n. First, for any game G on the uniform distribution, we show that \(\omega^*(G^n) = (1 - \epsilon^2)^{\Omega\left(\frac{n}{\log(|I||O|)} - |\log(\epsilon)|\right)}\), where |I| and |O| are the sizes of the input and output sets. From this result, we show that for any entangled game G, \(\omega^*(G^n) = (1 - {\epsilon^2})^{\Omega(\frac{n}{Q^4 \log(Q \cdot|O|)} - |\log(\epsilon/Q)|)}\) where p is the input distribution of G and \(Q = \max(\lceil \frac{1}{\min_{xy: p_{xy} \neq 0}\{\sqrt{p_{xy}}\}}\rceil, |I|)\).

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.