Quantifying Bell Nonlocality with the Trace Distance

PHYSICAL REVIEW A 97, 022111 (2018)

Quantifying Bell nonlocality with the trace distance

S. G. A. Brito,1 B. Amaral,1,2 and R. Chaves1 1International Institute of Physics, Federal University of Rio Grande do Norte, 59078-970, P. O. Box 1613, Natal, Brazil 2Departamento de Física e Matemática, CAP - Universidade Federal de São João del-Rei, 36.420-000, Ouro Branco, MG, Brazil

(Received 15 September 2017; published 21 February 2018)

Measurements performed on distant parts of an entangled quantum state can generate correlations incompatible with classical theories respecting the assumption of local causality. This is the phenomenon known as quantum nonlocality that, apart from its fundamental role, can also be put to practical use in applications such as cryptography and distributed computing. Clearly, developing ways of quantifying nonlocality is an important primitive in this scenario. Here, we propose to quantify the nonlocality of a given probability distribution via its trace distance to the set of classical correlations. We show that this measure is a monotone under the free operations of a resource theory and, furthermore, that it can be computed efﬁciently with a linear program. We put our framework to use in a variety of relevant Bell scenarios also comparing the trace distance to other standard measures in the literature.

DOI: 10.1103/PhysRevA.97.022111

I. INTRODUCTION it can stand before becoming local. Interestingly, these two measures can be inversely related as demonstrated by the With the establishment of quantum information science, the fact that in the Clauser-Horne-Shimony-Holt (CHSH) scenario often-called counterintuitive features of quantum mechanics [28], the resistance against detection inefficiency increases as such as entanglement [1] and nonlocality [2] have been we decrease the entanglement of the state [13] (also reducing raised to the status of a physical resource that can be used the violation the of CHSH inequality). to enhance our computational and information processing Bell’s theorem is a statement about the incompatibility capabilities in a variety of applications. To that aim, it is of probabilities predicted by quantum mechanics with those of utmost importance to devise a resource theory to such allowed by classical theories. Thus, it seems natural to quantify quantities, not only allowing for operational interpretations nonlocality using standard measures for the distinguishability but for the precise quantification of resources as well. Given its between probability distributions, with the paradigmatic exam- ubiquitous importance, the resource theory of entanglement [1] ple being the trace distance. Apart from being a valid distance is arguably the most well understood and vastly explored, and in the space of probabilities, it also has a clear operational for this reason has become the paradigmatic model for further interpretation [29]. developments [3–9]. In this paper we present an in-depth analysis of the trace as As discovered by Bell [10], one of the consequences of a quantifier of nonlocality, improving the exploratory results entanglement is the existence of quantum nonlocal correla- of [30,31]. In Sec. II, we describe the scenario of interest and tions, that is, correlations obtained by local measurements propose a quantifier for nonlocality based on the trace distance. on distant parts of a quantum system that are incompatible Further, in Sec. III, we show how our measure can be evaluated with local hidden variable (LHV) models. In spite of their efficiently via a linear program and, in Sec. IV, we show that it close connection, it has been realized that entanglement and is a valid quantifier by employing the resource theory presented nonlocality refer to truly different resources [2], the most in [5,27]. We then apply our framework for a variety of Bell striking demonstration given by the fact that there are entangled scenarios in Sec. V, including bipartite as well as multipartite states that can only give rise to local correlations [11]. In view ones. In Sec. VI, we discuss the relation of the trace distance of that and the wide applications of Bell’s theorem in quantum with other measures of nonlocality. Finally, in Sec. VII,we information processing, several ways of quantifying nonlocal- discuss our findings and point out possible venues for future ity have been proposed [2,5,12–27]. However, only recently research. a proper resource theory of nonlocality has been developed [5,27], thus allowing for a formal proof that previously introduced quantities indeed provide good measures of nonlocal II. SCENARIO behavior. Importantly, different measures will have different operational meanings and do not necessarily have to agree We are interested in the usual Bell scenario setup where on the ordering for the amount of nonlocality. For instance, a a number of distant parts perform different measurements natural way to quantify nonlocality is the maximum violation on their shares of a joint physical system. Without loss of of a Bell inequality allowed by a given quantum state. However, generality, here we will restrict our attention to a bipartite we might also be interested in quantifying the nonlocality of scenario (with straightforward generalizations to more parts; a state by the amount of noise (e.g., detection inefficiencies) see Sec. V) where two parts, Alice and Bob, perform mea-

