Geometric extension of Clauser-Horne inequality to more qubits

We propose a geometric multiparty extension of Clauser-Horne (CH) inequality. The standard CH inequality can be shown to be an implication of the fact that statistical separation between two events, $A$ and $B$, defined as $P(A\oplus B)$, where $A\oplus B=(A-B)\cup(B-A)$, satisfies the axioms of a distance. Our extension for tripartite case is based on triangle inequalities for the statistical separations of three probabilistic events $P(A\oplus B \oplus C)$. We show that Mermin inequality can be retrieved from our extended CH inequality for three subsystems. With our tripartite CH inequality, we investigate quantum violations by GHZ-type and W-type states. Our inequalities are compared to another type, so-called $N$-site CH inequality. In addition we argue how to generalize our method for more subsystems and measurement settings. Our method can be used to write down several Bell-type inequalities in a systematic manner.


I. INTRODUCTION
Intrinsic randomness of quantum mechanics has been a topic of a debate for many years. In 1935, Einstein, Podolsky, and Rosen (EPR) claimed that quantum mechanics is an incomplete theory [1], and hidden variables could be implemented to resolve the issue. In his pioneering work [2] Bell formulated an inequality that is satisfied by local hidden variable models, but can be violated by quantum mechanics for bipartite and two-level systems. Bell-theorem significantly improved our understanding of quantum intrinsic randomness with respect to the assumptions of reality and locality in local hidden variable models. Further studies have been done by considering more complicated systems. For examples, there have been Bell inequalities proposed for many qubits with two dichotomic observables per site [3][4][5][6][7] and for more than two alternative observables [8,9]. For higher dimensional systems, different forms of Bell inequalities have been introduced [10][11][12][13][14][15][16][17][18][19][20][21]. Other derivations of Bell inequalities can be found also in Ref. [22].
Experimental tests to invalidate Bell inequalities faced challenges such as detection and locality loopholes [22]. In particular, the detection loophole had to be dealt with by the use with the additional fair-sampling assumption, i.e., that the detected events give a fair representation of the entire ensemble [23,24]. In 1974, Clauser and Horne proposed another type of Bell inequality [24], which is very handy in dealing with such problems like detection inefficiency (see e.g the analysis by Eberhard [25,26]), and can be thought of as the most elementary Bell inequality. Some attempts to generalize the inequality to multipartite systems can be found in Refs. [27,28]. We shall continue here this effort, however our generalization will be based on different observations concerning the original CH inequality and therefore will take a dif- * Electronic address: hyoung@hanyang.ac.kr ferent form. Our extension is based on a geometric interpretation of Bell inequalities in Kolmogorov theory of probability as given by [33], further developed in e.g. Refs. [32,34].
An experimental falsification of Bell's inequality without the fair-sampling assumption was recently presented in Ref. [29]. Recently, a loophole-free violation of Bell inequality was reported by Hensen et al. [30], even though many attempts were previously made to close the two loopholes simultaneously (see Ref. [31] for important results). The possibility of having loophole free realizations of Bell experiments, opens the way for constructing device independent quantum information schemes and protocols, which would be fully secure. However, thus far we do not have a loophole free Bell experiment for three or more subsystems, and therefore one must continue research toward finding optimal approaches in this realm.
The essential traits of our generalization are as follows: • We first note that at the times in which CHSH and CH inequalities were first formulated, they looked as ad-hoc ones. Their starting points were certain, indeed ad-hoc, algebraic identities. Still as the CHSH-Bell inequality is implied by the CH one, the latter one seemed from the very beginning to be more fundamental. Much later, see ref. [33], it turned out that the CH inequality is derivable using a geometric notion of separation of two probabilistic events. The separation for two events A and B reads S(A, B) = P (A)+P (B)−2P (A, B). It has all properties of a distance, this includes most importantly the triangle inequality. The triangle inequality cannot be used to derive Bell inequalities, still suitable chaining of two triangle inequalities leads to a quadrangle one which, after it is re-written in terms of probabilities is the Clauser-Horne inequality. This geometric feature, we think, underlines the fundamental nature of the CH inequality, and singles it out. Therefore we postulate that three (or more) Bell inequalities which are direct generaliza-tions of the CH one should be derivable also using the geometric properties of separations of three (or more) probabilistic events.
