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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 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. 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(nlog(|I||O|)−|log(ε)|), where |I| and |O| are the dimensions of the input and output space. From this result, we show that for any entangled game G, ω∗(Gn)=(1−ε2)Ω(nQ4log(Q⋅|O|)−|log(ε/Q)|) where p is the input distribution of G and Q=max(⌈1minxy:pxy≠0(pxy)√⌉,|I|). This is the first time exponential decay is shown for the parallel repetition of any entangled game. 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.
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https://hal.inria.fr/hal-00927544
Contributor : André Chailloux <>
Submitted on : Thursday, January 16, 2014 - 3:29:58 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
Long-term archiving on: : Wednesday, April 16, 2014 - 11:48:26 PM

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André Chailloux, Giannicola Scarpa. Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost. QIP 2014 - Quantum Information Processing, Feb 2014, Barcelona, Spain. ⟨hal-00927544⟩

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