ABadmits a LHV decomposition (1). The most general way to solve that is to resort to a linear program (LP) formulation. First notice that we can represent a probability distribution x y q(a,b|x,y) as a vector q with a number of components given by n =|x||y||a||b| (|·|represents the cardinality of the random variable). Thus, (1) can be written succinctly as pC = A · λ, with λ being a vector with components λi = p(λ = i) and A being a matrix indexed by i and j = (x,y,a,b) (a multi-index = variable) with Aj,i δa,fa (x,λ=i)δb,fb(y,λ=i) (where fa and fb are deterministic functions). Thus, checking whether q is local amounts to a simple feasibility problem that can be written as the following LP (see [2,8,22,23,32] for other linear program formulations for the detection of nonclassicality in probability distributions): a b min v · λ, λ∈Rm FIG. 1. Bipartite Bell scenario where two parts, Alice and Bob, subject to q = A · λ, share a pair of correlated measurement devices, with inputs labeled λ 0, by x and y and outputs labeled by a and b, respectively. i λi = 1, (4) surements labeled by the random variables X and Y obtaining i measurement outcomes described by the variables A and B, where v represents a arbitrary vector with the same dimen- respectively (see Fig. 1). sion m =|x||a||y||b| as the vector representing the hidden A central goal in the study of Bell scenarios is the character- variable λ. ization of whether the distributions p(a,b|x,y) are compatible The measure we propose to quantify the degree of nonlo- with a classical description based on the assumption of local cality is based on the trace distance between two probability realism implying that = = distributions q q(x) and p p(x), p (a,b|x,y) = p(λ)p(a|x,λ)p(b|y,λ). (1) C 1 λ D(q,p) = |q(x) − p(x)|. (5) 2 All the correlations between Alice and Bob are assumed to x be mediated by common hidden variable λ that thus suffices The trace distance is a metric on the space of probabilities to compute the probabilities of each of the outcomes, that is, since it is symmetric and satisfies the triangle inequality. p(a|x,y,b,λ) = p(a|x,λ) (and similarly for b). Furthermore, it has a clear operational meaning since The central realization of Bell’s theorem [10] is the fact that there are quantum correlations obtained by local measurements = | − | x y D(q,p) max q(S) p(S) , (6) (Ma and Mb ) on distant parts of a joint entangled state , that S according to quantum theory are described as where the maximization is performed over all subsets S of | = x ⊗ y pQ(a,b x,y) Tr Ma Mb , (2) the index set {x} [29]. That is, D(q,p) specifies how well the distributions can be distinguished if the optimal event S is and cannot be decomposed in the LHV form (1). Moreover, an taken into account. Consider, for instance, distributions p(x) even more general set of correlations, beyond those achievable and q(x) for a variable X assuming d values as x = 1,...,d by quantum theory and called nonsignalling (NS) correlations, and such that p(x) = δ and q(x) = 1 (1 − δ ). In this can be defined. NS correlations are defined by the linear x,1 d−1 x,1 case, D(q,p) = 1, also meaning that q and p can be perfectly constraints distinguished in a single shot since, if we observe x = 1, we p(a|x) = p(a,b|x,y) = p(a,b|x,y), can be sure to have p(x)[orq(x) otherwise]. b b In our case, we are interested in quantifying the distance between the probability distribution generated out of a Bell | = | = | p(b y) p(a,b x,y) p(a,b x ,y), (3) experiment and the closest classical probability [compati- a a ble with (1)]. We are then interested in the trace distance that is, as expected from their spatial distance, the outcome of between q(a,b,x,y) = q(a,b|x,y)p(x,y) and p(a,b,x,y) = a given part is independent of the measurement choice of the p(a,b|x,y)p(x,y), where p(x,y) is the probability of the other. The set of classical correlations CC [those compatible inputs, which we choose to fix as the uniform distribution, C = 1 with (1)] is a strict subset of the quantum correlations Q that is, p(x,y) |x||y| . In principle, one could also optimize [compatible with (2)] that in turn is a strict subset of CNS over p(x,y) and we will do so in Sec. V. However, the [compatible with (3)]. assumption that the measurement choices are totally random Suppose we are given a probability distribution and want and identically distributed is a canonical choice in a Bell to test whether or not it is nonlocal, that is, whether it experiment.

022111-2 QUANTIFYING BELL NONLOCALITY WITH THE TRACE … PHYSICAL REVIEW A 97, 022111 (2018)

Using that, we can rewrite our problem as a linear program, q min 1n,t, t∈Rn,λ∈Rm d subject to −t q − A · λ t, CNS λi = 1, p∗ i λ 0, (11) C where q is a known vector of probability distribution to C which we want to quantify the nonlocality. This way, given an arbitrary distribution of interest, we can compute, in an efﬁcient manner, NL(q). Alternatively, we might be interested not in the full distri-

FIG. 2. Schematic drawing of a correlation q ∈ CNS and d = bution, but simply in a linear function of it, for example, the NL(q), the distance (with respect to the 1 norm) from q to the closest violation of a given Bell inequality in the form IBell · q = c. ∗ local correlation p ∈ CC. In this case, further linear constraints need to be added to the LP such as normalization and the fact that the distribution is nonsignalling (see Sec. V for examples), We are now ready to ﬁnally introduce our measure NL(q) for the nonlocality of distribution q = q(a,b|x,y), min 1n,t, t∈Rn,λ∈Rm,q∈Rn = 1 subject to −t q − A · λ t, NL(q) min D(q,p) |x||y| p∈CC λ = 1, 1 i = min |q(a,b|x,y) − p(a,b|x,y)|. i 2|x||y| p∈CC a,b,x,y I · q = c, Bell (7) q(ab|xy) = 1, a,b This is the minimum trace distance between the distribution under test and the set of local correlations. Geometrically it q(ab|xy) − q(ab|xy) = 0 ∀ (b,y), can be understood (see Fig. 2) as how far we are from the local a a polytope deﬁning the correlations (1). Therefore, the more we violate a Bell inequality, the higher will be its value. However, q(ab|xy) − q(ab|xy ) = 0 ∀ (a,x), as will be further discussed in Sec. VI, the violation of a given b b Bell inequality will in general only provide a lower bound to λ 0, its value. q 0. (12)