• The statistical separation between probabilistic events, A and B, can also be put as P (A⊕B), where . We find an extension of CH inequality for a tripartite system, by an extension of the separation measure to P (A ⊕ B ⊕ C). Most importantly we have A⊕(B ⊕C) = (A⊕B)⊕ C = A ⊕ (C ⊕ B). The symmetry property of the two event separation, S(A, B) = S(B, A), plays no obvious role in the derivation of the CH inequality. However its extension, shown above in the three party scenario, is essential to derive the initial triangle inequalities, with the use of which one can derive our generalization of the CH inequality for a tripartite system.
• The inequality can used to derive the Mermin inequality for three qubits [3], and of course, further on to the CHSH inequality of two qubits in Ref. [24,35]. So the generalization has an additional trait similar to the CH inequality. We show also that it can be reduced to a bipartite CH inequality if one party is eliminated.
• The inequality is tight, just as the CH one.
• The method can be further extended to more than three parties, however in this case non-trivial generalizations (along the lines: a polygon inequality for separations) of the CH inequality, in the case in which the parties play fully symmetric roles, involve more than two settings for each party.
The paper is organized as follows. We derive a geometric extension of CH inequality for three subsystems by introducing statistical separation of probabilistic events in Sec. II. There we also discuss the properties of the inequality such as it being tight, and reducible to a bipartite CH inequality and a tripartite Mermin inequality. In Sec. III, quantum violations of our inequality and Nsite CH inequality f Ref. [27] are discussed in terms of the degree of noise robustness and critical efficiency of detection. There is region of parameters in which our approach gives quantum violations which are more resistant to the imperfections. We also demonstrate the generality of our method by extending the inequality for a three-party system with two measurement settings per cite to a three-party system with three measurement settings per site and a four-party system with three measurement settings per site (Sec. IV). We summarize our results in Sec. V.

II. DERIVATION
A. Geometric approach with statistical separations In our derivation of CH-type Bell inequalities, statistical separation plays a crucial role. It is defined by the probability of symmetric difference between two events [36]. Let (Ω, F, P ) be a probability space, where Ω is a sample space, F an event space, and P a probability measure. The symmetric difference of two events X and Y is defined by For three events we have, most importantly, the permutation symmetry, that is X ⊕(Y ⊕Z) = Y ⊕(Z ⊕X) = Z ⊕(X ⊕Y ). The generalization to more events is inductive. As events X ∩ Y c and X c ∩ Y are mutually exclusive one has The probability P (X ⊕ Y ) of a symmetric difference is often called statistical separation between events X and Y. It was originally put as where Statistical separation has a geometric interpretation as it obeys the triangle inequality, Note that for events X, Y, and Z one has (X ⊕ Y ) ⊕ (Y ⊕ Z) = X ⊕ Z. The inequality (2) in terms of probabilities reads Let us consider probability P (X ⊕ Y ⊕ Z) of the measure of the symmetric difference of three events, X, Y, and Z. It because of the aforementioned permutation symmetry is a statistical separation between X and Y ⊕ Z, or between Y and Z ⊕ X, or finally between Z and X ⊕ Y. We define it as a statistical separation of three events X, Y, and Z. One can show that where P (X, Y, Z) ≡ P (X ∩ Y ∩ Z), and so on. Such considerations can be extended to more events that three. In the following subsection, we analyze statistical separations of certain combinations of symmetric differences of probabilistic events for a specific type of three-qubit experiments.

