Experimentally friendly approach towards nonlocal correlations in multisetting N -partite Bell scenarios

In this work, we study a recently proposed operational measure of nonlocality which describes the probability of violation of local realism under randomly sampled observables, and the strength of such violation as described by resistance to white noise admixture. While our knowledge concerning these quantities is well established from a theoretical point of view, the experimental counterpart is a considerably harder task and very little has been done in this field. It is caused by the lack of complete knowledge about the local polytope required for the analysis. In this paper, we propose a simple procedure towards experimentally determining both quantities, based on the incomplete set of tight Bell inequalities. We show that the imprecision arising from this approach is of similar magnitude as the potential measurement errors. We also show that even with both a randomly chosen N -qubit pure state and randomly chosen measurement bases, a violation of local realism can be detected experimentally almost 100% of the time. Among other applications, our work provides a feasible alternative for the witnessing of genuine multipartite entanglement without aligned reference frames.

In this work, we study a recently proposed operational measure of nonlocality which describes the probability of violation of local realism under randomly sampled observables, and the strength of such violation as described by resistance to white noise admixture. While our knowledge concerning these quantities is well established from a theoretical point of view, the experimental counterpart is a considerably harder task and very little has been done in this field. It is caused by the lack of complete knowledge about the local polytope required for the analysis. In this paper, we propose a simple procedure towards experimentally determining both quantities, based on the incomplete set of tight Bell inequalities. We show that the imprecision arising from this approach is of similar magnitude as the potential measurement errors. We also show that even with both a randomly chosen N -qubit pure state and randomly chosen measurement bases, a violation of local realism can be detected experimentally almost 100% of the time. Among other applications, our work provides a feasible alternative for the witnessing of genuine multipartite entanglement without aligned reference frames.

I. INTRODUCTION
Nonlocality is arguably one of the most striking aspects of quantum mechanics, dramatically defying our intuition about time and space [1]. Although this feature was initially thought to be an evidence of the incompleteness of the quantum theory [2], there is today overwhelming experimental evidence that nature is indeed nonlocal [3]. Nonlocality also plays a central role in quantum information science and has been recognized an essential resource for quantum information tasks [4], for instance, quantum key distribution [5,6], communication complexity [7], randomness generation [8], and device-independent information processing [9,10]. All such device-independent applications require states that strongly violate Bell inequalities. However, the concept of "strength of violation" is controversial in the literature [11]. Consequently, it is still unclear what is a good quantifier of nonlocality.
Yet another possibility to quantify the nonlocal correlations of complex states is based on the probability that random measurements generate nonlocal statistics. The probability of violation of local realism under random measurements, proposed in [12,13], has gained considerable attention as an operational measure of nonclassical-ity of quantum states [14]. It has been demonstrated both numerically [15][16][17][18] and analytically [14,19] that this quantity is a good candidate for a nonlocality measure. Furthermore, in [19] it was proved that this quantifier satisfies some natural properties and expectations for an operational measure of nonclassicality, e.g., invariance under local unitaries. The probability of violation is also often called volume of violation [14] or nonlocal fraction [18,19]. In this paper we will use the latter name.
The nonlocal fraction of the state ρ is defined as [12,13] P V (ρ) = f (ρ, Ω)dΩ, where we integrate over a space of measurement parameters Ω that can be varied within a Bell  What is important, in this approach the nonlocal correlations are quantified without any a prior assumptions about specific Bell inequalities.
Although definition in Eq. (1) fairly captures the nonlocality extent of a state, the nonlocal fraction does not provide much information about the strength of nonlocality. Therefore, it seems useful to put it together with another quantitative description, called nonlocality strength, which addresses the "fragility" of this nonlocality against noise [20].
While the study of the nonlocal fraction and the nonlocality strength offers a promising insight into a geometry of the set of quantum correlations, several crucial aspects towards experimental investigations were not addressed so far in the literature. The main limitation of these quantifiers is that the analysis of them requires a complete knowledge about local polytope (e.g. a complete set of tight Bell inequalities) to detect violation of local realism. But apart from the simplest cases, namely a Clauser-Horne-Shimony-Holt (CHSH) scenario [21] and Pitowsky-Svozil (PS) scenario [22,23], such complete set remains unknown. In general, the number of tight Bell inequalities is expected to grow exponentially with the number of parties and the number of measurement settings.