III. LINEAR PROGRAM FORMULATION Notice that instead of adding only NS constraints, one could also be interested in imposing quantum constraints [33]. Given a distribution q = q(a,b|x,y) of interest, in order to However, in this case, we would need to resort to a semidefinite compute NL(q) we have to solve the following optimization program (that asymptomatically converges to the quantum) problem: instead of a linear program. Finally, often we might be interested in having an analytical − · min q A λ 1 , λ∈Rm rather than numerical tool. To that aim, we can rely on the dual of the LP (11) and (12). We refer the reader to [23]for subject to λ 0, a very detailed account of the dualization procedure but, in λi = 1. (8) short, the optimum solution of (11) and (12) is achieved in one i of the extremal points of the convex set defined by the dual constraints. In this way, being able to compute such extremal n First notice that the 1 norm of a vector p (p ∈ R , with points vi , we have an analytical solution, valid for arbitrary test components pi ), distributions q, given by = | | = · p 1 pi , (9) NL(q) max q vi . (13) vi i can be written as a minimization problem in the form IV. PROVING THAT TRACE DISTANCE IS A NONLOCALITY MEASURE = p 1 min 1n,t , t∈Rn A resource theory provides a powerful framework for the subject to −t p t. (10) formal treatment of a physical property as a resource, enabling its characterization, quantification, and manipulation [3,4,7].

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Such a resource theory consists of three main ingredients: Regarding preprocessing operations, it is possible to show (i) a set of objects, specifying the physical property that may that NL is monotonous under uncorrelated input enlarging serve as a resource, and a characterization of the set of free operations defined in [5]. More generally, one can define a objects, which are the ones that do not contain the resource; preprocessing operation that transforms q into I(q), where (ii) a set of free operations, which map every free object into a free object; and (iii) resource quantifiers that provide a I(q)(a,b|χ,ψ) = q(a,b|x,y)I L(x,y|χ,ψ), (19) quantitative characterization of the amount of resource a given x,y object contains. One of the essential requirements for a resource quantifier and I L is a local correlation with inputs χ,ψ and outputs x,y. is that it must be monotonous under free operations, that is, the It was also shown in Refs. [9,34] that preprocessing operations quantifier must not increase when a free operation is applied. preserve the set of local distributions: if q ∈ CC, then I(q) ∈ Hence, to define proper quantifiers, one needs first to establish CC. In what follows, we will consider the restricted class of the set of free operations that will be considered, which can preprocessing operations that satisfy |x|=|χ|, |y|=|ψ| and vary depending on the applications or the physical constraints under consideration. I L(x,y|χ,ψ) 1. (20) In a resource theory of nonlocality, the set of objects is χ,ψ the set of nonsignalling correlations CNC, and the set of free objects is the set CC of local correlations. A detailed discussion Intuitively, these restrictions forbid one to artificially increase of several physically relevant free operations for nonlocality the number of inputs of the scenario and correlated input can be found in Refs. [5,27]. In what follows, we sketch the enlarging operations, defined in [5]. As a consequence of this proof that our measure NL(q) is a monotone for a resource restriction and convexity of the 1 norm, one can prove that if theory of nonlocality defined by a wide class of free operations. I is an output operation with |x|=|χ|, |y|=|ψ| satisfying The detailed proof of the results below can be found in the Eq. (20), then Appendix A. The first free operation we consider is relabeling R of inputs NL(I(q)) NL(q). (21) and outputs, defined by a permutation of the set of inputs x and y, and outputs a and b. This operation corresponds to a permutation of the entries of the correlation vectors q, and V. APPLICATIONS TO VARIOUS BELL SCENARIOS hence we have that A. Bipartite NL(R(q)) = NL(q). (14) We start considering the paradigmatic CHSH scenario [28], where each of the two parts has two measurement settings Another natural free operation is to take convex sums with two outcomes each, that is, x,y,a,b = 0,1. The only Bell ∈ C between a distribution q and a local distribution p C.In inequality (up to symmetries) characterizing this scenario is this case, the triangular inequality for the 1 norm implies that the CHSH one (equivalent under the no-signalling conditions if π ∈ [0,1], then (3) to the CH inequality [35]), which, using the notation in NL(πq + (1 − π)p) πNL(q). (15) [36], can be written as By combining monotonicity under relabelings and convexity CHSH 0, (22) of the 1 norm, one can show that NL is monotonous under convex combinations of relabeling operations. with the CHSH expression defined as Next, we sketch the proof of the monotonicity of NL under = 0,0 + 0,1 + 1,0 − 1,1 − 0 − 0 more sophisticated free operations, namely, postprocessing CHSH qAB qAB qAB qAB qA qB , (23) and preprocessing operations. Given q, we define a postprocessing operation as one that transforms q into O(q), where where in the expression above we have used the shorthand x,y = = = | notation qA,B q(a 0,b 0 x,y) and similarly to the other terms. Using the dualization procedure described in Sec. III and L O(q)(α,β|x,y) = O (α,β|a,b,x,y ) × q(a,b|x,y), assuming q to be a nonsignalling distribution [respecting (3)], a,b one can prove that (see [31] for a related version of this result) (16) L and O is a x,y-dependent correlation with inputs a,b and NL(q) = 1 max [0, (CHSH)], (24) outputs α,β that satisfies 2 where (CHSH) stand for all of the eight symmetries (under L | = L | × L | O (α,β a,b,x,y ) p(λ)OA(α a,x ) OB (β b,y ). permutation of parts, inputs, and outputs) of the CHSH inequal- λ (17) ity. As expected, in the CHSH scenario, the CHSH inequality As shown in Refs. [9,34], output operations preserve the completely characterizes the trace distance of a given test set of local distributions: if q ∈ C , then O(q) ∈ C .Asa distribution to the set of local correlations [37]. C C Moving beyond the CHSH scenario, we have also con- consequence of convexity of the 1 norm, one can prove that if O is a postprocessing operation, then sidered the Collins-Gisin-Linden-Massar-Popescu (CGLMP) scenario [38], where two parts perform two possible measure- NL(O(q)) NL(q). (18) ments with a number d of outcomes. The CGLMP inequality