B. A geometric tripartite extension of CH inequality
Here we derive a Bell-type inequality for statistical separations for tripartite systems. We consider a scenario of three qubits. The qubits are measured by three observers (Alice, Bob, and Charlie). Each partner choose between two measurement settings. Events associated with Alice's (Bob's and Charlie's) choice of measurement settings i are denoted by A i (B i and C i ), for i = 1, 2. For example, event A i for Alice represents the fact that Alice chooses a measurement related with a projectorÂ i and detects her qubit. We define the events B j and C k for Bob and Charlie, respectively, in a similar way. The respective outcomes, denoted as a i b j and c k ,, mean 'detection' if their value is 1 and 'no detection' is 0, i.e. a i , b j , c k = 0, 1.
In local realistic probabilistic model, the outcomes build elements of the sample space for the considered sce- Note that is such a treatment, we avoid introduction of other hidden variables. Only the possible full sets of hidden results suffice. Events E are subsets of the sample space, E ⊆ Ω, and their probabilities are denoted by P (E). For instance, P (A 1 ) is the probability of event that is, the detection of Alice's qubit with the choice of projectorÂ 1 . Joint probabilities of two events A i and B j , that Alice's and Bob's qubits are all detected for local settings i and j, are denoted by P (A i , B j ). This applies to the other pairs, Bob and Charlie, and Charlie and Alice. Similarly, detections of all qubits by the observers (Alice, Bob, and Charlie) in settings (i, j, and k, respectively) are associated with joint probability P (A i , B j , C k ) of the three detection events A i and B j and C k .
Note that the following triangle inequality holds In the above the additional inside brackets we introduced to point that we in fact deal here with the original triangle inequality for separation of pairs of events (2). However they can be dropped, as we have the symmetries mentioned earlier. The second triangle inequality which we need is and it is written in a similar convention. Note that both inequalities involve events which form the quantum point of view are inaccessible experimentally, if the projectors associated with A 1 and A 2 (also B 1 and B 2 ) do not commute.
With the two triangle inequalities we get an inequality which applies only to operationally accessible situations This is the geometric tripartite extension of CH inequality.
To get a form of it which is expressed in terms of probabilities of events and their coincidences, we expand this inequality by using Eq. (1), to get where L is the sum of local (single-site) probabilities, Q XY and T are certain combinations of pair-and triplesite joint probabilities, respectively. More explicitly, The tripartite inequality (8) is reduced to a bipartite CH inequality if one party is eliminated. Assume that Charlie has no events of detections in his measurements, i.e. all probabilities of events C 1,2 vanish. Then, In other words, the tripartite inequality in Eq. (8) becomes a bipartite CH inequality, if the third party does not measure anything, i.e. its events are represented in the sample space by empty sets [24]. It is worth to mention that the left hand side in Eq. (8) is upper bounded by 1.
We may formulate the inequality (8) in a form of Eberhard inequality for 3 qubits, as shown in Appendix C 2. Eberhard inequality [25] includes explicitly non-detection events. Its derivation from CH inequality is presented in Appendix C 1, showing the algebraic equivalence between the two inequalities [26].

C. Mermin inequality
Our method can be applied to deriving Mermin inequality, just like CH ones lead to CHSH ones. Let us consider an experiment where each of three observers possesses a two-channel analyzer and two detectors, each with two outcomes (±1). Suppose the experiment is described by a local hidden variable model with sample In this scenario we assume that every subsystem is detected with one of two outcomes ±1, contrary to the previous subsections. Event A 1 is defined by the detection with outcome +1 of Alice's qubit for a projectorÂ 1 chosen and event A c 1 by the one with outcome −1. That is, events Similarly, other events are defined. To this end, we obtain an inequality, as in Eq. (7), by applying the procedure similar to Sec. II B. The inequality, in the form of Eq. (7), is then Mermin inequality. To show this, we construct correlation functions in terms of probabilities. Note first i.e. a normalization condition. The correlation functions are given by where α i , β j , and γ k parameterize the local settings, respectively. Note from Eq. (4) that P (A ⊕ B ⊕ C) contains the joint probabilities of even numbers of outcomes with −1, while P (A ⊕ B ⊕ C c ) does the ones of odd numbers of outcomes with −1, so that E αiβj γ k are the same correlation functions as those in Ref. [3]. Applying the normalization condition, we obtain (7), we obtain Mermin inequality [3], Here, we have the lower bound of Mermin inequalities. The upper bound 2, can be obtained in a similar way.

