We now view the bipartite correlation setting of the previous section from the per-spective of optimization. We consider linear optimization over the sets of classical and quantum correlations. For a given objective function, a difference in the optimal value over these sets can be used to conclude that the set of classical correlations is strictly contained in the set of quantum correlations. With a simple change of language, linear optimization over the set of bipartite correlations arises naturally in games. We now refer to Alice and Bob as players. The game consists of a referee giving each of the players a question, afterwards the players have to respond with an answer. The players decide on a strategy before the game starts: to each question
3.3. Nonlocal games 33 they associate a measurement device, and the outcomes of those measurements form their answers. To each pair of outcomes we associate a payoff (which may depend on the pair of questions). Linear optimization over the set of bipartite correlations then corresponds to Alice and Bob trying to maximize their expected payoff. When the two players are not allowed to communicate during the game, such a game is called a nonlocal game.
Formally, a nonlocal game G is defined by two finite sets of questions S and T , two finite sets of answers A and B, a probability distribution π : S × T → [0, 1]
and a predicate3f : A × B × S × T → {0, 1}. The predicate f determines the rules of the game: given question pair (s, t) ∈ S × T , the pairs of answers (a, b) ∈ A × B such that f (a, b, s, t) = 1 are called correct, all other pairs are wrong. Alice and Bob receive a question pair (s, t) ∈ S × T with probability π(s, t) and they win the game if their answers are correct. They know the game parameters π and f , but they do not know each other’s questions, and they cannot communicate after they receive their questions. Their answers (a, b) are determined according to some correlation P ∈ RΓ, called their strategy, on which they may agree before the start of the game, and which can be classical or quantum depending on whether P belongs to Cloc(Γ), Cq(Γ), or Cqc(Γ). Then their corresponding winning probability is given by
X
(s,t)∈S×T
π(s, t) X
(a,b)∈A×B
P (a, b|s, t)f (a, b, s, t). (3.8)
A strategy P is called perfect if the above winning probability is equal to one, that is, if for all (a, b, s, t) ∈ Γ we have
π(s, t) > 0 and f (a, b, s, t) = 0
=⇒ P (a, b|s, t) = 0. (3.9) In other words, the probability of giving a wrong answer equals zero.
Computing the maximum winning probability of a nonlocal game is the problem of finding a bipartite correlation that maximizes (3.8). This is an instance of linear optimization (of the function (3.8)) over Cloc(Γ) in the classical setting, and over Cq(Γ) or Cqc(Γ) in the quantum setting. Since the inclusion Cloc(Γ) ⊆ Cq(Γ) can be strict, the maximum winning probability can be higher when the parties have access to entanglement. A famous example of such a game is due to Clauser, Horne, Shimony, and Holt [CHSH69]; we will discuss this game in detail in Section 3.3.2.
In fact there are nonlocal games that can be won with probability 1 by using entan-glement, but only with probability strictly less than 1 in the classical setting; see for example the Mermin-Peres magic square game [Mer90, Per90]. In general it is hard to determine the maximum (quantum) winning probability of a game4: as we will see below, certain hard combinatorial problems such as max-cut can be phrased
3To make the above analogy with linear optimization correct one would need to consider a real-valued function f . For this thesis it suffices to only consider 0/1-valued functions f .
4Slofstra showed that for a certain class of games called linear system games it is undecidable to determine if such a game has a perfect strategy in Cqc(Γ) [Slo16], or in Cq(Γ) or its clo-sure cl(Cq(Γ)) [Slo19]. In particular, the problem of determining whether the maximum winning probability of a game equals 1 over Cqc(Γ) is undecidable. As we point out later, this implies that there is no general stopping criterion for the noncommutative sum-of-squares hierarchy that we will mention in Chapter 4.
34 Chapter 3. Quantum information theory as linear optimization over the set of classical correlations. However, for certain classes of games and strategies determining the maximum winning probability be-comes easy. Below we introduce one such class, called XOR games, for which the maximum winning probability when using quantum strategies can be determined using semidefinite programming.
3.3.1 XOR games
A nonlocal game G is called an XOR game when Alice and Bob each output a single bit, that is, A = B = {0, 1}, and the predicate f is of the form f (a, b, s, t) = 1 if and only if a ⊕ b = g(s, t) for some function g : S × T → {0, 1}. In other words, the rules of the game are such that whether a pair of answers is correct or wrong only depends on the logical XOR of the answers. XOR games are special in the sense that the maximum winning probability over quantum strategies can be expressed using semidefinite programming, as we explain below.
Let P ∈ Cq(Γ) be a quantum strategy and suppose that ψ, {Esa}, {Ftb} are as in (3.2), that is, P (a, b|s, t) = ψ∗(Esa⊗ Ftb)ψ for all a, b, s, t. Then one can verify5 that the winning probability of P in the XOR game G (cf. (3.8)) can be written as
X
(s,t)∈S×T
π(s, t) X
a,b∈{0,1}
f (a, b, s, t) ψ∗(Esa⊗ Ftb)ψ
=1 2 +1
2 X
(s,t)∈S×T
π(s, t)(−1)g(s,t)ψ∗ (E0s− Es1) ⊗ (Ft0− Ft1)ψ.
This suggests changing variables and working with the matrices Es= Es0− Es1and Ft= Ft0− Ft1: We consider
X
(s,t)∈S×T
π(s, t)(−1)g(s,t)ψ∗(Es⊗ Ft)ψ. (3.10)
This change of variables does not lose any information: A Hermitian matrix whose eigenvalues lie in [−1, 1] can be written, uniquely, as the difference between two Hermitian positive semidefinite matrices that sum to the identity.