From theoretical point of view, the problem can be lifted by applying a linear programming approach and directly considering the space of behaviors (space of joint probabilities), which local polytopes inhabit. In this case, the only context information required being the number of parties and the number of measurements per party [15]. However, this approach has no direct experimental implementation what causes a lack of experimental studies of the subject. The only known experimental results are related with the CHSH [24] and PS scenario [18] mentioned above.
A question which naturally arises is whether the nonlocal fraction and the nonlocality strength can be measured experimentally with a partial knowledge about a full set of Bell inequalities. So far only a few attempts the have been made to solve the problem, showing usually a great underestimation of the exact results. For instance, it was shown in Ref. [13] that the nonlocal fraction estimated by means of the Mermin-Ardehali-Belinskii-Klyshko (MABK) and Weinfurter-Werner-Wolf-Żukowski-Brukner (WWWŻB) inequalities is few times smaller than that of the full set of Bell inequalities. However, it turns out recently that in some cases, it is possible to indicate an incomplete set of inequalities, which significantly improves the estimation [18]. In other words, determining the nonlocality within incomplete set looks promising if choosing Bell's inequality class appropriately. Moreover, it seems that, from an experimental point of view, just such an approach with an incomplete set of Bell inequalities is desirable. By the very definition, P V is solely related to the fact of violation of local realism, not to the strength of such violation. Therefore, any Bell inequality which is violated with the strength of being close to the upper threshold, is unsuitable for experimental verification of e.g. the nonlocal fraction since a violation might be simply accidental due to shot noise.
In this work, we tackle these problems and analyze the statistical relevance of various classes of Bell inequalities for several Bell scenarios. We show that even in the very general scenario, one such class provides results that are close to those of the full set of Bell inequalities. In other words, one can considerably simplify the procedure towards determining the nonlocal fraction by using only one suitable inequality instead of the complicated linear programming method. Surprisingly, the imprecision arising from this approach can be made negligible. Therefore, these results open a door for experimental verification of many known theoretical predictions for multipartite qubit states. Our predictions were also investigated experimentally for a three-qubit case, showing a good agreement with theoretical results.

II. DESCRIPTION OF THE PROBLEM
In this paper we consider the most general Bell experiment with N spatially separated observers performing measurements on a given state of N qubits. Each observer can choose among m i arbitrary ob- where U i j denotes a general unitary transformations belonging to U (3) group and |r i stands for computational basis state of the ith observer. The measurement in each bases provides the observer with one out of two possible outcomes, denoted r i j = {0, 1}. For simplicity, we will refer to this scenario as m 1 × · · · × m N .
With the above assumption, a local realistic description of a Bell experiment is equivalent to the existence of a joint probability distribution p lr (r 1 1 , . . . , r 1 m1 ; . . . ; r N 1 , . . . , r N m N ), where r i ji denotes the result of the measurement performed by the ith observer when they choose j i th measurement setting. If the model exist, quantum predictions for the probabilities are given by the marginal sums: where P (r 1 , · · · , r N |Ô 1 k1 , · · · ,Ô N k N ) denotes the probability that all observers simultaneously obtain the respective result r i while measuring observables O i ki . It can be shown that for some quantum entangled states the marginal sums cannot be satisfied, which is an expression of Bell's theorem. Determining the existence of the local realistic description, for a given state and set of observables, is a typical linear programming problem [25]. However, in the case of experimental studies, one should follow a different direction.