D. N -site CH inequality
In this subsection we briefly discuss one of the multipartite extensions of CH-inequality, so-called a N -site CH inequality that Larsson and Semitecolos proposed [27]. For three qubits with N = 3, the 3-site CH inequality reads where Q 1 is the sum of a specific set of pair-site probabilities, and T 1 and T 2 are of triple-site probabilities. More explicitly, They proposed the N -site CH inequality to find the minimum detection efficiency required for quantum mechanics to violate the inequality. In later sections, we will discuss the results related to the 3-site CH inequality and compare them to our geometric tripartite extension of CH inequality.

E. Tightness of geometric tripartite extension of CH inequality
We show that our geometric tripartite extension of CH inequality is tight, i.e. it is a facet inequality of a local realistic polytope so that it can discriminate sharply the domains of the local realistic and quantum correlations [22]. For the purpose we consider the local realistic polytope with the experimental scenario in which the inequality is derived. The local realistic polytope, called a Bell polytope B, is a collection of probability vectors , and the others similarly. Here, the deterministic probabilities P v (X) are either 0 or 1 for X = A, B, C. Then, the vertices p i have entries 0 and 1 (see Appendix. B). In other words, the set of deterministic configurations are the vertices of Bell polytope [38] and their number is 2 6 .
Every facet inequality of the Bell polytope B is given in a form of where the lower bound C 0 is a real number and vector C ∈ B is the normal vector to the facet hyperplane. The equality holds for the facet. The facet is identified by d independent vertices p i such that C · p i = C 0 , if C 0 = 0. If C 0 = 0, on the other hand, the necessary number of linearly independent vertices drops down to d − 1, as the null vertex trivially satisfies the equality with all components being zero. We test the linear independence of the vertices by the matrix rank, once the matrix is composed of row vectors with the vertices [39]. The tripartite inequality (8) can be cast in the form of Eq. (10) with the lower bound C 0 = 0, as shown in Appendix. B. The equality to the lower bound is satisfied by 2 5 vertices. Among them we find d − 1 (= 25) linearly independent vertices (see Table III in Appendix. B). Reminded that the dimension of Bell polytope B is d = 26, these imply that geometric tripartite extension of CH inequality (8) is a facet inequality of the Bell polytope so that it is tight. With the same function as in the inequality (8), we obtain another inequality which is upper bounded by 1, i.e. C · P ≤ 1. Among 2 5 vertices which satisfy the equality to the upper bound, d (= 26) vertices are found to be linearly independent, implying that this is also a facet inequality of the Bell polytope (see Table IV in Appendix. B). We get the two tight tripartite inequalities with the given function. This fact is similar to the case of CH inequality, which defines two facets with a single function [22]. On the other hand, the 3-site CH inequality (9) defines a single facet.

III. QUANTUM VIOLATION
Quantum mechanics violates the geometric tripartite extension of CH inequality (8) and 3-site CH inequalities (9) due to the combined effects entanglement of states and measurement incompatibility of observables. Here we illustrate the quantum violations of both the inequalities for GHZ-type (12) and W-type entangled states (13). Quantum violations are characterized by robustness of violations and a critical detection efficiency, below which no quantum violation is obtained.
