Sampling Quantum Nonlocal Correlations with High Probability

It is well known that quantum correlations for bipartite dichotomic measurements are those of the form γ = (〈ui, vj〉)i,j=1, where the vectors ui and vj are in the unit ball of a real Hilbert space. In this work we study the probability of the nonlocal nature of these correlations as a function of α = m n , where the previous vectors are sampled according to the Haar measure in the unit sphere of R. In particular, we prove the existence of an α0 > 0 such that if α ≤ α0, γ is nonlocal with probability tending to 1 as n → ∞, while for α > 2, γ is local with probability tending to 1 as n → ∞. Introduction It is well known that local measurements on entangled bipartite quantum states can lead to correlations which cannot be explained by Local Hidden Variable Models (LHVM) [7]. This phenomenon, known as quantum nonlocality, is one of the most relevant features of quantum mechanics. In fact, though initially discovered in the context of foundations of quantum mechanics, during the last decade quantum nonlocality has become a crucial resource in many applications; some of them are quantum cryptography ([1], [2], [15]), communication complexity ([8]) and random number generators ([14], [16]). In this work, we will consider a particularly simple but very interesting context, where two spatially separated observers, Alice and Bob, perform dichotomic (two-outcome) measurements on a bipartite quantum state ρ, each on their part of the system. The simplicity of this scenario has made it the natural one to start developing the previously mentioned applications and also in the experimental verification of the quantum nonlocality phenomenon (see for instance [4], [5]). According to the postulates of quantum mechanics, a two-outcome measurement for Alice (resp. Bob) is given by {A+, A−} (resp. {B+, B−}), where A± (resp. B±) are projectors acting on a Hilbert space and summing to the identity. We define the observable corresponding to Alice’s (Bob’s) measurement as A = A − A− (B = B − B−). The joint correlation of Alice’s and Bob’s measurement results, denoted by a and b respectively, is 〈ab〉 = tr(A⊗ Bρ). Motivated by this, we say that γ = (γi,j) n i,j=1 is a quantum correlation matrix and denote by γ ∈ Q, if there 1 2 C. E. GONZÁLEZ-GUILLÉN, C. H. JIMÉNEZ, C. PALAZUELOS, AND I. VILLANUEVA exist a density matrix ρ acting on a tensor product of Hilbert spaces H1 ⊗ H2 and two families of contractive self-adjoint operators {Ai}i=1, {Bi}i=1 acting on H1 and H2 respectively such that γi,j = tr(Ai ⊗ Bjρ) for every i, j = 1, · · · , n. (0.1) That is, γ is a matrix whose entries are the correlations obtained in an AliceBob scenario where each of the observers can choose among n different possible dichotomic measurements. On the other hand, we say that γ = (γi,j) n i,j=1 is a local correlation matrix if it belongs to the convex hull L = conv { (αiβj) n i,j=1, αi = ±1, βj = ±1, i, j = 1, · · · , n } . (0.2) Local correlation matrices are precisely those whose entries are the correlations obtained in an Alice-Bob scenario when the measurement procedure can be explained by means of a LHVM. It is well known ([17]) that L and Q are convex sets satisfying L Q KGL, where 1.67696... ≤ KG ≤ 1.78221... is the so called Grothendieck’s constant. Indeed, the first strict inclusion exactly means that there exist quantum correlations which cannot be explained by means of a LHVM (what we have called quantum nonlocality above) while the second inclusion is a consequence of Grothendieck’s inequality (see Theorem 1.4 below) and a result proved by Tsirelson ([17]) which states that γ = (γi,j) n i,j=1 is a quantum correlation matrix if and only if there exist a real Hilbert space H and unit vectors u1, · · · , un, v1, · · · , vn in H such that γi,j = 〈ui, vj〉 for every i, j = 1, · · · , n. (0.3) As we just mentioned, we know of the existence of quantum correlations which are nonlocal. A natural question appears now: how common is nonlocality among quantum correlations? That is, if we pick “randomly” a quantum correlation, which is the probability that it is nonlocal? To study this problem, we first need to choose a probability distribution on the set of quantum correlations, in other words, a way of sampling these matrices. We see at least two natural candidates for this. At first sight, it would seem from expression (0.1) that a natural procedure would be sampling on the set of states ρ and on the set of families of self-adjoint and contractive operators A1, · · · , An, B1, · · · , Bn. The problem with this approach is twofold. First, we do not know a natural probability measure on the set of selfadjoint contractive operators. Second, it seems that we would need to allow for Hilbert spaces of very high dimension. 1A density matrix is a positive operator ρ : H → H acting on a Hilbert space H with tr(ρ) = 1. 2The exact value of the Grothendieck’s constant is still unknown. 