We consider one-round games between a classical referee and two players. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of players? Classically, efforts to resolve this question, open for many years, have culminated in Raz's celebrated parallel repetition theorem on one hand, and in efficient product testers for PCPs on the other. In the case where players share entanglement, the only previously known results are for special cases of games, and are based on techniques that seem inherently limited. Here we show for the first time that the maximum success probability of entangled games can be reduced through parallel repetition, provided it was not initially 1. Our proof is inspired by a seminal result of Feige and Kilian in the context of classical two-prover one-round interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest.
A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples n independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than
(1−ϵ), what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141–172], who showed that the players cannot win all copies in the repeated game with probability better than
(1−ϵ/2)Ω(nϵ2/c) (here c is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is
(1−ϵ/2)Ω(ϵn) as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant
0<δ<ϵ, the probability that the players win a
(1−ϵ+δ) fraction of the games in the parallel repetition is at most
exp(−Ωϵ(δ3n/c)) (here the constant may depend on
ϵ); our result has applications to testing Bell inequalities, since it implies that the parallel repetition of the CHSH game can be used to get an experiment that has a very large classical versus quantum gap. Our first bound is independent of the answer length and has a better dependence on
ϵ. By the recent work of Raz [Proceedings of the
49th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 369–373], this bound is tight. Our bound gives a generic way to improve the soundness of a probabilistically checkable proof (PCP), in a way that is independent of the answer length of the PCP. Using it, for every k, one can convert any q query PCP with answer length c, size
sc, and soundness
(1−ϵ) into a two-query PCP with answer length
ck, size
O(ck(2s)k), and soundness
(1−ϵ/2q)Ω(ϵk/q). Another consequence of our bound is that the unique games conjecture of Khot [Proceedings of the
34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] can now be shown to be equivalent to the following a priori weaker conjecture: There is an unbounded increasing function
f:R+→R+ such that for every
ϵ>0, there exists an alphabet size
M(ϵ) for which it is NP-hard to distinguish a unique game with alphabet size M in which a
(1−ϵ2) fraction of the constraints can be satisfied from one in which a
(1−ϵf(1/ϵ)) fraction of the constraints can be satisfied.
We present a strong parallel repetition theorem for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our result is the first parallel repetition theorem for entangled games involving more than two players. Furthermore, our theorem applies to games where the players are allowed to output (possibly entangled) quantum states as answers. More specifically, let
G be a
k-player free game, with entangled value
val∗(G)=1−ε. We show that the entangled value of the
n-fold repetition of
G,
val∗(G⊗n), is at most
(1−ε)Ω(n/k2). In the traditional setting of
k=2 players, our parallel repetition theorem is optimal in terms of its dependence on
ε and
n. For an arbitrary number of players, our result is nearly optimal: for all
k, we exhibit a
k-player free game
G and
n>1 such that
val∗(G⊗n)≥val∗(G)n/k. Hence, exponent of the repeated game value cannot be improved beyond
Ω(n/k). Our parallel repetition theorem improves on the prior results of [Jain, et al. 2014] and [Chailloux, Scarpa 2014] in a number of ways: (1) our theorem applies to a larger class of games (arbitrary number of players, quantum outputs); (2) we demonstrate that strong parallel repetition holds for the entangled value of free games: i.e., the base of the repeated game value is
1−ε, rather than
1−ε2; and (3) there is no dependence of the repeated game value on the input and output alphabets of
G. In contrast, it is known that the repeated game value of classical free games must depend on the output size. Thus our results demonstrate a seperation between the behavior of entangled games and classical games.
Consider a game where a refereed chooses (x,y) according to a publiclyknown distribution PXY, sends x to Alice, and y to Bob. Withoutcommunicating with each other, Alice responds with a value "a" and Bobresponds with a value "b". Alice and Bob jointly win if a publiclyknown predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probabilitythat Alice and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998)shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most v'(n/log(s)), where s is the maximal number of possible responses from Alice and Bob in the initial game, and v' is a constant depending only on v. In this work, we simplify Raz's proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication among them.
We show a parallel repetition theorem for the entangled value ω*(G) of any two-player one-round game G where the questions (x, y) ∈ X × Y to Alice and Bob are drawn from a product distribution on X × Y. We show that for the k-fold product Gkof the game G (which represents the game G played in parallel k times independently) ω*(Gk) = (1 - (1 - ω*(G))3)Ω(k/Iog(|A|·|B|)where A and B represent the sets from which the answers of Alice and Bob are drawn. The arguments we use are information theoretic and are broadly on similar lines as that of Raz [1] and Holenstein [2] for classical games. The additional quantum ingredients we need, to deal with entangled games, are inspired by the work of Jain, Radhakrishnan, and Sen [3], where quantum information theoretic arguments were used to achieve message compression in quantum communication protocols.