A three qubit mixed state can be written aŝ where |ψ is a pure three-qubit state and v is a parameter, called a visibility, with 0 ≤ v ≤ 1. We consider two classes of three-qubit pure states, e.g, GHZ state [40], and W state, |GHZ(α) and |W (θ, φ) are maximally entangled for α = π 4 , and θ = arccos( 1 √ 3 ), φ = π 4 , respectively. Three qubits are measured by spatially separated observers. Each of them locally measures with an analyzer, which is oriented alternatively in two different directions. The measurements are represented by projectors, where σ = (σ x ,σ y ,σ z ) is the vector of Pauli operatorsσ i , and Bloch unit vector b X i stands for i-th setting of the analyzer at site X.
When considering Bell inequality in terms of symmetric differences, as in Eq. (7), it is convenient to define their positive operators, each bŷ where 1 1 X is an identity operator at site X,Π X a projector as in Eq. (14), and '⊗' is the tensor product. The symmetric differences X ⊕Y in quantum theory are nominal and their expansions are inapplicable to the set operations. Eq. (15) can be extended to more sites recur- Then, those for the pair-and triple-site joint probabilities are given aŝ where b X are Bloch vectors of the settings at site X = A, B, C. Here we assume the perfect detection.
The quantum joint probability of triple sites, that all the three qubits are detected for given settings, is given by whereρ v is a quantum state, as in Eq. (11). Singleand pair-site probabilities are obtained similarly. Here, the detectors together with the analyzers are assumed to work perfectly with unit efficiency η = 1. In realistic situations, however, detectors can be imperfect with less efficiencies than the perfect detectors. Then the quantum probabilities are functions of efficiencies, where are quantum probabilities with perfect detections, as in Eq. (18). Here we assume that the detectors have the same efficiency η with 0 ≤ η ≤ 1. Also, projectors in Eqs. (15) and (16) are replaced by ηΠ X .
We check the violations of the inequality (8) by replacing Kolmogorov probabilities with the quantum ones, as in Eqs. (19). We characterize quantum violations in terms of experimental imperfections. In particular, two types of imperfections are accounted in measurements and states, i.e., the loss of particles in terms of detection efficiency η, as in Eq. (19), and the depolarization by environment in terms of 1 − v, as in Eq. (11). Quantum violations can be achieved only if v > v crit . We find the critical visibility v crit for a given pure state |ψ and detection efficiency η = 1, as the local measurement settings are optimized. For, a given pure state, we define the robustness of violation against the white noise as It is also considered as a violation degree (or the maximal fraction of white noise for which the violations are not found). For the given pure state |ψ , we decrease the detection efficiency down to η ≤ η crit , such that no quantum violation can be observed even with the maximal visibility v = 1. We also find the minimum of the critical detection efficiencies η min crit over all possible pure states. η min crit is obtained by looking for the lowest eigen value of Bell operator associated to geometric tripartite extension of CH inequality (8). Similar kind of analysis was done for CH inequality in Ref. [25].

A. Critical visibilities and detection efficiencies
The Bell operatorB corresponding to the inequality (8) is written aŝ whereL is the sum of local (single-site) projectors,Q XY is a combination of pair-site projectors for subsystems X and Y , andT is a combination of triple-site projectors. Their explicit forms, including the detection efficiency η, are given bŷ The quantum violation of the inequality (8) is witnessed when B ≡ Tr(ρB) < 0. Let |ψ be a given pure state andρ v be the mixed state, as in Eq. (11). The critical visibility v crit is defined by v, such that where B v is minimized over local measurements settings. Here, η is a given detection efficiency.

Critical visibility for GHZ states
At first, we analyze the violations of the inequality (8) by GHZ states (12). Assuming detectors with perfect efficiency η = 1, we find the critical visibility v crit from Eq. (23). Here, B v is found to be minimized, if the local settings b in Eq. (18) are chosen along the directions in the x-y plane, where θ X i are the angles. For a GHZ state (12) the critical visibility reads v crit = min where Then, the critical visibility v crit is given as From this, we obtain the violation robustness n in Eq. (20), In Fig. 1, we present the violation robustness n as a function of the entanglement degree S A , where S A is the von-Neunmann entropy of the reduced density operatorρ A for Alice's qubit, i.e.