In the space of probabilities, the set of local correlations P (r 1 , · · · , r N |Ô 1 k1 , · · · ,Ô N k N ) which satisfy 2 (hereafter denoted as L N ) is convex with finitely many vertices and called the local polytope [26]. The L N polytope is bounded by facets (hyperplanes) which can be described by a linear function of the probabilities where wÔ r and C LHV are real coefficients, and we have simplified the notation by introducing P = {P (r 1 , · · · , r N |Ô 1 k1 , · · · ,Ô N k N )},Ô = {Ô 1 k1 , · · · ,Ô N k N }, and r = {r 1 , · · · , r N }. Correlations which do not admit the decomposition in Eq. (2) are referred to as nonlocal and lie outside the local polytope L N . In other words, they must violate at least one inequality I (N ) (P) ≤ C LHV . Such inequalities are called tight Bell inequalities and C LHV depicts the upper threshold of one inequality for local realism. In order to determine the nonlocal fraction P V for a given state, we calculate how many sets of settings (in percents) lead to violation of local realism, i.e. whether the decomposition in Eq. (2) exists or alternatively whether all Bell inequalities I (N ) for given Bell scenario are satisfied. As in general, the full set of tight Bell inequalities is unknown, in the rest of the text by P L N V we emphasize the fact (if necessary) that results were obtained with linear programming method and refer to the whole polytope L N while P I V corresponds to a subset of Bell inequalities.
Usually for experimental purpose, an alternative parameterization of I (N ) (P) is used. It is based on correlation coefficients, e.g E i ki , E i ki E j kj etc., which satisfy the relation and have a clear experimental interpretation. For instance, for an experimental setup based on correlated photons each correlation coefficient can be expressed as a function of coincidence counts measured on the detectors [16].
The degree of violation of the Bell inequality I (N ) (P) is also directly related with the so-called resistance to noise i.e. the amount of white noise admixture required to completely suppress the nonlocal character of the original correlations of a given state ρ. Specifically, if for the state ρ and particular choice of measurement settings I (N ) (P) > C LHV then a new state Following this observation, a new quantity called nonlocality strength, S, can be defined [20]. It is given by S = 1 − v crit . Furthermore, it is convenient to use the average value of nonlocality strength: where g(S) is a nonlocality strength distribution and S max depict a highest attainable nonlocal strength with respect to the full set of tight Bell inequalities and measurement settings. Our results are normalized such that the areas of the regions bounded by the plots directly provide the nonlocal fraction,

III. NUMERICAL RESULTS AND DISCUSSION
A. Three-qubit states Let us start with the three spatially separated observers performing one out of two dichotomic measurements, i.e. the 2 × 2 × 2 case. This scenario is completely characterized by 46 classes (families) of Bell inequalities derived by Pitowsky and Svozil [22]. First, we determine the statistical relevance of these classes concerning the nonlocal fraction. We calculate P V of each class independently for two inequivalent types of tripartite entangled  Fig. 1, where the families of Bell inequalities are numbered in like manner as in Ref. [23] and listed in Appendix B: Table B.1.
As we see, depending on the chosen entangled state the statistical relevance of individual families may vary but the best three items remain unchanged. They are given by the class of 4th, 5th, and 6th facet inequality [23] (hereafter I 4 , I 5 , and I 6 , respectively). Moreover, apart from these three classes, the nonlocal fraction for other families of Bell inequalities do not exceed 1 2 P L3 V neither for the GHZ 3 nor for the W 3 state (see Table I). In particular, the MABK and WWWŻB inequalities, previously discussed in Refs. [12,13], provide results much smaller than these of the best three items. Interestingly, even if we consider such 43 classes simultaneously (the complete set of Bell inequalities, excluding I 4 , I 5 , and I 6 ), the resulting nonlocal fraction is not greater than 57% and 30% for the GHZ 3 and W 3 state, respectively. To highlight this phenomenon, we note that these 43 classes contain 51712 tight Bell inequalities while I 4 , I 5 , and I 6 corresponds only to 96, 512, and 1536 inequalities, respectively. Therefore, all the upper-mentioned 43 classes are rather unsuitable for a potential experimental measure of the nonlocal fraction.
Among the I 4 , I 5 , and I 6 families, a clear dominant position is reserved for the 4th facet inequality. As shown in Table I, this family provides a very good approximation of P L3 V with a gap smaller than 5.3 p.p., . Furthermore, the I 4 family can also be used to estimate the nonlocal fraction for other kinds of states, like the generalized nonmaximally entangled GHZ 3 states (see Ref. [18]).
