Parallel repetition for entangled k-player games via fast quantum search

Abstract

We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val * (G) = 1 − ǫ, the n-fold repetition of G has entangled value val * (G ⊗n ) at most (1 − ǫ 3/2 ) Ω(n/sk 4 ) , where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G ⊗n ) ≤ (1 − ǫ 2 ) Ω(n/s) , due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition.Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1 − ǫ 2 ) Ω(n/sk 2 ) . We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n ≥ 1 such that val * (G ⊗n ) ≥ val * (G) n/k .Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.val * (G) of game G, which is the maximum success probability over all possible entangled strategies for the players.Recently, there has been significant interest in the parallel repetition of entangled games [KV11, CS14a, CS14b, JPY14, DSV14]. More formally, the n-fold parallel repetition of a game G is a game G ⊗n where the referee will sample n independent pairs of questions (x 1 , y 1 ), . . . , (x n , y n ) from the distribution µ. Alice receives (x 1 , . . . , x n ) and Bob receives (y 1 , . . . , y n ). They produce outputs (a 1 , . . . , a n ) and (b 1 , . . . , b n ), respectively, and they win only if V (x i , y i , a i , b i ) = 1 for all i. We call each i a "coordinate" of G ⊗n or "repetition" of G.Suppose we have a game G where val * (G) = 1 − ǫ. Intuitively, one should expect that val * (G ⊗n ) should behave as (1−ǫ) n . Indeed, this would be the case if the game G were played n times sequentially. However, there are counterexamples of games G and n > 1 where val * (G ⊗n ) = val * (G) (see Section 7). Despite such counterexamples, it has been shown that the classical value val(G ⊗n ) (i.e. where the players are restricted to using classical strategies) of a repeated game G ⊗n goes down exponentially with n, for large enough n [Raz98, Hol07]. This result is known as the Parallel...