3It is known ([17]) that every quantum correlation γ = (γi,j) n i,j=1 can be written as in (0.1) by using a Hilbert space of dimension exponential in n and, furthermore, such a dimension is required in order to describe the extreme points of Q. SAMPLING QUANTUM NONLOCAL CORRELATIONS WITH HIGH PROBABILITY 3 So, we look for the second candidate: looking at the equivalent reformulation (0.3) of a quantum correlation, we do have a natural sampling procedure: we can sample the vectors u1, · · · , un, v1, · · · , vn independently uniformly distributed on the unit sphere of R. It is well known that this is exactly the same as sampling independent normalized m-dimensional gaussian vectors. Our results will depend on the relation between the dimensionm and the number of questions n. As we will show later, it is very easy to see that if one fixes any finite m, the probability that a quantum correlation matrix sampled according to the previous procedure is nonlocal tends to one as n tends to infinity. However, this kind of sampling, though interesting to obtain quantum nonlocal correlations, does not say much about our problem, since the set of quantum correlation matrices of order n which can be obtained with a fixed m is very small. We are interested in the case where m and n are of the same order. In that case we are sampling on a representative set of quantum correlation matrices. The main result of our work can be condensed as: Theorem 0.1. Let n andm be two natural numbers and α = m n . Let us consider 2n vectors u1, · · · , un, v1, · · · , vn sampled independently according to the Haar measure on the unit sphere of R and let us denote by γ = (〈ui, vj〉)i,j=1 the corresponding quantum correlation matrix. a) If α ≤ α0 ≈ 0.004 then γ is nonlocal with probability tending to one as n tends to infinity. b) If α > 2, then γ is local with probability tending to one as n tends to infinity. This result shows clearly the need of studying the problem as a function of the parameter α = m n . One possible way to think of this problem is the following: say that we want to sample our vectors on a space of large dimension m. In that case, how many vectors u1, · · · , un, v1, · · · , vn will we need to sample in order to have nonlocality with high probability? Our results show that n = m 2 will be too few vectors, whereas n = m α0 will be enough. There is a considerable gap between α0 and 2. Our techniques could be refined to slightly increase the bound α0, but they will never reach the relevant case α0 = 1. From the other side, our proof of part b) suggests that a more clever argument could lead to replace 2 by KG, but again our present approach does not seem to allow for further improvement. Along these lines, it is plausible that a relation between α and KG describes interesting behaviors of our correlation matrices. It would be very interesting to understand the problem for the values α ∈ (α0, 2) both by reducing this gap and by studying the existence, or not, of a sharp threshold behaviour of the probability of nonlocality. 4 C. E. GONZÁLEZ-GUILLÉN, C. H. JIMÉNEZ, C. PALAZUELOS, AND I. VILLANUEVA Interestingly enough, we will see below that if one samples normalized vectors whose entries are independent Bernoulli variables, the probability of obtaining a nonlocal correlation matrix is zero, since all of them will be local. This means that, in contrast to many other contexts in random matrix theory, sampling gaussian and Bernouilli random variables in our problem leads to completely different conclusions. In order to prove Theorem 0.1 we will use a result previously proved in [10] on random matrix theory. Similar techniques were previously used in [3] in order to study the dual problem; that is, how likely it is for a random (in some sense) XOR game to have a maximum quantum value strictly bigger than a maximum classical value. In that case, the authors studied the values ω∗(A) and ω(A) for random matrices A = (ai,j) n i,j=1, where ω∗(A) = sup { n ∑ i,j=1 ai,jγi,j : γ ∈ Q } and ω(A) = sup { n ∑ i,j=1 ai,jγi,j : γ ∈ L } . They concluded that, for any given ǫ > 0, ω∗(A) ≥ (2 − ǫ)n 32 and ω(A) ≤ 1.6651 . . . n 3 2 with probability 1 − o(1) as n → ∞ in both cases. This result is the starting point for the proof of our Theorem 0.1. Note that stating ω ∗(A) ω(A) > 1 for some A’s is a reformulation (in a quantitive way) of the fact that L Q. The elements A’s are usually called correlation Bell inequalities (or XOR-games in the context of computer science) and the fact that ω ∗(A) ω(A) > 1 is usually referred to as a Bell inequality violation. The paper is organized as follows. In the first section we briefly introduce some basic results which will be used along the whole paper. The proof of Theorem 0.1 is presented in Section 2 and Section 3. The proof of part a) of the theorem, based on some results on random matrix theory, is given in Section 2, while Section 3 deals with the proof of part b). 1. Preliminary results For completeness and to simplify the reading of the paper, we state in this section the known, or essentially known, previous results which we use along the paper. The following proposition can be easily deduced from [11, Lemma 2.2]. Proposition 1.1. Let Gn be the gaussian measure on R and let L ⊂ R be a k-dimensional subspace. For a vector g = (g1, · · · , gn) ∈ R, let ḡ = g ‖g‖ and let PL(ḡ) denote the orthogonal projection of ḡ onto L. Then, for any 0 < ρ < 1 we 4Although the authors focused on sign matrices, the same proof works in the case of more general random matrices. SAMPLING QUANTUM NONLOCAL CORRELATIONS WITH HIGH PROBABILITY 5 have Gn ( (g1, · · · , gn) ∈ R : ‖PL(ḡ)‖ ≥ 1 1− ρ √