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0.25 CHSH 0.5 I2222 0.10355 CGLMP ∀d 0.125 I3322 I4422 0.20 0.4 I5522 I6622

0.15 ) 0.3

q 0.10355 0.11435 ) 0.12155 q

NL( 0.10 0.12695

NL( 0.2

0.05 0.1

0.00 0.00.10.20.30. .5 0.0 CGLMP 0.00.51.01.52.02.5 Inn22 FIG. 3. Value of NL(q) as a function of the value of the CGLMP d = inequality ICGLMP for d 2,3,4,5. The inset shows the maximum FIG. 4. Value of NL(q) as a function of the value of Inn22 for n = known quantum violation for each of these values of d,whichin 2,3,4,5,6. The inset shows the maximum known quantum violation turn implies that NL(q) = 0.1035 (d = 2), NL(q) = 0.1143 (d = 3), (achieved with qubits) for n = 2,3. Interestingly, even if the maximum NL(q) = 0.1215 (d = 4), and NL(q) = 0.1269 (d = 5). NS violation of the inequality increases with the number of settings n, its trace distance to the set of local correlations decreases. can be succinctly written for any d as [d/2]−1 The maximum nonsignalling violation of these inequalities d 1 2k max = − I = 1 − grows linearly with the number of settings n as Inn22 (n CGLMP 4 d − 1 k=0 1)/2. Again, by fixing the value of I and imposing the × [p(a = b + k|00) − p(a = b − k − 1|00) nn22 nonsignalling constraints, we have obtained the corresponding + p(a + k = b|01) − p(a − k − 1 = b|01) value of NL(q), with the results shown in Fig. 4. Interestingly, + p(a + k + 1 = b|10) − p(a − k = b|10) even achieving the maximal nonsignalling violation of the Inn22, we obtain a value for NL(q) that decreases with the + p(a = b + k|11) − p(a = b − k − 1|11)] number of settings n. Therefore, when restricted to this class 1 of inequalities, the best situation is already achieved with n = 2 − 0, (25) 2 (the CHSH scenario). Furthermore, this illustrates well the fact that different measures of nonlocality do not coincide in = + | ≡ where [d/2] means the integer part, and p(a b k xy) general: even though the violation of a Bell inequality can grow d−1 = = + | j=0 p(a j,b j k mod d xy). The maximum value with the number of setting considered, the trace distance of the d of ICGLMP is 1/2 and the maximum value for the local variable corresponding distribution might decrease. theories is 0 ∀d [39]. In this case, we have solved for d = 2,...,5 a LP where instead of fixing the test distribution | d B. Tripartite q(a,b x,y), we only fix the value of ICGLMP and also impose nonsignalling constraints (3) over it [see Eq. (12)]. Similarly, In the tripartite scenario, considering that each of the parts to the CHSH case, we obtain the same expression given by measures two dichotomic observables, all the different classes of Bell inequalities have been classified [40]. There are 46 of NL(q) = 1 max 0,I d , (26) 2 CGLMP them, under the name of Sliwa inequalities. and which we conjecture to hold true to any d. Given that the Following a similar approach to the CGLMP and Inn22 quantum violation of the CGLMP inequality increases with d discussed above, we have computed the value of NL(q)as [38], it follows that the maximum quantum value of NL(q) will a function of the various Sliwa inequalities. The result in follow a similar trend, as shown in Fig. 3. shown in Fig. 5 together with the values associated to the Finally, another scenario we have considered is the one maximum violation of these inequalities. Regarding quantum introduced in [36], where each of the two parts measure a violations, we have used the probability distributions presented number n of observables with two outcomes each. We analyze in [41] where the maximum quantum violation of the Sliwa inequalities has been considered. The results are shown in the Inn22 inequality that has the form [36] Fig. 5(b), where we can see that the optimum quantum value Inn22 0, (27) of NL(q) = 1/8, which is higher than the one obtained for the maximum quantum violation of CHSH but smaller than the one = where, for example, for n 3wehave for CGLPM already for d = 5. The maximum nonsignalling = I = q0,0 + q0,1 + q0,2 + q1,0 + q1,1 − q1,2 violation of these inequalities leads to NL(q) 0.25, the same 3322 AB AB AB AB AB AB value obtained for the CGLMP inequality (any d)inthe + 2,0 − 2,1 − 0 − 1 − 0 qAB qAB 2qA qA qB . (28) bipartite case.

