Near-Optimal and Explicit Bell Inequality Violations

Entangled quantum systems can exhibit correlations that cannot be simulated classically. For historical reasons such correlations are called"Bell inequality violations."We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a strategy using a maximally entangled state with local dimension $n$ (e.g., $\log n$ EPR-pairs), while we show that the winning probability of any classical strategy differs from ${1}/{2}$ by at most $O((\log n)/\sqrt{n})$. The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here $n$-dimensional entanglement allows the game to be won with probability $1/(\log n)^2$, while the best winning probability without entanglement is $1/n$. This near-linear ratio is almost optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.

the inputs x ∈ {0, 1} and y ∈ {0, 1} are uniformly distributed, and Alice and Bob win the game if their respective outputs a ∈ {0, 1} and b ∈ {0, 1} satisfy a ⊕ b = x ∧ y; in other words, a should equal b unless x = y = 1. The classical value of this game is easily seen to be ω(G) = 3/4, while the entangled value is known to be ω * (G) = 1/2 + 1/(2 √ 2) ≈ 0.85. The entangled value is achieved already with 2-dimensional entanglement (i. e., one EPR-pair), so ω * (G) = ω * 2 (G) for this game [27]. One common figure of merit of a game is that of the Bell inequality violation exhibited by a game, which is defined as the ratio of entangled and classical values. More generally, we allow to replace the values (which are the maximum winning probabilities) by biases around some arbitrary "center" p ∈ [0, 1], where by bias we mean the maximum distance of the winning probability from p. For instance, by using p = 1/2 as the center, one can see that the CHSH game above exhibits a Bell inequality violation of √ 2. 1 In Section 2.2 we explain the origin of the term "Bell inequality violation," define it more formally, and explain the close relationship between games and Bell inequalities.
In two recent papers, Junge et al. [14,13] studied how large a Bell inequality violation one can obtain. In terms of upper bounds, [14] proved that the maximum Bell inequality violation ω * n (G)/ω(G) obtainable with entangled strategies of local dimension n, is at most O(n), and [13,Theorem 6.8] proved that if Alice and Bob have at most k possible outputs each, then the violation ω * (G)/ω(G) is at most O(k), irrespective of the amount of entanglement they can use. (This improved an earlier O(k 2 ) upper bound due to Degorre et al. [7], and was also obtained for the special case of games by Dukaric [8,Theorem 4].) In terms of lower bounds, [14] showed the existence of a Bell inequality violation of Ω( √ n/(log n) 2 ), where n is both the entanglement dimension and the number of outputs of Alice and Bob. This was improved to √ n/ log n in [13]. Both constructions are probabilistic, and the proofs show that with high probability the constructed games exhibit a large violation, yielding the existence of such games, without giving an explicit formulation. Their proofs are heavily based on connections to the mathematically beautiful areas of Banach spaces and operator spaces, but as a result are arguably somewhat inaccessible to those unfamiliar with these areas, and it is difficult to get a good intuition for them. (It is actually possible to analyze their game and reprove many of their results-often with improved parameters-using elementary probabilistic techniques [26].) Our main result in this paper is to exhibit two fully explicit games with strong Bell inequality violations. The first achieves the same violation as [13], namely √ n/ log n, where n is both the number of possible outputs and the dimension of entanglement. The second achieves the much stronger violation of n/(log n) 2 , which is optimal up to a polylogarithmic factor by the results of [14,13]. Even though the second game gives a much stronger violation, the first one still has some merit; for instance, entanglement allows the players to win it with certainty. Interestingly, although addressing a question in mathematical physics, both games are inspired by earlier work in theoretical computer science (communication complexity and unique games, respectively), and so is their analysis. In the remainder of this introduction we provide an overview of the two games, followed by some discussion and comparison.