Notice that one possible strategy for Alice and Bob is to each base their answer on an unbiased coin flip, this strategy has a winning probability of 1/2. The above quantity (3.10) represents the bias that the strategy P has towards winning. It is equal to the probability of winning minus the probability of losing.
The problem of maximizing the bias of the game G over quantum strategies P ∈ Cq(Γ) is thus
maxn X
(s,t)∈S×T
π(s, t)(−1)g(s,t)ψ∗(Es⊗ Ft)ψ : d ∈ N, ψ ∈ Cd⊗ Cd unit, Es, Ft∈ Hd+, (Es)2= I = (Ft)2 for all s ∈ S, t ∈ To
. (3.11)
5Here we use the identity f (a, b, s, t) =12+12(−1)a+b+g(s,t)for all a, b ∈ {0, 1}, s ∈ S, t ∈ T .
3.3. Nonlocal games 35 We can observe that ψ∗(Es⊗ Ft)ψ = h(Es⊗ I)ψ, (I ⊗ Ft)ψi and that both vectors xs = (Es⊗ I)ψ and yt = (I ⊗ Ft)ψ have norm at most one. It follows that the quantum bias of the game (3.11) can be upper bounded by the semidefinite program
maxkxsk,kytk≤1 X
(s,t)∈S×T
π(s, t)(−1)g(s,t)hxs, yti. (3.12)
To see that this indeed an SDP, let us define the S × T matrix B with entries Bs,t= π(s, t)(−1)g(s,t). (3.13) Then (3.12) can be written as the following SDP in primal form:
max 1 2
0 B
BT 0
, X
(3.14) s.t. X ∈ SS∪T+
Xii ≤ 1 for i ∈ S ∪ T.
For notational convenience, here we assumed that the sets S and T are disjoint.
Remarkably, Tsirelson [Tsi87] showed that the quantum bias of the game in fact equals the value of the SDP (3.14). That is, from a solution X = Gram({xs}, {yt}) to the semidefinite program (3.14) one can construct a quantum strategy P that has a bias equal to the value of the SDP. We recall this construction in Chapter 6, Theorem 6.13. In that chapter we exploit this connection between optimal strategies for XOR games and semidefinite programming to construct quantum correlations which require a lot of entanglement.
What about the bias of an XOR game over classical strategies? By considering the bias of a deterministic strategy, one can show that the maximum classical bias can be computed by restricting to 1-dimensional vectors in the above SDP:
maxxs,yt∈{±1} X
(s,t)∈S×T
π(s, t)(−1)g(s,t)xsyt. (3.15)
Notice that this problem is exactly the max-cut problem in a complete bipartite graph where the edge weights are given by π(s, t)(−1)g(s,t), which is known to be an NP-hard problem [MRR03, Lem. 3].
3.3.2 The Clauser-Horne-Shimony-Holt game
We now illustrate the above concepts and the relation to this thesis via a classical example of an XOR game: the CHSH game [CHSH69]. In this game each player both receives and responds with a single bit, that is, A = B = S = T = {0, 1}.
The distribution π is the uniform distribution on S × T . The rules of the game are such that the players win if the logical XOR of their answers equals the logical AND of their questions. That is, g : {0, 1}2 → {0, 1} is the function g(s, t) = st, and f (a, b, s, t) = 1 if and only if a ⊕ b = g(s, t) = st.
Using the formulation of Equation (3.15), it is easy to see that the classical bias is at most 12. On the other hand, the quantum bias of the game is at least
36 Chapter 3. Quantum information theory
√1
2 (which is strictly larger than 12) which can be seen from the following feasible solution to (3.12):
Using the dual of the SDP (3.14) one can show that the quantum bias of the CHSH game is in fact equal to 1/√
2. Translating this back to the language of winning probabilities, the CHSH game can be won with probability at most 34 = 0.75 using classical strategies, but it can be won with probability 12+ 1
2√
2 ≈ 0.85 using quantum strategies. Note that the classical winning probability is attained for the deterministic strategy where Alice and Bob always output 0. The difference between the maximum quantum and classical winning probabilities suggests a way to test whether classical mechanics is the correct model for the physical world: come up with an experiment that wins the CHSH game with probability strictly larger than 0.75. This is precisely what has been done in laboratories around the world, eventually leading to the first loop-hole free Bell inequality violation [Hea15].
What about the smallest entanglement dimension needed to win the CHSH game with probability 12 + 1
2√
2? Tsirelson showed that the SDP for the quantum bias of an XOR game is tight by constructing a quantum strategy from unit vectors xs, yt (s ∈ S, t ∈ T ) achieving the same bias. His construction, which we recall in Theorem 6.13, combined with the above 2-dimensional vectors, shows that this winning probability can be achieved using a strategy with local dimension equal to 2.6 In this case it is easy to see that the same bias cannot be realized in a smaller local dimension (since such a strategy would be classical).
In Chapter 6 we construct XOR games with which we can certify that certain quantum correlations have a large minimal entanglement dimension Dq(P ). The techniques used there combine the fundamental work of Tsirelson with the dual-ity theory of semidefinite programming and the theory of universal rigiddual-ity. As an example of hitting a small object with a huge hammer, let us give an alternative way to show that 2 is the smallest local dimension in which the quantum strat-egy corresponding to x0, x1, y0, y0 can be realized. One can use Theorem 6.6 and Theorem 6.10 (pick λ0 = λ1 = µ0 = µ1) to show that the corresponding correla-tion matrix is extreme, the lower bound on the local dimension then follows from Corollary 6.17.
6The fact that there exists an optimal strategy with local dimension equal to 2 was already known [CHSH69].