Analogical conclusion can be drawn when the nonlocality strength, S, is taken into consideration. Once again we see (Table I) that the best approximation of the average strength,S, is provided by the I 4 family, what suggests a similar shape of the nonlocality strength distribution for both cases, L 3 and I 4 . Intuitively, one may expect such outcome, assuming that the better estimation of nonlocal fraction, the closer approximation of g(S). Although that assumption is correct for the cases in question (see Fig 2), in general, it is not true as will be discussed later. Despite of that another very interesting remarks towards an experimental implementation can be made on the distribution g(S). As we see in Fig. 2, the nonlocality strength distribution for I 4 has a similar shape as g(S) for the politope L 3 . Any differences are either small in magnitude or appear for small S. For instance, the function g I4 (S) for GHZ 3 state vanishes above S > 0.29 what stands in contrast to g L3 (S), due to the strong violation of MABK inequality. On the other hand, the greatest difference between g I4 (S) and g L3 (S) for W state is observed for S < 0.1. Bearing in mind that an experimental detection of the nonlocality (violation of Bell inequality) is ambiguous when the nonlocal strength is close to the measurement accuracy (say S = ±0.015 [28]), this tendency should imply a positive impact on the potential experimental results by decreasing experimental errors. The classes of the 5th and 6th facet inequality, on the other hand, indicate an opposite trend and they overestimate g L3 (S) in the regime of small S while strongly underestimate g L3 (S) for S > 0.06 (Fig 2).
Note that some of the considered inequalities can be seen as genuine multipartite entanglement witnesses. If for a given state the nonlocal fraction P V > 2(π − 3) ≈ 28.319% than the state is genuine multipartite entangled [20]. Based on this fact we can conclude that the inequalities: I 4 , I 5 , I 6 , I 13 and I 19 can detect genuine three qubit entanglement of GHZ 3 state. For the three qubit W state the set of such inequalities is smaller and contains only the inequalities: I 4 , I 5 and I 6 .
Next, we analyze the scenario when the number of measurement settings increases. In particular, we extend our studies to the 3 × 2 × 2, 3 × 3 × 2, and 3 × 3 × 3 scenario. As a complete set of tight Bell inequalities for any of these cases is unknown, we employed a linear programming method [25] to identify an explicit form of the most relevant Bell inequalities, which were the most frequently violated for given (random) measurement settings. Some of them naturally overlap with the inequalities derived by Pitowsky and Svozil [22] but genuine m 1 × m 2 × m 3 inequalities also belong to that group. All identified genuine inequalities are listed in Appendix B: Table B.2. Every such expression represents a distinct class of Bell inequalities, equivalent under permutation of parties, inputs, and outputs [29,30].
Based on our identification, the nonlocal fraction and the average strength of nonlocality has been calculated for each m 1 × m 2 × m 3 scenarios mentioned above. The results are collected in Table II from which the following remarks can be drawn: (i) As we see, for each scenario the highest P V and S is always achieved for the I 4 family. Moreover, the gap between P L3 V and P I4 V decreases with the number of measurement settings (see Fig. 3). In other words, for three observers the I 4 family seems to be sufficient tool for experimental determination of P V .
(ii) The last observation also implies that the rapid increase of the nonlocal fraction with the number of measurement settings observed in Ref. [15] has a statistical explanation (at least for three-qubit states), i.e. by increasing the number of settings, we increase the number of Bell inequalities (equivalent under permutation of parties, inputs, and outputs) that belong to the I 4 family and hence, the probability that some of them are violated, involving only two settings. It is worth emphasizing that TABLE III. Nonlocal fraction PV and average nonlocality strengthS for various four-qubit states and I 4 opt inequality. In the last column we present the threshold values of PV which can be achieved with two-producible states (i.e. states which involve only two-party entanglement. They can be calculated using the formalism presented in [15] with P ij V being the nonlocal fraction for the two-qubit GHZ state for the scenario i × j and equal to: etc.) provide the nonlocal fraction significantly smaller than that of the I 4 family.
