Provable Advantage for Quantum Strategies in Random Symmetric XOR Games
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 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 -player symmetric XOR game the entangled value of the game is adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is for almost any symmetric XOR game.
Non-local games provide a simple way to test the difference between quantum mechanics and the classical world. A prototypical example of a non-local game is the CHSH game  (based on the CHSH inequality of ). In the CHSH game, we have two players who cannot communicate between themselves but may share common random bits or a bipartite quantum state (which has been exchanged before the beginning of the game). A referee sends one uniformly random bit to the player and another bit to the player. Players respond by sending one-bit answers . They win in the following 2 cases:
If at least one of is equal to 0, players win if they produce such that ;
If , players win if they produce such that ;
Classically, CHSH game can be won with probability at most 0.75. In contrast, if players use an entangled quantum state, they can win the game with probability .
Other non-local games can be defined by changing the number of players, the number of possible questions and answers and the winning condition. Many non-local games have been studied and, in many cases, strategies that use an entangled quantum state outperform any classical strategy.
Recently , it has been shown that, for a large class of non-local games, quantum strategies are better than any classical strategy for almost all games in this class. Namely,  considered 2-player games in which the questions are taken from the set and the winning condition is either or , depending on . (Games with a winning condition of such form are called XOR games.)  showed that, for fraction of all such games, the entangled value of the game is at least 1.2… times its classical value.
Then , it was discovered that a similar effect might hold for another class of games: -player symmetric XOR games with binary questions. Namely,  showed a gap between entangled and classical values of order - assuming that a non-rigorous argument about the entangled value is correct.
In this paper, we make this gap rigorous, by proving upper and lower bounds on the entangled value of a random game in this class. We show that, with a high probability, the entangled value is equal to . The quantum-vs-classical gap of follows by combining this with the fact that the classical value is of the order (shown in ).
To prove this result, we use an expression for the entangled value of a symmetric -player XOR game with entangled answers from . This expression reduces finding the entangled value to maximizing the absolute value of a polynomial in one complex variable. If conditions for the XOR game are chosen at random, this expression reduces to random trigonometric polynomials studied in .
Although maxima of random trigonometric polynomials have been studied in , they have been studied under different conditions. For this reason, we cannot apply the results from  directly. Instead, we adapt the ideas from  to prove a bound on maxima of random trigonometric polynomials that would be applicable in our setting.
A non-local game with players proceeds as follows:
Players are separated so that they cannot communicate – hence the name non-local,
The players receive inputs where is the set of possible inputs. -th player receives ,
The players respond with outputs where is the set of possible outputs.
The winning condition is consulted to determine whether the players win or lose. The condition is known to everyone at the start of the game.
The players are informed of the rules of the game and they can agree upon a strategy and exchange other information. In the classical case players may only use shared randomness. In the quantum case they can use an entangled quantum state which is distributed to the players before the start of the game.
We will restrict ourselves to the case when and the vector of inputs is chosen uniformly at random. In an XOR game, the winning condition depends only on and the parity of the output bits . A game is symmetric if the winning condition does not change if are permuted.
The winning conditions of a symmetric XOR game can be described by a list of bits: , where the players win if and only if when .
The entangled value of the game is the probability of winning minus the probability of losing in the conditions that the players can use a shared quantum-physical system. In this paper, we study the value of symmetric XOR games when the winning condition is chosen randomly from the uniform distribution of all -bit lists. We use the following lemma (which follows from a more general result by Werner and Wolf for non-symmetric XOR games ):
Lemma 1 (See ).
The entangled value of a symmetric XOR game  is
where is the probability that players are given an input vector with variables .
In our case, since is uniformly random, we have .
In the following sections we introduce additional notation to keep the proofs more concise as well as to keep in line with the original proofs in :
The Rademacher system is a set of functions for over such that , where is the -th digit after the binary point in the binary expansion of . Rademacher system will turn out to be a convenient way to state that are random variables that follow a uniform distribution: if is chosen randomly from a uniform distribution on , then generates a uniformly random element from . That in turn corresponds to coefficients in eq. (1) being picked randomly.
Furthermore, we define
3 Main Result
By adapting the work of Salem and Zygmund  on the asymptotics of random trigonometric polynomials, we show
Our proof yields and .
We will now show how these two theorems lead to an asymptotic bound for the entangled value of a random game.
For almost all -player symmetric quantum XOR games the value of the game is asymptotically .
4 Proof of Upper and Lower Bounds
We now proceed to prove theorems 3.1 and 3.2. Our proof is based on an old result by Salem and Zygmund , in which they prove bounds on the asymptotics of random trigonometric polynomials in a different setting (in which the coefficients are not allowed to depend on ).
Due to the difference in the two settings, we cannot immediately apply the results from . Instead, we prove corresponding theorems for our setting, re-using the parts of proof from  which also work in our case and replacing other parts with different arguments.
Lemma 2 (From ).
Let , where is the Rademacher system and are real constants. Let and let be any real number. Then
Lemma 3 (From ).
Let , , , be a bounded real function. Suppose that
Then, for any positive number ,
Furthermore, when ,
Lemma 4 (From ).
Let be real and be a trigonometric polynomial of order , with real or imaginary coefficients. Let denote the maximum of and let be a positive number less than 1. Then there exists an interval of length not less than in which .
Lemma 5 (From ).
Let , and suppose that
(clearly, ). Let . Then
If is even:
If is odd:
Proof of Theorem 3.1.
First, the lower bound clause of Lemma 2 applied to gives for any real (we will assign its value later, at our convenience),
The second step is to establish an upper bound for . We start out in a similar fashion, by applying Lemma 2:
One can verify that
And by Lemma 6,
So far we have established the two prerequisites for Lemma 5:
The third step is to apply Lemma 5 with , , and . This results in
Finally we show that for suitably chosen , and the claim follows. Set having such that . We deal with the two claims separately:
But . We can choose arbitrarily close to and arbitrarily close to 0 to obtain . ∎
Since is positive, .
Proof of Theorem 3.2.
We will examine . By Lemma 4 there exists such that:
On the other hand, by Lemma 2 we obtain:
Have and multiply both sides by , where . Then
The sum over all converges:
Since the exponent function is non-negative and the whole sum converges, it is safe to interchange sum and integral:
Therefore, for almost all
Hence, for almost all there exists such that for all
It follows that
We have proven that the entangled value of almost any -player symmetric XOR game is and therefore by a factor of greater than its classical value. However, our numerical experiments indicate that neither of the coefficients and in eq. 2 are tight.
In Fig. 1 there is plotted the mean value of the coefficient over a sample of games for each up to 100. We speculate that the actual constant is approaching .
In this paper we have dealt with a small portion of non-local games. In particular, the case of random non-symmetric games is still open and there has been little progress in multiplayer XOR games with input. The primary hurdle in the -player -input setting is at the moment it lacks a description in terms of algebraic and analytic expressions. Recently an approach using the theory of operator norms has been successful in proving the entangled value of 3-player, -input XOR game .
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