The Hidden Matching game
The "Hidden Matching" problem was introduced in quantum communication complexity by Bar-Yossef et al. [1], and many variants of it were subsequently studied [12,11,10]. The original version is as follows, where it should be noted that now we allow communication, in contrast to the setting of non-local games. Let n be a power of 2. Alice is given input x ∈ {0, 1} n and Bob is given a perfect matching M, i. e., a partition of the set [n] = {1, . . . , n} into n/2 disjoint pairs {i, j}. Both inputs are uniformly distributed. 2 We allow one-way communication from Alice to Bob, and Bob is required to output a pair {i, j} ∈ M and a bit v ∈ {0, 1}. They In Section 3.1 we show that if Alice sends Bob a c-bit message, then their optimal winning probability is 1/2 + Θ(c/ √ n). Bar-Yossef et al. [1] earlier proved this for c = Θ( √ n), using information theory. However, their tools seem unable to give good bounds on the success probability for much smaller c. Instead, the main mathematical tool we use in our analysis is the so-called "KKL inequality" [15] from Fourier analysis of Boolean functions. Roughly speaking, this inequality implies that if the message that Alice sends about x is short, then Bob will not be able to predict the parity x i ⊕ x j well for many {i, j} pairs. His matching M is uniformly distributed, independent of x, and contains only n/2 of all n 2 possible {i, j} pairs. Hence it is unlikely that he can predict any one of those n/2 parities well. The KKL inequality was used before to analyze another variant of Hidden Matching in [11], though their analysis is different and more complicated because their variant of Hidden Matching is a promise problem with a non-product input distribution.
The following non-local version of the Hidden Matching problem (and the entangled strategy for it) is originally due to Buhrman, and related problems were studied in [12,10].
where the dot indicates inner product (modulo 2) of two log n-bit strings.
Observe that Alice has n possible outputs a and Bob has 2 · n/2 = n possible outputs ({i, j}, d) given his matching.
A classical strategy that wins this game induces a protocol for the original Hidden Matching problem with communication c = log n bits and the same winning probability p, as follows. Alice sends Bob the log n-bit output a from the non-local strategy, Bob computes v = (a·(i⊕ j))⊕d and outputs ({i, j}, v). We have that v = x i ⊕x j with probability p. Hence, our bound for the original communication problem implies that no classical strategy can win with probability that differs from 1/2 by more than O((log n)/ √ n).
2 All our results also hold with minor modifications for the case that Bob's matching is chosen uniformly from the set {M k | k ∈ {0, . . . , n/2 − 1}}, where the matching M k consists of the pairs {i, j} where i ≤ n/2 and j = n/2 + 1 + (i + k − 1 mod n/2). This has the advantage of lowering the number of possible inputs to Bob to n/2. The main thing is to notice that equation (3.1) still holds with respect to this new distribution on Bob's matching if we replace the right-hand side by 2/n, and the rest of the proof goes through.
In contrast, there is a strategy that wins with probability 1 using log n EPR-pairs, which shows ω * n (G) = 1. 3 This game therefore exhibits a Bell violation of Ω( √ n/ log n) (by measuring the maximal deviation of the winning probability from 1/2). This order is the same as that obtained by Junge et al. [14,13], but our game is fully explicit and arguably simpler (which would help any future experimental realization). One might feel that our reduction to a communication complexity lower bound is responsible for losing the log n factor; however in Theorem 3.9 we exhibit a classical strategy with winning probability 1/2 + Ω( (log n)/n). This shows that at least the square root of the log-factor is really necessary.