(iii) In the case ofS, an increase of the number of measurement settings implies a growth of the gap between S L3 andS I4 , although very slight if the W 3 states are taken under consideration. This behavior is caused by a considerable reduction in the number of Bell inequalities describing local polytopes and will be further discussed in the next section.

B. N-qubit states
In this section we present our main result. To do that let us consider the most general scenario when the Bell experiment is performed by N spatially separated observers with m 1 × · · · × m N measurement settings. Let us also restrict our calculations to a family of Bell inequalities with a form where I stands for the CHSH expression [21] and E (i) j denotes an observable measured by ith observer when he/she chooses jth measurement setting. In other words, the inequality I

Highly entangled states
Let us now investigate the statistical aspects of nonlocal correlations for several four-and five-qubit states, using only the Bell inequality I (N ) opt . As for N −parties systems, numerous inequivalent kinds of entanglement exist, in this subsection we investigate the behavior of some archetypal four-and five-qubit maximally entangled states. They are explicitly defined in the Appendix A.
We started by discussing the case in which the quantum state under study is a product of two states, |ψ = |GHZ 2 ⊗ |00 , what clearly illustrates the nature of the I (N ) opt inequality. In this case, the nonlocal fraction takes one of several recurring values, depending on the number of measurement settings m 1 and/or m 2 , e.g. 28.318%, 52.401%, and 78.219% as in Table III. The number of measurement settings m 3 and m 4 is irrelevant and has no influence on P V . It is because the second part of the analyzed state cannot reveal any kind of nonlocality, regardless of the projectors E (i) j . The only valuable projection of the state |ψ onto two-qubit state is the one which gives maximally entangled state |GHZ 2 . Therefore, the nonlocal fraction of |ψ coincides exactly with the corresponding results of |GHZ 2 (see Ref. [15]). However, the value of m 3 and m 4 undoubtedly influences the resulting nonlocal strength (see Table III). Furthermore, the average S of |ψ is around two times greater than that of |GHZ 2 [20]. For instance,S 2×2×1×1 (ψ) = 0.0536 whilē S 2×2 (GHZ 2 ) = 0.028. This observation can be easily explained based on the very definition of the inequality I (N ) opt . As mentioned above, although its violation always requires I (2) opt > 2, the final degree of violation depends on the (N − 2) single measurements. Such correction has been studied in details in Ref. [31], exposing several interesting properties like higher robustness against white noise for biseparable states than for maximally entangled states. Consequently, the larger m 3 and m 4 , the greater chance to find a better correction for given m 1 and m 2 , what implies an increase ofS. On the other hand, for quantum states revealing multipartite entanglement, such as those in Table III, the value of m 3 and m 4 affects both the nonlocal fraction and average nonlocal strength. Specifically, any increase in the number of measurement settings entails a fast growth of P V andS. Consequently, for a scenario with 3 × 3 × 3 × 2 measurement settings the nonlocal fraction is close to unity. It means that our results satisfy the theorem of Lipinska et al. [19] that P V → 1 with infinitely many settings. The explanation of this fact once again has a purely statistical background, i.e. the number of settings determines the number of two-qubit state achieved after single-qubit projections and so, it becomes more likely that at least one of these states violates the CHSH inequality. Furthermore, except the simplest scenario, 2 × 2 × 1 × 1, the nonlocal fraction for multipartite entangled states surpasses P V of |ψ . It is because, the multipartite entangled states may reveal nonlocal correlations between any pair of qubits, in contrast to the biseparable state |ψ .