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(a) 0.30 0.5 N =2 0.25 N =3 N =4 0.4 0.20 N =5 N =6 ) q 0.3 ) NL(

0.10 q

NL( 0.2

0.00 02468101214161820 Sliwa 0.1

(b) 0.40 max NS violation max Q violation 0.0

0.30 MN 0.25 ) FIG. 6. NL(q) as a function of the values of MN for N = q 0.20 2,3,4,5,6. We have used the straight line for odd cases and dashed

NL( lines for even cases. The circles correspond to NL(q) evaluated 0.10 assuming that each possible measurement setting has a probability 1/2N . The diamonds correspond to the case where only the measure- 0.00 ment settings appearing in MN have a probability different from zero 12 5 1015202530354045 (happening for the N odd case). Sliwa Class

FIG. 5. (a) Values of NL(q) as a function of the violation of the to decrease the value of NL(q). The reason for that comes 46 Sliwa classes of inequalities. (b) The values of NL(q) for the from the fact that from all possible 2N measurement settings maximal quantum [41] and nonsignalling violation of each one of allowed by the scenario, only 2N−1 enter in the evaluation of = these inequalities. The maximum value of NL(q) 1/8isachievedby the MN . Since we are fixing the probability of the inputs to be the maximum violation of the Mermin inequality [42] (corresponding identically distributed, that amounts to reducing NL(q)bya to the second Sliwa inequality [40]). factor of one-half. If, instead, we now choose the probability N−1 of inputs to be 1/2 for all those appearing in MN , and zero C. Mermin inequality for more parts otherwise, we then recover a monotonically increasing value of NL(q) with N. In particular, notice that by doing this, we The results presented above show that considering the = paradigmatic and Sliwa [40] inequalities, the best values we achieve a value of NL(q) 1/4 for the maximum quantum could achieve for our quantifier were NL(q) = 1/4 in the case violation of the Mermin inequality in the tripartite scenario. of nonsignalling correlations and NL(q) = 1/8 for quantum In this case, we can also provide an analytical construction, providing an upper bound for NL(q), that perfectly coincides correlations. As we show next, these values can be improved = by analyzing multipartite scenarios beyond three parts; more with the LP results with N 2,...,6 and that, for this reason, precisely, by considering the generalization of the Mermin we conjecture provides the actual value of NL(q) for any N. inequality [42–45] in its form given by [46], For simplicity in what follows, we will restrict our attention to the case of N even. M 1, (29) = + − 1 N We choose a fixed distribution q vqmax (1 v) 2N , where qmax is the distribution given the maximum NS of the that is, defined recursively starting with M1 = A1 by inequality. Notice that MN only contain full correlators and M − M¯ − can be succinctly written as M = i 1 (A + A¯ ) + i 1 (A − A¯ ), (30) i 2 i i 2 i i 1 x1 xN MN = cx ,...,x A ...A , (31) where M¯ i−1 is obtained from Mi−1 by exchanging all the ob- N 1 N 1 N 2 = servables A ↔ A¯. By choosing suitable projective observables x1,...,xN 0,1 in the X-Y plane of the Bloch sphere and Greenberger-Horne- =± with cx1,...,xN 1. This means that qmax is such that Zeilinger (GHZ) states [47], the maximum quantum violation | = N−1 ⊕···⊕ ⊕ N−1 p(a1,...,aN x1,...,xN ) 1/2 if a1 aN is given by |MN | = 2 2 .ForN odd, this is also the algebraic δ− and p(a ,...,a |x ,...,x ) = 0 otherwise. 1,cx1,...,xN 1 N 1 N or nonsignalling maximum of the inequality. For N even, the As the probability p entering in q − p ,we N 1 | | = 2 = + − 1 algebraic or nonsignalling maximum is given by MN 2 . choose p vpmax (1 v) 2N , where pmax is defined N−1 N−1 Succinctly, the maximum NS is |MN | = 2 2 . by p(a1,...,aN |x1,...,xN ) = 1/2 if a1 ⊕···⊕ ⊕ Following the same approach as before, we have computed aN δ−1,cx ,...,x (and zero otherwise). We see that for 1 N=+ − = the value of NL(q) as a function of MN . The results are shown every cx1,...,xN 1, it follows that q p 1 0. For =− − = in Fig. 6. Interestingly, we see a clearly increase of NL(q)as cx1,...,xN 1, we have q p 1 2v. So, basically, we we consider the maximum violation of MN with increasing have to count the number of positive and negative coefficients + N N. Notice, however, that going from N even to N 1 seems cx1,...,xN and multiply each of these by 1/2 (assuming all

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0.25 I only gives a nontight lower bound to NL(q). Moreover, we see 2222 that optimizing over p(x,y) such that p(x,y) = 1/4iffx,y = I3322 2 (in such a way that we recover the CHSH inequality) gives 0.20 us a higher value for NL(q).

