Advantage of Quantum Strategies in Random Symmetric XOR Games

Abstract

Non-local games are widely studied as a model to investigate the properties of quantum mechanics as opposed to classical mechanics. In this paper, we consider a subset of non-local games: symmetric XOR games of n players with 0-1 valued questions. For this class of games, each player receives an input bit and responds with an output bit without communicating to the other players. The winning condition only depends on XOR of output bits and is constant w.r.t. permutation of players. We prove that for almost any n-player symmetric XOR game the entangled value of the game is Θ √ ln n n 1/4 adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is Θ( √ ln n) for almost any symmetric XOR game.

Random Constructions in Bell Inequalities: A Survey

“…This theorem implies that there does not exist a uniform constant C such that LV 3 (N ) ≤ C independently of the number of inputs N ; giving in this way a negative answer to the question posed by Tsirelson in [46]. It is advisable to extend the previous definition to LV 3…”

“…Symmetric XOR games. In the works [2], [3] and [4] Ambainis and coauthors studied the scenario of binary inputs correlation Bell inequalities with many parties T = (T x 1 ,··· ,xn ) x 1 ,··· ,xn∈{0,1} . In this case, it is well known that the quotient between ω * (T ) and ω(T ) can be equal to 2 n 2 for some particular inequalities T ( [5], [33]).…”

“…The reader should immediatly note that this problem is completely different from the one considered in the previous section, where unbounded Bell violations were proved by considering only a fixed number of parties n (say three) and increasing the number of inputs N . In order to clarify the connection between the works [2], [3], [4] and the explanation in this survey, let us recall the reader that any correlation Bell inequality T = (T x 1 ,··· ,xn ) x 1 ,··· ,xn verifying x 1 ,··· ,xn |T x 1 ,··· ,xn | = 1 can be trivially written as T = π(x 1 , · · · , x n )f (x 1 , · · · , x n ) x 1 ,··· ,xn , where π is a probability distribution over the set of inputs and f : X 1 × · · · × X N → {−1, 1} is a function. This gives a correspondence between the classical (resp.…”

“…Then, one can define a probability distribution on the set of these Bell inequalities by picking the (n + 1)-bit string (M 0 , M 1 , · · · , M n ) uniformly at random. The main results in [2] and [3] imply that for every ǫ > 0 there exist positive constants C 1 (ǫ), C 2 (ǫ) such that for a high enough n we have…”

“…B y (ω) ∈ {−1, 1} for every y); and quantum correlation matrices 1 Formally, Bell talked about a local hidden variable model (LHVM). 2 Note that P is not a probability distribution itself. For every x, y fixed we have that (P (a, b|x, y)) K a,b=1 is a probability distribution.…”