The Khot-Vishnoi game
Our second non-local game derives from the work of Khot and Vishnoi [20] on the famous Unique Games Conjecture (UGC), which was introduced by Khot [19]. Although not necessary for the rest of this paper, we now provide some background and motivation. Roughly speaking, the UGC says that approximating the classical value of so-called unique games is a hard problem, even if we are only interested in a very rough approximation that can tell the difference between value less than ε and value more than 1 − ε. This conjecture implies many other hardness-of-approximation results that do not seem obtainable using the more standard techniques based on the PCP theorem. Khot and Vishnoi considered the standard semidefinite programming (SDP) relaxation of the classical value and showed that there are games for which it provides a very poor approximation, in the sense that the classical value is close to 0, yet the SDP relaxation is close to 1. This so-called integrality gap demonstrates that the standard SDP relaxation, which can be computed efficiently, does not lead to an algorithm contradicting the UGC.
Kempe, Regev, and Toner [18] already observed that they could combine their "quantum rounding" technique with the game of [20] to get a game with n possible outputs exhibiting a Bell inequality violation of n ε for some small constant ε > 0, using entanglement dimension n. Our main contribution in the second part of this paper is a refined (and at the same time simpler) analysis of both the Khot-Vishnoi game and of the quantum rounding technique. We show that, somewhat surprisingly, nearly optimal violations can be obtained using this method.
We first give a precise definition of the Khot-Vishnoi (KV) game.
Definition 1.2 (Khot-Vishnoi Game (KV n )). The game is parametrized by an integer n, which we assume to be a power of 2, and a "noise parameter" η ∈ [0, 1/2]. Consider the group ({0, 1} n , ⊕) of n-bit strings together with bitwise addition mod 2, and let H be the subgroup containing the n Hadamard codewords. 4 This subgroup partitions {0, 1} n into 2 n /n cosets of n elements each. Alice receives a uniformly random coset x as input, which we can think of as u ⊕ H for uniformly random u ∈ {0, 1} n . Bob receives a coset y obtained from Alice's by adding a string of low Hamming weight, namely y = x ⊕ z = u ⊕ z ⊕ H, where each bit of z ∈ {0, 1} n is set to 1 with probability η, independently of the other bits. Addition of z gives a natural bijection between the two cosets, mapping each element of the first coset to a relatively nearby element of the second coset; namely, the distance between the two elements is the Hamming weight of z, which is typically around ηn. Each player is supposed to output one element from its coset, and their goal is for their elements to match under the bijection. In other words, Alice outputs an element a ∈ x, Bob outputs b ∈ y, and they win the game if and only if a ⊕ b = z. 5 Notice that the number of possible inputs to each player is 2 n /n and the number of possible outputs for each player is n.
Based on the integrality gap analysis of Khot and Vishnoi, in Section 4 we show that no classical strategy can win this game with probability greater than 1/n η/(1−η) . We also sketch a classical strategy that achieves this winning probability up to lower order terms. In contrast, using a simplified version of the "quantum rounding" technique of [18], we exhibit an entangled strategy that uses the n-dimensional maximally entangled state and wins with probability at least (1 − 2η) 2 . This strategy follows from the observation that each coset of H defines an orthonormal basis of R n in which we can do a measurement. Summarizing, we have entangled value ω * n (G) ≥ (1 − 2η) 2 and classical value ω(G) ≤ 1/n η/(1−η) for this game. Setting the noise-rate to η = 1/2 − 1/ log n, the entangled value is Ω(1/(log n) 2 ) while the classical value is O(1/n), leading to a Bell inequality violation ω * n (G)/ω(G) = Ω(n/(log n) 2 ). Up to the polylogarithmic factor, this is optimal both in terms of the local dimension, and in terms of the number of possible outputs.
Palazuelos [24] recently used our result to prove an interesting super-activation result. He identified a constant-dimensional quantum state (a mixture of the maximally entangled state of local dimension 8 with the completely mixed state) that cannot be used to violate any Bell inequality, but a sufficiently large number of copies of which can be used to violate a Bell inequality-namely the one associated with our KV game.

Discussion and open problems
Although Bell violations provide an elegant way to quantify the non-locality exhibited in a game, in experimental realizations of such games it is often important to take into account the actual classical and entangled values, and not just their ratio, especially when one tries to take into account possible imperfections in the experimental set-up. The large Bell violation and the tiny success probability achievable by classical players seem to make the KV game attractive to an experimental realization. One should keep in mind, though, that the success probability achievable by entangled players is 1/(log n) 2 , which is somewhat low in absolute terms, and might not be visible if the experiment has too many false positives. It also means that an experiment must be repeated about (log n) 2 times before we expect to see the first win. In the HM game, on the other hand, entangled players can win with certainty, which seems beneficial in case there are few false negatives. Another advantage of the HM game over KV is its somewhat simpler description.
One natural open question is to improve the Bell violation of n/(log n) 2 achieved by the KV game either by tweaking the game or defining another game, possibly even matching the O(n) upper bound up to a constant factor. 6 Throughout this paper we considered the Bell violation as a function of the number of outputs of the players and/or of the dimension of entanglement. One can also analyze the violation in terms of the number of possible inputs. We recall that in the KV game both players have inputs taken from an exponentially large set, and that in the HM game (when modified as in footnote 2) Bob has only n/2 possible inputs, but Alice still has an exponentially large set of inputs. The Bell inequality violation of √ n/ log n presented by Junge and Palazuelos [13] has the advantage that the number of inputs is only O(n). Accordingly, another open question presents itself: can we find a game with a (near-)linear Bell inequality violation, and linear number of inputs and outputs for both Alice and Bob?
Finally, while this paper focuses on the two-party setting, obtaining stronger Bell inequality violations for settings with three or more parties is also a worthwhile goal. Pérez-García et al. [25] (see also [4]) gave a randomized construction of a three-party XOR game (in such a game each party outputs a bit, and winning or losing depends only on the XOR of those three bits) that gives a Bell inequality violation of Ω( √ d) using an entangled state in dimensions d × D × D with D d. 7 In contrast, it is a known consequence of Grothendieck's inequality that such non-constant separations do not exist for two-party XOR games. We do not know how large Bell inequality violations can be for arbitrary three-party games.
Note that it is easy to make a three-party version of the Hidden Matching game: Alice gets input x ∈ {0, 1} n , Bob gets input y ∈ {0, 1} n , and Charlie gets a matching M as input, all uniformly distributed. The goal is that Alice outputs a ∈ {0, 1} log n , Bob outputs b ∈ {0, 1} log n , Charlie outputs d ∈ {0, 1} and By modifying the two-party proofs in this paper, it is not hard to show that the winning probability using an n-dimensional GHZ state is 1, while the best classical winning probability deviates from 1/2 by at most O((log n) 2 /n). So going from two to three parties squares the Bell inequality violation for Hidden Matching. This improvement unfortunately does not scale up with more than three parties, because one can show the classical winning probability is always at least 1/2 + Ω(1/n).