0.15 ) B. The EPR-2 decomposition q Any probability distribution q = q(a,b|x,y)inaBellex- NL( 0.10 periment can be decomposed into a convex mixture of a local part qL and a nonlocal and nonsignalling part qNL as q = 0.05 (1 − wNL)qL + wNLqNL, with 0 wNL 1. The minimum nonlocal weight over all such decompositions, 0.00 . 0.00.20.40.60. .0 w˜ NL(q) = min wNL, (33) qL,qNL Inn22 defines the nonlocal content of q, a natural quantifier of the FIG. 7. The black line shows NL(q) for the distribution q = nonlocality in q. Nicely, the violation of any Bell inequality L L vqmax + (1 − v)1/4 as a function of I3322 = 2v − 1 considering that I I (with I the local bound) yields a nontrivial lower = = p(x,y) 1/4iffx,y 2 (in which case we basically recover I2222). bound tow ˜ NL given by The red solid line shows NL(q) as a function of I3322 = 2v − 1 − L considering the full distribution q. The purple dashed line shows I(q) I = − w˜ NL(q) , (34) NL(q) as a function of I3322 2v 1 considering only the value I NL − I L of I (thus optimizing over all q compatible with it). Clearly, 3322 NL imposing the value of a given Bell inequality only provides a (in with I being the maximum value of I obtainable with general, nontight) lower bound for NL(q). nonsignalling correlations. Similarly to the trace distance, the nonlocal content can also be computed via a linear program. However, differently from the trace distance, we see that any the inputs are equally likely) and by the distance 0 or 2v. extremal nonlocal point of the NS polytope will achieve the The number of elements with negative coefficients can be maximum according to this measure, independently of its found by a recursive relation. Given that MN had αN negative N−2 actual distance to the set of local correlations. coefficients, it is easy to see that α + = 2α + 2 with N 2 N Considering the CHSH scenario and following an identical the initial condition that α = 1. So our quantifier for N even 2 approach as the one used to get (24), one can show thatw ˜ = is given by NL 2max[0, (CHSH)]. That is, in this particular case, we have N 3N/2 = NL(q) = v(1/2 )αN = MN (1/2 )αN , (32) a simple relationw ˜ NL(q) 4NL(q). This picture, however, changes drastically already at the tripartite scenario. For each = →∞ = = which tends to NL(q) 1/2 with N and v 1(MN of the 45 classes of extremal NS points (those maximally N/2 2 ). violating the Sliwa inequalities), it follows from (34) that they achieve the maximum,w ˜ NL(q) = 1. So, from the perspective VI. RELATION TO OTHER NONLOCALITY MEASURES of the nonlocal content, all nonlocal extremal points of the NS polytope display the same amount of nonlocality. That is not the In the following, we will compare the trace distance measure case for NL(q), as can be clearly seen in Fig. 5 as the different with three other standard measures of nonlocality: the amount extremal points have different values for it, illustrating the fact of violation of Bell inequalities, the Elitzur-Popescu-Rohrlich that they are at different distances from the local polytope. (EPR)-2 decomposition [12,48], and the relative entropy [17].

A. The violation of Bell inequalities C. The relative entropy The violation of Bell inequalities is a standard way to quan- Another important nonlocality quantifier is defined in terms tify the degree of nonlocality and, in some cases, can as well of the relative entropy (also known as the Kullback-Leibler find operational interpretations, for instance, as the probability divergence) given by [16,17] of success in some distributed computation protocols [49,50]. = 1 Clearly, one can expect that the more a given distribution NLKL(q) min KL(q,p), (35) |x||y| p∈CC violates a Bell inequality, the higher is the trace distance from = the local set. with KL(q(x),p(x)) x q(x)log2 q(x)/p(x). Similarly to However, in general, the violation of a given Bell inequality (7), we have also chosen an identically distributed distribution only provides a lower bound to such trace distance, as can be for the outcomes. This quantity is also a monotone for the clearly seen in Fig. 7. In this figure, we consider the I3322 resource theory of nonlocality defined by the free operations inequality [see (29)]. Using the LP formulation, we have com- discussed in Sec. IV. The relevance of this measure comes puted NL(q) for a distribution of the form q = vqmax + (1 − from the fact that it is a standard statistical tool quantifying v)1/4, that is, a mixture of the distribution maximally violating the average amount of support against the possibility that an the I3322 with white noise rendering I3322 = 2v − 1. Clearly, apparent nonlocal distribution can be generated by a local simply imposing the value of the violation of the inequality model [16].

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3.0 0.25 2 I3 ME γ =0.577 4CNL(q) CGLMP 0.10 ∗ ∗ ME γ =0.577 MV γ =0.617 KL(q , p ) 2.5 MV γ =0.617 OS γ =0.642 NL (q∗) 0.20 KLbf OS γ =0.642 NL (q )=0.058 γ =0.369 KL me c 0.08 NL (q )=0.072 2.0 KL mv NLKL(qos)=0.077 0.15 )