It is worth mentioning that all presented inequalities in Tab. III detect the Cluster state as non two-producible (i.e. at least three partite entangled). The nonlocal fractions are greater than the respective thresholds. This is also the case for the GHZ state and the W state in all scenarios except the 2 × 2 × 1 × 1 scenario. Let us also mention the possibility of detecting genuine four-partite entanglement in these emblematic four-qubit states. For 2×2×2 settings according to Tab. I the three-qubit GHZ state gives the threshold P V = 74.688 (whereas W state gives the smaller value of P V = 54.893). Moreover Rosier et al. [15] supports numerically (see Table I in Ref. [15]) that among all biproduct four qubit-states the largest threshold for the nonlocal fraction is P V = 74.688. From this threshold it follows that any P V value higher than 74.688 in the 2 × 2 × 2 × 2 scenario indicates genuine four-qubit entanglement in the quantum state. Then according to Tab. III and relying on the validity of the this threshold, the GHZ 4 , W 4 and the Cluster states are all detected as genuinely four-qubit entangled. Finally, a very interesting observation can be made when comparing our results with the previous calculations based on linear programming method [15,20]. As illustrated in Fig. 4, the nonlocal fraction for I (4) opt is in quite good agreement with P L4 V . The best approximation of exact results is obtained for the Cluster state, with accuracy not greater that 2 p.p., while the weaker estimation appears for the Dicke state |D 2 4 (accuracy not greater that 10 p.p.). Naturally, the gap between P L4 V and P I (4) opt V decreases with the number of measurement settings. It is worth mentioning that previous calculation made for the MABK and WWWWŻB inequalities yield a much smaller results. For instance, the nonlocal fraction for 2 × 2 × 2 × 2 scenario is equal to P I MABK V (GHZ 4 ) = 13.410% and P I WWWŻB V (GHZ 4 ) = 23.407% [12,13]. When analyzing the histograms for the four-qubit states and 2 × 2 × 2 × 2 scenario we noticed an interesting behavior, i.e. an atypical character of the nonlocality strength distribution for the GHZ 4 state (see Fig.  5). While the g(S) functions for |Cluster 4 , |W 4 , and even |D 2 4 state quite well correspond to the exact results [20] (up to the slight shift towards zero), the probability of violating I  Table III). A pos- opt family provides the highest violation strength in 14.6% of a random set of settings while 79.04% of them require a genuine 2 × 2 × 2 × 2 Bell inequality. On the contrary, for the W 5 state 55.06% of highest violation involve I (4) opt and just 17.68% of them engage 2 × 2 × 2 × 2 measurements settings.
For the W 5 state it is also important to mention about a presence of a dip for nonlocality strength close to 0.02. As we see in Fig. 5, when the analysis is restricted only to the inequality I (4) opt the function g(S) takes the sharper minimum compared to the entire polytope. However, due to the ambiguity of violation for S ≤ 0.015 such dip is rather meaningless from the experimental point of view.
The results for the five-qubit states expose the above observations even more strongly. Specifically, when the number of measurement settings increases, a rapid growth of P V andS is observed (see Table IV). In particular, our numerical simulation shows that for a scenario with 2 × 2 × 2 × 2 × 2 measurement settings the nonlocal fraction P I (5) opt V is near 100% for any of the studied fivequbit states. Furthermore, a comparison with previous calculations based on linear programming [15] reveals a very good agreement between these two sets of outcomes, with the gap not greater then 2 p.p.. As illustrated in Fig.  6, the best compatibility is found for the linear-and ringcluster states [32] while the weaker estimation occurs for |W 5 .
As before, the atypical character of the nonlocality strength distribution for the GHZ state takes place also here, i.e. the functions g L5 (S) and g I (5) opt (S) for a 2×2×2× 2 × 2 scenario, presented in Fig. 7, have markedly different shapes. For other states, g I (5) (S) distribution agrees qualitatively with results perform for the whole polytope, though exposing the shift towards zero (stronger than previously) and usually higher maximum compared to g L5 (S). The greater shift causes a higher difference be-tweenS L5 andS I 1503 what provides the difference betweenS L5 andS I (5) opt of around 0.072 for the GHZ 5 and less than 0.034 for other states (see Table IV).