Fourier analysis
The crucial technical tool used in the analysis of both of our games is Fourier analysis on the Boolean cube. We will just introduce what we need here, referring to [22,28] for more details and references. Unless mentioned otherwise, expectations and probabilities are taken over a uniformly random x ∈ {0, 1} n . Define the inner product between functions f , g : {0, 1} n → R as For S ⊆ [n], the function χ S (x) = (−1) ∑ i∈S x i is the parity (represented as a sign) of the variables indexed in S. These 2 n functions (one for each S) form an orthonormal basis for the space of all real-valued 6 Interestingly, very recently Palazuelos [23] showed that this cannot be done using the maximally entangled state (which is the state used in all our entangled strategies): he proved that the ratio between the optimal value of entangled strategies using the maximally entangled state of local dimension n, and the classical value, is at most O(n/ √ log n). 7 They also showed that using GHZ states, there is no superconstant Bell inequality violation for XOR games (see also [3].) functions on the Boolean cube. The Fourier coefficients of f are f (S) = f , χ S , and we can write f in its Fourier decomposition Since the χ S form an orthonormal basis, it is easy to show (2.1) For p ≥ 1, the p-norm of f is defined as This is monotone non-decreasing in p. For p = 2, equation (2.1) with g = f gives Parseval's identity: For ρ ∈ [−1, 1], the noise-operator T ρ adds "η-noise" to each of the input bits, where η = (1 − ρ)/2. More precisely, the function T 1−2η f is defined as where z ∈ {0, 1} n is an "η-biased noise string," i. e., each bit z i is set to 1 with probability η, independently of the other bits. The linear operator T ρ is diagonal in the Fourier basis: it just multiplies each χ S by the factor ρ |S| . Equivalently, Since T ρ f is a convex combination of shifted copies f z (x) = f (x ⊕ z) of f , by the triangle inequality we see that T ρ is a contraction: T ρ f p ≤ f p for all p ≥ 1, ρ ∈ [−1, 1], and f . The Bonami-Beckner hypercontractive inequality implies T ρ f q ≤ f p even for q somewhat bigger than p.
An important consequence of the hypercontractive inequality is the so-called KKL inequality [15].