Critical visibility for W states
Assuming perfect detection efficiency η = 1, we check the critical visibility associated to W state (13) and the geometric tripartite extension of CH inequality (8), as done for the GHZ state. For W states, the local settings b, in Eq. (18), lie in the x-z plane, where φ X i are angles. Similarly to Eqs. (25) and (26), the critical visibility v crit = min where (13), M W ≈ 3.0459 [28] so that v crit ≈ 0.6566 and n ≈ 0.3434.
The maximum of robustness for the maximally entangled W state is less than (n = 0.5) of the maximally entangled GHZ state but larger than (n = 1 − 1/ √ 2) of the maximally entangled EPR state. For other W states, we perform numerical analyses.

Critical efficiency of detection
We find the critical efficiency of detection, η crit , above which a pure state |ψ violates the geometric tripartite extension of CH inequality (8). For the purpose we employ Bell operator in Eq. (21) and investigate its lowest eigenvalue. The critical detection efficiencies for GHZ and W states are to be found by restricting to the subspaces of GHZ and W. Their minima are compared to the critical detection efficiency η min crit to be found on the whole Hilbert space. Here, we shall obtain critical detection efficiencies by assuming particular sets of local settings and numerically confirm them without the assumption.
We consider Bell operatorB in Eq. (21), assuming that the three observers choose the same alternating local measurements with Bloch vectors b X i = b i for X = A, B, C. Furthermore b i = (cos φ i , 0, sin φ i ) lies in the x-z plane, where φ 1 = π/2 and φ 2 = π/2 − φ. One might release this constraint for more general settings and obtain the same result. LettingB =B(φ), the eigenvaluesB(φ) are determined by the roots of the characteristic polynomial χ(λ, φ) = det(λ1 1 −B(φ)). Let the roots λ r be functions of φ, λ r = λ r (φ). If φ = 0, all the settings are equal to each other and they are compatible, revealing no violation. However, for small φ > 0, one finds the lowest eigenvalue ofB, that is negative. In the case we apply a perturbation theory by expanding χ(λ, φ) and its roots λ r (φ) in powers of φ upto 4th orders. The approximate equation for the roots is given as where the coefficients λ r,n come from the expansion of λ r (φ) ≈ 4 n=0 λ r,n φ n . By solving the equations q n ({λ r,m }) = 0 for all n, we obtain λ r,n and thus the approximate roots λ r (φ).
We first find the approximate roots on the subspace, spanned by {|001 , |010 , |100 , |111 }, which includes W states and EPR states such as |1 ⊗ (c 00 |00 + c 11 |11 ) . The subspace can be called as W-EPR. Note that our tripartite inequality is reduced to a bipartite CH inequality, and with the latter the critical detection efficiency η crit → 2/3 in the limit of product states, c 11 → 1. On the W-EPR subspace, the lowest approximate root (i.e., eigenvalue) is given by Thus, on the W-EPR subspace, the critical detection efficiency The eigenvector with the lowest eigenvalue is of the form where r = r 2 1 + r 2 2 + r 2 3 . The critical detection efciency η crit = 2/3 remains unchanged for states with any r 1,2,3 being non-zero. In other words, η crit = 2/3 for W states as well as for EPR states. We apply the similar method to the whole Hilbert space and find the same value of critical detection efficiency, η min crit = 2/3. This is equal to that with a bipartite CH-inequality [25].