Typicality of nonlocal correlations
All results presented above clearly demonstrate a promising role of I   [20]. question is whether for an arbitrarily generated N −qubit state the quality of such determination is closer to that of e.g. cluster states or rather GHZ states. Therefore, in order to make our main conclusion more general, let us study the typical nonlocal fraction T V for a randomly sampled pure state. In other words, in this problem we specify only the number of qubits N and the Bell scenario, without choosing a prior any quantum state. Moreover, we will also compute the averaged strength T S in this more general situation. As expected, the typicality T V grows as the number of settings increases, reaching the value close to 100%, although m i ≤ 3 (see Table V). This means that the violation of local realism can be detected for almost all states by employing Bell scenario with not more than three randomly chosen measurement settings. Furthermore, by detection we understand the violation only the very simple inequality I (N ) opt . When comparing our results with the known outcomes for the whole polytope [20], we see that such detection underestimate T V of around 3 − 4 p.p. and T S by about 0.02, depending on the number N . Naturally, when the number of observers grows the underestimation of T S further increases, after all our approach of detection is based on the strong limitation of the set of Bell inequalities. However, in our opinion the results presented here suffice to consider the I (N ) opt family of Bell inequalities as a simple tool for experimental studies of the subject. Furthermore, complementary to [12,13,15,20], our paper gives insight into the geometry of Bell correlations in the case of multiqubit systems, showing the the majority of the phenomenon can be explain by a scenario which effectively involve two measurement settings.

A. Measurement device
We have experimentally tested theoretical ideas for the GHZ state contained in this paper using the platform of linear optics with discrete photons as qubit cariers (see Fig. 8). Quantum state was encoded into both their polarization state as well as into their spatial mode. To generate photons in an entangled state, we have adopted the idea by Kwiat et al. [33]. A crystal cascade is used that consists of two BBO (β-BaB 2 O 4 ) crystals both cut for Type-I spontaneous parametric downconversion placed in contact so that their optical axes lie in perpendicular planes. The crystals are 1 mm thick and generate photon pairs at 710 nm when pumped by a laser beam at 355 nm. First crystal generates horizontally (H) polarized photons when pumped by vertically polarized laser beam. Second crystal generates vertically (V ) polarized photon pairs when subject to horizontally polarized pumping. Rotating the laser beam polarization by an angle ϑ (with respect to the vertical polarization), we pump both the crystals coherently and obtain a superposition state where letters denote polarization of the first and second photons respectively. Note that the probability that two pairs are generated simultaneously is negligible. At this point we associate horizontal and vertical polarization state with logical states |0 and |1 . To generate a three-qubit entangled GHZ state, we need to make use of an additional degree of freedom of the first photon and entangle this degree of freedom with the photon's polarization state. This is experimentally achieved by using a beam displacer where horizontally polarized light continues propagating along the input spatial mode, but vertically polarized light is displaced into a parallel well separated spatial mode (see beam displacer BD in Fig.  8). One can easily check that when labeling the original spatial mode logical |0 and the displaced mode |1 , the overall state of the photon pair in (5) transforms into the form of where the first symbol in each bracket labels the spatial mode of the first photon, the second and third symbols correspond the polarization states of the first and second photon respectively. To observe the correlations between individual qubits, we perform local projections on all three of them simultaneously and record coincident detections within a 2 ns interval. The polarization-encoded qubit of the first photon is projected by subjecting simultaneously both its spatial modes to a half and quarter-wave plates followed by a polarizer. Subsequently, the spatial mode encoded into the first photon is converted to its polarization mode using a second beam displacer. We recall that the original polarization qubit of the first photon has already been projected before this conversion happens. Then a sequence of half and quarter-wave plates and a polarizer is used to perform projection. The projection of the third qubit is implemented by projecting polarization state of the second photon using again, as usual, the sequence of half and quarter-wave plates and a polarizer.

B. Experimental violations with random measurements
The experiment is composed of two steps. In the first one, we use the setup presented in Fig. 8 to prepare the three-qubit GHZ states. It is know that in any experimental preparation of the quantum state, various kinds of imperfections are inevitably present. The imperfections are caused, e.g., by improper setting of individual experimental components or by depolarization effects (presence of noise). To get an information about their presence in the generated state ρ expt , the quantum state tomography and maximum-likelihood estimation have been used to reconstruct the output-state density matrix [34,35]. From this data the output-state can be approximated by where the parameter ϑ = 45 • with a precision of ±0.5 • and the visibility v = 0.97 ± 0.01. The error bars are determined by Monte Carlo simulations of Poissonian noise distribution.