A more formal look at Bell violations
Before we analyze the two games mentioned above, let us first say something more about the mathematical treatment of general Bell inequalities. Readers who are content with the above (more concrete) approach in terms of winning probabilities of games, may safely skip this section. Consider a game with n possible inputs to each player and k possible outputs. The behavior of the players (irrespective of whether they use a classical or an entangled strategy) can be summarized in terms of n 2 probability distributions, each on the set [k] × [k]. We denote by P(ab | xy) the probability of producing outputs a and b when given inputs x and y, with respect to a fixed strategy. As described in the introduction, a game is defined by a probability distribution π on the input set where the supremum now is over all entangled strategies (using an entangled state of arbitrary dimension). If the entangled state is restricted to local dimension n, the value is denoted ω * n (M). We note that if M is the Bell functional corresponding to a game, then these definitions coincide with our definitions from the introduction, and in this case the absolute value is unnecessary since M is non-negative.
A Bell inequality is an upper bound on ω(M) for some Bell functional M; it shows a limitation of classical strategies. 8 The Bell inequality violation demonstrated by a Bell functional M is defined as the ratio between the entangled and the classical value This provides a convenient quantitative way to measure the extra power provided by entanglement. This definition of Bell violation enjoys a rich mathematical structure, as witnessed by the numerous connections found to Banach space and operator space theory [14,13,8], and also has a beautiful geometrical interpretation as the "distance" between the set of all classical strategies and the set of all entangled strategies. (See Section 6.1 in [13].) Clearly, any game G for which ω * (G) ≥ Kω(G) gives a Bell violation of K by just taking the functional corresponding to G. But recall that in the introduction we said that one is also allowed to look at the ratio of biases around some center probability p. We now explain why this still agrees with the above definition of Bell violation. We claim that if G is a game for which the winning probability of any classical strategy cannot deviate from p by more than δ 1 and, moreover, there is an entangled strategy obtaining winning probability at least p + δ 2 (or at most p − δ 2 ), then we obtain a Bell violation of δ 2 /δ 1 . To see why, let M be the functional corresponding to the game, and let M be the functional obtained by subtracting from each M ab xy the probability of input pair (x, y) times p. Then it is easy to see that for any strategy P, M , P = M, P − p. Hence, ω(M ) and ω * (M ) are exactly the bias around p of classical and entangled strategies, respectively, and the claim follows. The converse to this statement is also true: any Bell functional M can be converted to a game in such a way that the ratio between the entangled bias and the classical bias of the game (both around 1/2, say) is exactly the Bell violation demonstrated by the functional. To prove this, consider the game in which an input pair (x, y) is chosen uniformly, and outputs a, b are accepted with probability 1/2 + δ M ab xy for some sufficiently small δ > 0 so that all these probabilities are in [0, 1].

Hidden Matching game
In this section we define and analyze the Hidden Matching game. Here and below, unless stated otherwise, all probabilities and expectations are taken over the distributions on x and M specified by the game.

The Hidden Matching problem in communication complexity
While our focus is non-locality, it will actually be useful to first study the original version of the Hidden Matching problem in the context of protocols where communication from Alice to Bob is allowed. Both the problem and the efficient quantum protocol below come from [1]. Theorem 3.2. For every n that is a power of 2, there is a protocol for HM n with log n qubits of one-way communication that wins with probability 1 (i. e., v = x i ⊕ x j always holds).
Proof. The protocol is the following: For each {i, j} ∈ M, the probability to get (1/ √ 2)(|i + | j ) equals 2/n if x i ⊕ x j = 0, and equals 0 otherwise, and similarly for (1/ √ 2)(|i − | j ) with the parity flipped. Hence Bob's output is always correct.

Limits of classical protocols for HM n
Here we show that classical protocols with little communication cannot have good success probability. To start, note that a protocol that uses shared randomness is just a probability distribution over deterministic protocols, hence the maximal winning probability is achieved by a deterministic protocol.
The intuition behind the proof is the following. If the communication c is small, the set X m of inputs x for which Alice sends message m will typically be large (of size about 2 n−c ), meaning Bob has little knowledge of most of the bits of x. The KKL inequality implies that for most of the n 2 pairs {i, j}, Bob cannot guess the parity x i ⊕ x j well. Of course, Bob has some freedom in which {i, j} he outputs, but that freedom is limited to the n/2 pairs {i, j} in his matching M, and it turns out that on average he will not be able to guess any of those parities well. We have Define ε such that and ε m such that Then ε = ∑ m p m ε m . The best Bob can do when guessing x i ⊕ x j given message m, is to output the value of x i ⊕ x j that occurs most often among the x ∈ X m . Define Intuitively, if X m (and hence p m ) is large, then most of these β mi j should be small. We now use the KKL inequality (Theorem 2.2) to make this intuition precise.
Using the KKL inequality with a δ ∈ [0, 1] to be specified later, we get For p m > 1/2 simply choose δ = 1. For p m ≤ 1/2, choose δ = 1/ log 2 (1/p m ), which is in [0, 1], so that the above is The fraction of x ∈ X m where x i ⊕ x j = 0 is 1/2 + β mi j /2, hence Bob's optimal success probability when guessing x i ⊕ x j is 1/2 + |β mi j |/2. This implies, for fixed m, where "{i, j} ∼ q m " means that {i, j} is distributed according to distribution q m . This allows us to upper bound ε m for m where p m ≤ 1/2: where the second inequality is by Cauchy-Schwarz and the third uses both equation (3.2) and the first part of Claim 3.4. Since the p m sum to 1, there can be at most one m for which p m > 1/2. For that m we have ε m ≤ 1/ √ n − 1 by an analogous argument combined with the second part of Claim 3.4. Finally we can bound ε, treating the (at most one) m with p m > 1/2 separately: where H(p) = ∑ m p m log 2 (1/p m ) denotes the binary entropy function, which is at most c since the distribution {p m } is on 2 c elements.