Some GHZ states do not violate the geometric tripartite extension of CH inequality inequality not apply the previous method. Instead we directly diagonalize Bell operatorB on the subspace of GHZ states, spanned by {|000 , |111 }. Here we assume the settings b i = (cos θ i , sin θ i , 0), where θ 1 = θ and θ 2 = π/2. The settings were chosen in Eq. (24) to find the critical visibility for the GHZ states. One might release the constraint for more general settings and obtain the same result. The lowest eigenvalue of Bell operatorB is given by It follows from Eq. (32) that on the GHZ subspace, the critical detection efficiency for which θ = 0. Using Garg-Mermin approach [45] for Mermin inequality and GHZ state, similar bound for η crit is obtained in Ref. [46]. The respective eigenvector is a maximally entangled GHZ state, |GHZ( π 4 ) = (|000 + |111 ) / √ 2. The minimum critical detection efficiency for the GHZ states is 0.7913, which is larger than η min crit = 2 3 , for W-EPR states.  (9), respectively. The inequality (8) provides larger robustness to white noise than 3-site CH inequality, even though there is a crossover at low η. To the contrary, the critical detection efficiency, the minimum η to obtain the violation is larger in case of the inequality (8). , violation robustness n as a function of detection efficiency η for the geometric tripartite extension of CH inequality (8)* and 3-site CH inequality (9), respectively. The violation robustness is larger in the inequality (8) than 3-site CH inequality over all η. The critical detection efficiency is smaller in the geometric tripartite extension of CH inequality, contrary to the GHZ state in Fig. 2.
We present our numerical results for the geometric tripartite extension of CH inequality (8) and 3-site CH inequality (9). In Fig. 2, we plot the robustness of violation n as a function of detection efficiency η for the maximally entangled GHZ state with the geometric tripartite extension of CH inequality and 3-site CH inequalities. The violation robustness n is a monotonically increasing function of detection efficiency η for the maximally entangled GHZ state, while it vanishes if η ≤ η crit . It is shown that the inequality (8) provides larger robustness of violation than 3-site CH inequality, even though there is a crossover at low η. To the contrary, the critical detection efficiency, the minimum η to see the violation is larger in the geometric tripartite extension of CH inequality in-equality. Fig. 3 analyses robustness of violations for the maximally entangled W state. From Fig. 3 we see that for the maximally entangled W state, the violation robustness is larger in the inequality (8) than 3-site CH inequality over all values of η between 0 and 1. The critical detection efficiency is smaller in the geometric tripartite extension of CH inequality, contrary to the GHZ state in Fig. 2.
Comparing the two states with the geometric tripartite extension of CH inequality in Figs. 2 and 3, it is remarkable that the maximally entangled GHZ state is more robust against white noise than the maximally entangled W state over all values of detection efficiency η between 0 and 1. In particular, the critical detection efficiency η crit = ( √ 21 − 3)/2 ≈ 0.7913 for the GHZ state, slightly smaller than η crit ≈ 0.8090 for the W state. With the 3-site CH inequality, to the contrary, the maximally entangled GHZ state is less robust than that of W state.   For a set of GHZ states (12), and both types of inequalties the violation robustnesses n, critical visibilities v crit , and critical detection efficiencies η crit are presented in Table I. The analyses are done for α = 2π/18, 3π/18, and 4π/18, for which the entanglement degree S A ≈ 0.52, 0.81, and 0.98, respectively. For η = 1, the robustnesses of violations n increase for the geometric tripartite extension of CH inequality and 3-site CH inequality as we increase the degree of entanglement S A (or α). The violations of the inequality (8) have the larger robustnesses than the 3-site CH for all values of α. However, critical detection efficiency η crit decreases in case of the ineaquality (8), whereas it increases in the 3-site CH inequality. The maximum of critical detection efficiencies of 3-site CH, η max crit ≈ 0.7631, is smaller than the minimum of detection efficiencies of the geometric tripartite extension of CH inequality, η min crit ≈ 0.7913. The minimum of critical detection efficiencies η min crit associated to 3-site CH inequality is estimated to be 0.6 in the GHZ subspace [43]. Table II presents the results for a set of W states (13). For η = 1, the robustnesses of the violations of the geometric tripartite extension of CH inequality n are larger than those of the 3-site CH inequality. Furthermore, critical detection efficiencies η crit for the inequality (8) are lower than those associated to 3-site CH inequality. For the biseparable EPR state (θ = π 2 and φ = π 4 , or specifically, |0 ⊗ (|01 + |10 ) / √ 2) and the geometric tripartite extension of CH inequality inequality, the critical detection efficiency η crit = 2/(1 + √ 2) ≈ 0.8284, which coincides with that of the critical detection efficiency associated to CH inequality and the EPR state [24,25,42]. The value of η crit for 3-site CH inequality and the biseparable EPR state is obtained as 0.8724, which is greater than the critical detection efficiency obtained for the geometric tripartite extension of CH inequality.