Next, for the state ρ expt the violation of I opt family has been studied for n = 5 · 10 4 randomly generated sets of measurement settings. Each set denotes an ensemble of projective measurement E i ki = e i ki · σ, where i = 1, 2, 3, σ = {σ x , σ y , σ z } corresponds to the vector of the Pauli operators associated with three orthogonal directions, and all unit vectors are represent in spherical coordinates, e i ki = (sin 2φ i ki cos ξ i ki , sin 2φ i ki sin ξ i ki , cos 2φ i ki ). The projective measurement are generated by random sampling the angles {ξ i ki , φ i ki } according to the Haar measure [36], namely, ξ i ki is taken from uniform distribution on the intervals 0, 2π), while φ i ki = arcsin( ω i ki ) and ω i ki is distributed uniformly on 0, 1). erated independently for each observer and measurement number k i . To measure any correlation coefficient E 1 k1 · · · E N k N the six wave plates in the projection part of the setup are adjusted accordingly to the angles {ξ i ki , φ i ki }. For each sets of angles, the coincidence counts on the two detectors are measured for a fixed amount of time and then, the maximal value of I (3) opt family is computed, taking into account all permutation of parties, inputs and outputs as detailed above. The value of I (3) opt is determined with precision ±0.015. Dividing the number of detected violation of local realism by n, the nonlocal fraction is estimated. Similarly, the average nonlocality strength is estimated.
The experimental results are collected in Table VI. As we see, our measurements are in good agreement with theoretical predictions for v = 0.96. It proves strong nonlocal properties of the GHZ state. Since the nonlocal fraction for the scenario 2 × 2 × 2, P V = 56 ± 5, is clearly greater than 2(π − 3) ≈ 28.319 the experiment revealed genuine three gubit entanglement of the GHZ state. Furthermore, to present more details about our experimental results, the histograms of the nonlocality strength have been studied. The experimental distributions are presented in Fig. 9, where each point denotes g expt (S) for the nonlocality strength in the interval (S, S +0.005). For all analyzed m 1 ×m 2 ×m 3 scenario, the function g expt (S) has a similar shape as its theoretical counterpart, despite reaching the smallest maximal value. A potential explanation could be the fact that white noise is only a first approximation of the imperfections occurring in the experimental setup. Furthermore, the wave plates (6 in total) are subject to experimental imperfections, especially measurable when using simultaneously all of them at completely random settings. This explains the slight discrepancy between theoretical predication and the experiment when measuring the g(S) quantity. Note that quantum tomography seams more robust again this sort of problems because of the limited set of wave plates settings used to obtain it.

V. CONCLUSIONS
In this paper we investigated the nonlocal fraction and the nonlocality strength as two important quanti-ties characterizing nonlocal correlations of the quantum states. Most of the conclusions were presented in the previous sections. Here we want to stress that the overall message of the obtained results is that both quantities can be accurately estimated using a greatly simplified model of nonlocality based only on the violation of one class of tight Bell inequality, namely I opt expressed by the correlation coefficients makes these inequalities of paramount importance for practical experimental investigation of all problems discussed in this paper. In particular, the nonlocal fraction can be used as a witness of genuine multipartite entanglement without having the distant parties share a common reference frames. In contrast to the witnesses based on the MABK and WWZB inequalities, our procedure provides a significant increase in entanglement-detection efficiency.
On the other hand, our results shade a new lights on the geometry of the quantum correlations, showing that statistically the most relevant one involve effectively two measurements settings per party. Interestingly, the inequality I (N ) opt provides also a paradoxical result. It amounts to the observation that the products of k qubit GHZ states and (N −k) pure single qubit states are more nonclassical than the N qubit GHZ state, if we employ the robustness of correlations against white-noise admixture as a measure of nonclassicality [31]. Although the resistance against noise is not a good quantifier of nonclassicality [15], our results clearly shows that the I (N ) opt has another important meaning.

VI. ACKNOWLEDGEMENTS
Below we present the set of states for which statistical properties of the nonlocality strength have been analyzed

Appendix B: Three-qubit Bell inequalities
Below we present the the most relevant three-qubit Bell inequalities with respect to the nonlocal fraction.