Classical protocol for HM n
Here we design a classical protocol that achieves the above upper bound on the success probability. This protocol has no bearing on the large Bell inequality violations that are our main goal in this paper, but it is nice to know the previous upper bound on the maximal success probability is essentially tight.
Theorem 3.5. For every n that is a power of 2, and every positive integer c ≤ √ n, there exists a classical protocol for HM n with c bits of one-way communication, such that for all inputs x, M, Proof. Assume for simplicity that √ n is integer, and that c is even and sufficiently large. Alice and Bob use shared randomness to choose two disjoint subsets S 1 , S 2 of [n] of size √ n each. Let y denote the bits of x located in the indices given by the first subset, and z the bits located in the indices given by the second subset. Alice and Bob use shared randomness to produce 2 c/2 random √ n-bit strings y (1) , . . . , y (2 c/2 ) . For each , the distance d(y, y ( ) ) is distributed binomially, as the sum of √ n fair coin flips. The following well-known fact about the tail of binomial distribution can be seen for instance by estimating k k/2−β √ k using Stirling's approximation. Fact 3.6. There exists a universal constant γ > 0 such that if X is the sum of k fair coin flips, then for all Thus we have Pr d(y, y ( ) ) ≤ √ n/2 − β n 1/4 ≥ 2 −γ(1+β 2 ) .
Hence by choosing β = Θ( √ c), with probability close to 1, there will be an such that y and y ( ) are at relative distance ≤ 1/2 − Ω(c 1/2 /n 1/4 ). If so, Alice sends Bob the first such , and otherwise she tells him there is no such . This costs c/2 bits of communication. Similarly, at the expense of another c/2 bits of communication, Bob obtains an approximation of z with relative distance at most ≤ 1/2 − Ω(c 1/2 /n 1/4 ).
It is easy to see that with probability at least 1/2, Bob's matching M contains an {i, j} with i ∈ S 1 and j ∈ S 2 . Bob can predict x i with success probability 1/2 + Ω(c 1/2 /n 1/4 ) from his approximation of y, and can predict x j with success probability 1/2 + Ω(c 1/2 /n 1/4 ) from his approximation of z. These success probabilities are independent, hence he can predict x i ⊕ x j with success probability 1/2 + Ω(c/ √ n). If there is no such {i, j} ∈ M, or if he did not get good approximations to y or z, then Bob just outputs any {i, j} ∈ M and a random bit for v, giving success probability 1/2. Putting everything together, we have a protocol that wins with probability 1/2 + Ω(c/ √ n).

Entangled value for HM NL
n We now port our results to the non-local setting, referring to Definition 1.1 for the game associated with the Hidden Matching problem.
Theorem 3.7. For every n that is a power of 2, there exists an entangled strategy for HM NL n using a maximally entangled state with local dimension n, such that condition (1.1) is always satisfied.
1. Alice performs a phase-flip according to her input x. The state becomes 2. Bob performs a projective measurement with projectors P i j = |i i| + | j j|, with {i, j} ∈ M. The state collapses to 1 for some {i, j} ∈ M known to Bob.
3. Both players apply Hadamard transforms H ⊗ log n , and the state becomes Notice that in the latter state, any pair a, b with nonzero amplitude must satisfy that Hence, if the players measure the state, Alice outputs a, and Bob outputs {i, j} and the bit d = b · (i ⊕ j), then they win the game with certainty.