q 0.06 1.5 (p∗, q∗) → min NL(q) 3 CGLMP

I p ∈ L, NL( 0.10 1.0 0.04 q ∈ NS

0.05 0.5 Relative Entropy 0.02

0.0 0.00 0.1 0.3 γ 0.5 0.7 0.37 0.50 0.60 0.70 c 0.00 γ γ 0.00 0.05 0.10 0.15 0.20 0.25 I3 3 CGLMP FIG. 8. The behavior of the ICGLMP (left) and NL(q) (right) inequality as a function of γ for the distribution q arising from FIG. 9. Plot of minp∈C KL(q,p) as a function of the CGLMP in- | = | + − 2| + | C measurements on the state φ γ 00 1 2γ 11 γ 22 . equality for d = 3. The black dashed curve represents the lower bound 3 From the critical value of γc > 0.369, we have that ICGLMP > 0and provided the Pinsker inequality, deﬁned by KL(q,p) 4CNL(q)2, NL(q) > 0. In both panels, the circle, square, and diamond correspond = 1 = 1 − where C 2 log2 e and NL(q) 2 q p 1 . The red curve provides to the maximal entangled state (ME), the state that maximally violates ∗ ∗ an upper bound for min ∈C KL(q,p) since it computes KL(q ,p ), 3 p C (MV) the ICGLMP, and the optimal state (OS) that provides the maximal where q∗ and p∗ are the solutions provided by the LP minimization of q p 3 KL( , ), respectively. As we can see, the maximal ICGLMP violation NL(q) subjected to NS constraints and a given value of the CGLMP = is achieved by γ 0.617 and the maximal of NL(q) is achieved for inequality. The blue curve is obtained by a brute force minimization the same γ . This is opposed to the result found in [17], where the ∗ of minp∈C KL(q ,p). The color points represent the three points q p 3 C quantum distribution with maximum KL( , ) does not violate ICGLMP found in [17] and discussed in the main text. Interestingly, for the maximally. We also notice that according to the measure proposed in green (circle) and pink (diamond) points, there are NS correlations [21,22], the maximal nonlocality is achieved exactly to the maximally (possibly postquantum) with the same value of the CGLMP inequality entangled state. This shows how different measures can lead to very but providing a higher value of minp∈C KL(q,p). different orderings. C

VII. DISCUSSION In [17], special attention has to be given to the relation Apart from its primal importance in the foundations of between NLKL(q) and the violation of the CGLMP inequality quantum physics, nonlocality has also found several applica- [38]. It has been shown that quantum correlations maximally tions as a resource in quantum cryptography [52], randomness violating the CGLMP inequality do not necessarily imply the generation and amplification [53,54], self-testing [55], and optimum relative entropy. For that, the set of considered quan- distributed computing [49]. Within both these fundamental and tum distributions comes from a fixed set of measurements on applied contexts, quantifying nonlocality is undoubtedly an a two-qutrit state of the form |φ=γ |00+ 1 − 2γ 2|11+ important primitive. γ |22. Inspired by that result, we have analyzed what is the Here we have introduced a natural quantifier for nonlocality, value of NL(q) in the same setup. The results are shown in a geometrical measure based on the trace distance between Fig. 8 with three distributions being especially relevant [17]: the probability distribution under test and the set of classical (i) the distribution obtained from the maximum√ violation of distributions compatible with LHV models. Nicely, our quan- CGLMP with maximally entangled states (γ = 1 3), (ii) the tifier can be efficiently computed (as well as closed-form an- maximum violation of the CGLMP (obtained with γ = 0.617), alytical expressions can be derived) via a linear programming and (iii) the maximum value of NLKL(q) (achieved with γ = formulation. We have shown that it provides a proper quantifier 0.642). As expected from the results in Sec. V and differently since it is monotonous under a wide class of free operations from the results obtained for the relative entropy, the more we defined in the resource theory of nonlocality [5,27]. Finally, violate the CGLMP inequality the higher is the trace distance we have applied our framework to a few scenarios of interest measure NL(q). and compared our approach to other standard measures. Interestingly, via the Pinsker inequality [51], we can relate It would be interesting to use the trace distance measure the trace and relative entropy measure. That is, the trace to analyze previous results in the literature, for instance, the distance provides a nontrivial bound to the relative entropy. nonlocality distillation and activation protocols proposed in Furthermore, the LP solutions to the minimization of the trace [56] or the role of the trace distance in cryptographic protocols distance naturally give an ansatz solution providing an upper [52]. From the statistical perspective, one can investigate the bound for NLKL(q). These results are shown in Fig. 9, where relation of the trace distance to the relative entropy [16,17] we plot NLKL(q) as a function of the CGLMP violation. In and their interconnections to p values [57] and the statistical this plot, we also show the curve obtained using the optimal significance of Bell tests with limited data [58,59]. Finally, nonsignalling distribution q∗ obtained from the LP used to we point out that even though here we have focused on minimize NL(q) subjected to a specific value of the CGLMP Bell nonlocality, the same measure can also be applied to inequality (see the caption in Fig. 9 for more details). quantify nonclassical behavior in more general notions of

022111-8 QUANTIFYING BELL NONLOCALITY WITH THE TRACE … PHYSICAL REVIEW A 97, 022111 (2018) nonlocality [60–62] as well as in quantum contextuality [8,9]. 1 q − p∗ (A11) We hope our results might motivate further research in these 2|x||y| 1 directions. 1 = q − p∗ | || | πk k πk k (A12) ACKNOWLEDGMENTS 2 x y k k 1 We thank D. Cavalcanti for valuable comments on the 1 ∗ π q − p (A13) manuscript and T. O. Maciel for interesting discussions. 2|x||y| k k k 1 k S.G.A.B., B.A., and R.C. acknowledge ﬁnancial support from the Brazilian ministries MEC and MCTIC. B.A. also acknowl- = πkNL(qk). (A14) edges ﬁnancial support from CNPq. k