Classical value for HM NL
n In contrast to the entangled value, the optimal classical winning probability is not much better than 1/2: Proof. A strategy that wins HM NL n with success probability 1/2 + ε can be turned into a protocol for HM n with log n bits of communication and the same winning probability: the players play HM NL n , with Alice producing a and Bob producing {i, j}, d; Alice then sends a to Bob, who outputs {i, j}, (a · (i ⊕ j)) ⊕ d. The latter bit equals x i ⊕ x j with probability 1/2 + ε. This requires c = log n bits of communication, so Next we show that our upper bound on the success probability of classical strategies for HM NL n is nearly optimal: we can achieve advantage at least Ω( (log n)/n). In the Appendix we also give a simple alternative strategy with a slightly weaker advantage of Ω(1/ √ n). The correctness of that more elementary strategy can be proven from first principles, unlike the one presented here. Theorem 3.9 and the Appendix have no bearing on our separation, but are included here mostly for completeness. Theorem 3.9. For every n that is a power of 2, there exists a classical deterministic strategy for HM NL n with winning probability 1/2 + Ω (log n)/n (under the uniform input distribution).
Proof. The strategy is as follows. Bob finds the j ∈ {2, . . . , n} that is matched to 1 in M, and outputs {1, j} and d = 0. Since the number i = 1 corresponds to the string 0 log n , the winning condition (a · (i ⊕ j)) ⊕ d = x i ⊕ x j is now equivalent to a · j = x 1 ⊕ x j . Alice, given her input x ∈ {0, 1} n , outputs the value a ∈ {0, 1} log n that maximizes the winning probability subject to j being uniformly distributed over {2, . . . , n}. That is, she selects an a that maximizes the number The winning probability of this strategy, for fixed x and uniformly random M, is max a J ax /(n − 1). In the remainder of this proof we show that which implies the theorem. We use the following result due to Talagrand [21,Proposition 4.13]. For a finite set T ⊆ R n , define This is the expected maximal overlap between z and the elements of T , expectation taken over uniformly random z ∈ {±1} n . Let N(T, d 2 ; ε) denote the minimal number of (open) balls of radius ε > 0 (in Euclidean distance d 2 ) needed to cover T . The definition of r(T ) takes the absolute value of ∑ n i=1 z i t i , but by symmetry, with probability 1/2 this quantity is positive for the maximizing t, and moreover the sum can never be smaller than − √ n since T forms an orthogonal basis. Hence we also have This means that we expect z to be relatively close in Hamming distance to some Hadamard codeword: the expected number of positions where z agrees with the closest t ∈ T is n/2 + Ω( √ n log n). In our application, since x itself is uniformly random, the string y := ((−1) x 1 ⊕x j ) j∈{0,1} log n is uniformly random except for its first bit, which is fixed to 1. Our quantity max a J ax measures the number of positions where y agrees with the closest t ∈ T , ignoring the first position. We conclude that E x∈{0,1} n max a∈{0,1} log n J ax ≥ n/2 + Ω( n log n) − 1 .