APPENDIX: DETAILED PROOF OF THE RESULTS IN Corollary 1. If p ∈ CC and π ∈ [0,1], then SECTION IV NL(πq + (1 − π) p) πNL(q). (A15) 1. Relabeling operations Theorem 1. If R is a relabeling of inputs or outputs of q, 3. Postprocessing operations R q = q . then NL( ( )) NL( ) Theorem 3. If O is an postprocessing operation, deﬁned as Proof. p∗ ∈ C Let C be such that in Eq. (16), then NL(O(q)) NL(q). ∗ 1 Proof. Let p ∈ CC be the distribution satisfying Eq. (A1). NL(p) = q − p∗ . (A1) 2|x||y| 1 Then, ∗ 1 The distributions R(q) and R(p ) are obtained, respectively, NL(O(q)) = min O(q) − p (A16) ∗ | || | ∈C 1 from q and p by a permutation of entries. Then, it follows that 2 x y p C ∗ ∗ 1 ∗ R(q) − R(p ) =q − p . (A2) O(q) − O(p ) (A17) 1 1 2|x||y| 1

Since relabeling operations preserve the set of local distribu- ∗ 1 L tions, R(p ) ∈ CC and hence = O (α,β|a,b,x,y)[q(a,b|x,y) 2|x||y| α,β,x,y a,b R = 1 R − NL( (q)) min (q) p 1 (A3) 2|x||y| p∈CC ∗ − p (a,b|x,y)] (A18) 1 R(q) − R(p∗) (A4) | || | 1 1 2 x y OL α,β|a,b,x,y |q a,b|x,y | || | ( ) ( ) 1 2 x y = q − p∗ (A5) α,β,a,b,x,y | || | 1 ∗ 2 x y − p (a,b|x,y)| (A19) = NL(q). (A6) 1 = |q(a,b|x,y) − p∗(a,b|x,y)| (A20) 2|x||y| Relabeling operations are invertible, and hence there is a a,b,x,y relabeling operation R−1 such that q = R−1[R(q)]. A similar 1 argument shows that = q − p∗ = NL(q). (A21) 2|x||y| 1 = R−1 R R NL(q) NL( [ (q)]) NL( (q)), (A7) which proves the desired result. 4. Preprocessing operations 2. Convex combinations In Ref. [5], the author deﬁnes the uncorrelated input enlarg- = = ing operation, which consists of one or more parts adding an Theorem 2. If q k πkqk, where πk 0 and k πk 1, then uncorrelated measurement locally. Without loss of generality, we can assume that this measurement is deterministic since NL(q) π NL(q ). (A8) k k convexity of the 1 norm implies that NL is also a monotone k under the addition of a nondeterministic uncorrelated measure- ∗ ∈ C ment. Proof. Let pk C be such that Given a correlation q, suppose part A adds one uncorrelated 1 | |= NL(q ) = q − p∗ , (A9) measurement at her side. Denoting x mA, we deﬁne k 2|x||y| k k 1 q(a,b|x,y)ifx mA ∗ ∗ q (a,b|x,y) = (A22) = ∈ C f | = + and let p k πkpk C. Then, we have q(b y)δa,a if x mA 1, 1 where a is the deterministic output of the additional measure- NL(q) = min q − p (A10) 1 + 2|x||y| p∈CC ment mA 1.

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Theorem 4. If qf is obtained from q by the input enlarging Theorem 5. If I is an preprocessing operation such that | |=| | | |=| | L | operation deﬁned in Eq. (A22), then χ x , ψ y , and χ,ψ I (x,y χ,ψ) 1, then NL(q ) NL(q). (A23) f NL(I(q)) NL(q). (A29) ∗ Proof. Let p ∈ CC be the distribution satisfying Eq. (A1). + ∗ For any pair of inputs mA 1,y, we have that Proof. Let p ∈ CC be the distribution satisfying Eq. (A1). Then, | | + − ∗ | + | qf (a,b mA 1,y) pf (a,b mA 1,y) (A24) a,b NL(I(q)) = |q(b|y) − p∗(b|y)| (A25) = 1 I − min (q) p (A30) b 2|χ||ψ| p∈L 1 1 ∗ = q(a,b|x,y) − p∗(a,b|x,y) ∀x m (A26) I(q) − I p (A31) A 2|χ||ψ| 1 b a | | − ∗ | |∀ 1 ∗ q(a,b x,y) p (a,b x,y) x mA. (A27) = [q(a,b|x,y) − p (a,b|x,y)] 2|χ||ψ| a,b a,b,χ,ψ x,y

Hence,

× L | | | + − ∗ | + | I (x,y χ,ψ) (A32) qf (a,b mA 1,y) pf (a,b mA 1,y) a,b 1 I L(x,y|χ,ψ)|q(a,b|x,y) ∗ 2|χ||ψ| min q(a,b|x,y) − p (a,b|x,y) . (A28) a,b,x,y χ,ψ xmA a a,b − p∗(a,b|x,y)| (A33) This implies that qf is obtained from q by adding pairs of 1 ∗ inputs for which the distance of the probability distributions = |q(a,b|x,y) − p (a,b|x,y)| (A34) ∗ 2|x||y| qf and pf decreases. Since NL is deﬁned by taking the average a,b,x,y over the pairs of inputs, this implies the desired result. 1 ∗ We now proceed to prove monotonicity under the prepro- = q − p = NL(q). (A35) 2|x||y| 1 cessing operations deﬁned in Eq. (19).

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