.1 The classical value
In this section we analyze the classical value of the Khot-Vishnoi game (Definition 1.2). Our main result is an upper bound on the classical value of 1/n η/(1−η) , based on the analysis from [20].
Before we give that upper bound, let us first argue that it is essentially tight, i. e., there exists a strategy whose winning probability is approximately 1/n η/(1−η) . To get some intuition for this game, first think of η as some small constant (even though we will eventually choose it close to 1/2), and consider the following natural classical strategy: Alice and Bob each output the element of their coset that has highest Hamming weight.
The idea is that if a is the element of highest Hamming weight in Alice's coset x, we expect a ⊕ z to also be of high Hamming weight (because it is close to a in Hamming distance), and since a ⊕ z is in his coset y, Bob is somewhat likely to pick it as his output. We now give a brief back-of-the-envelope calculation suggesting that the winning probability of this strategy is approximately 1/n η/(1−η) ; since it is not required for our main result, we will not attempt to make this argument rigorous.
Let t ≥ 0 be such that the probability that a binomial B(n, 1/2) variable is greater than (n + t)/2 is 1/n. Recalling that the cumulative distribution function of the binomial distribution B(n, p) can be approximated by that of the normal distribution N(np, np(1 − p)), and that the probability that a normal variable is greater than its mean by s standard deviations is approximately e −s 2 /2 , we can essentially choose t to be the solution to e −t 2 /(2n) = 1/n (so t = √ 2n ln n). Then we expect Alice's n-element coset to contain exactly one element of Hamming weight greater than (n + t)/2. Since the element a that Alice picks is the one of highest Hamming weight, we assume for simplicity that its Hamming weight is (n + t)/2. The players win the game if and only if a ⊕ z has the highest weight among Bob's n elements, which we heuristically approximate by the event that a ⊕ z has Hamming weight at least (n + t)/2. The Hamming weight of a ⊕ z is distributed as the sum of two independent binomial distributions B((n + t)/2, 1 − η) and B((n − t)/2, η), which can be approximated as above by the normal distribution N((n +t)/2 − ηt, nη(1 − η)). Hence, for the Hamming weight of a ⊕ z to be at least (n +t)/2, the normal variable needs to be greater than its mean by ηt/ nη(1 − η) standard deviations, and the probability of this happening is approximately e −η 2 t 2 /(2nη(1−η)) = 1 n η/(1−η) , as claimed. Now we show that no classical strategy can be substantially better. The main technical tool used in the proof is the hypercontractive inequality (Theorem 2.1), which is applicable to our setting because we choose u uniformly and u ⊕ z may be viewed as a "noisy version" of u. Proof. Recall that the inputs are generated as follows: we choose a uniformly random u ∈ {0, 1} n and an η-biased z ∈ {0, 1} n , and define the respective inputs to be the cosets u ⊕ H and u ⊕ z ⊕ H. We can assume without loss of generality that Alice and Bob's behavior is deterministic. Recall that the players win if and only if the sum of Alice's output and Bob's output equals z. Hence for all u, z, ∑ h∈H A(u ⊕ h)B(u ⊕ z ⊕ h) equals 1 if the players win on input pair u ⊕ H, u ⊕ z ⊕ H, and equals 0 otherwise. Therefore, the winning probability is given by where the second equality uses the fact that for all h, u ⊕ h is uniformly distributed.
We use the Fourier analysis framework introduced in Section 2.1. We now have Here the third equality follows from equations (2.1) and (2.2), the first inequality is by Cauchy-Schwarz, the second is the hypercontractive inequality (Theorem 2.1 with q = 2, p = 2 − 2η and ρ = √ 1 − 2η) applied to each of A and B, and the second to last equality uses the fact that A, B have range {0, 1}. We complete the proof by noting that n/n 1/(1−η) = 1/n η/(1−η) .

Lower bound on the entangled value
In this section we describe a good entangled strategy for the Khot-Vishnoi game, following the ideas of Kempe, Regev, and Toner [18] and the SDP-solution of [20].
Theorem 4.2. For every n that is a power of 2, and every η ∈ [0, 1/2], there exists an entangled strategy that wins KV n with probability at least (1 − 2η) 2 , using a maximally entangled state of local dimension n.
Proof. For a ∈ {0, 1} n , let v a ∈ R n denote the unit vector ((−1) a i / √ n) i∈ [n] . Notice that for all a, b, we have v a , v b = 1 − 2d(a, b)/n, where d(a, b) denotes the Hamming distance between a and b. In particular, the n vectors v a , as a ranges over a coset of H, form an orthonormal basis of R n .
The entangled strategy is as follows. Alice and Bob start with the n-dimensional maximally entangled state. Alice, given coset x = u ⊕ H as input, performs a projective measurement in the orthonormal basis given by {v a | a ∈ x} and outputs the value a given by the measurement. Bob proceeds similarly with the basis {v b | b ∈ y} induced by his coset y = x ⊕ z ⊕ H. A standard calculation now shows that the GIANNICOLA SCARPA is a Ph. D. student at CWI, Amsterdam, supervised by Ronald de Wolf. In 2009, he received a Master's degree in Computer Science from the University of Salerno, Italy. His research interests include quantum computing, non-locality, combinatorial optimization, and game theory. In his free time, he is a devoted movie-goer but unsuccessful movie maker, he devours short stories, writes some, and he often claims he is going to lose weight.
RONALD DE WOLF received his Ph. D. from the University of Amsterdam and CWI in 2001. His advisors were Harry Buhrman and Paul Vitányi. After doing a postdoc at the University of California, Berkeley, he now holds a permanent position at CWI and a part-time position as full professor at the University of Amsterdam. His CS interests include quantum computing, complexity theory, and learning theory. He also holds a degree in philosophy, and enjoys classical music and literature.