IV. EXTENSION OF CH INEQUALITY FOR MORE MEASUREMENT SETTINGS THAN TWO PER SITE AND MORE SUBSYSTEMS THAN THREE
First, we present an extension of CH inequality for three subsystems and three measurement settings per site. Then, following the similar method we obtain a CH inequality for four qubits with three measurement settings per site.
Let's consider a set of events A l , B m , C n for Alice, Bob, and Charlie, respectively, where l, m, n = 1, 2, 3. For detailed descriptions, the reader is referred to subsection II B. As discussed in case of three subsystems and two measurement settings per site, we can find similar kind of relations between symmetric differences of three events for the situation at hand. For example, for a pair of statistical separations, the following inequality holds: Also, due to permutation symmetry, Using the fact, we can have another triangle inequality Following the similar argument, we obtain and finally, With the four inequalities: (33), (34), (35), and (36), a CH inequality for three qubits and three settings per site reads It is worth to mention that we can have several invariant forms of the inequality (37) by choosing different sets of symmetric differences associated with statistical separations.
Let's consider a set of events A l , B m , C n , D p for Alice, Bob, Charlie and Daniel, respectively, where l, m, n, p = 1, 2, 3. From earlier discussions we know that the statistical separations obey triangle inequalities. Thus following set of inequalities can be obtained by chaining using permutation symmetry of symmetric difference, we also have next, and finally, From these four inequalities (38), (39), (40), and (41) we obtain (42) Quantum mechanics violates the inequalities (37), and (42). It is interesting to note that because the basic formalism leading to the inequalities (7), (37), and (42) is same, in principle, we may obtain several CH inequalities for arbitrary number of subsystems with many measurement settings per site. It will be discussed elsewhere.

V. REMARKS
We propose a geometric method to construct a set of CH inequalities in Kolmogorov theory of probability. We show that the inequalities results from a set of triangle inequalities in terms of the statistical separations, defined by the probability of symmetric differences between local detection events. The method can be generalized to obtain inequalities for more qubits than three and more measurement settings than two per site. Also, a new set of inequalities, which lead to Mermin inequalities, are presented (see Appendix. D) upto five subsystems. It is also shown that our inequality (8) can be reduced to the CH inequality of two subsystems and the Mermin inequality of 3 qubits under the appropriate conditions.
The inequalities are characterized in terms of two parameters, e.g., detection efficiency and robustness of quantum violation against white-noise. The later is employed as a degree of violation. For the three qubits, we show that the geometric tripartite extension of CH inequality can be quantum-mechanically violated by the pure entangled states of GHZ and W types, even though no violation is found for some partially entangled GHZ states. In terms of the violation robustness and critical efficiency of detection, we compare our inequality to the 3-site CH inequality (9), another generalization of CH inequalities for tripartite systems. Our inequality (8) shows violation robustness stronger than the 3-site CH inequality by both types of GHZ and W states with the perfect detections. We find that, for the given GHZ states, the 3-site CH inequality has the critical efficiencies lower than the geometric tripartite extension of CH inequality, whereas it is opposite for the given W states. In particular, the minimal critical efficiency of detection over all the states is lower in the 3-site than the the geometric tripartite extension of CH inequality. It is worth to mention that the authors in Ref. [27] did not account background noises, even though they conjectured that the inclusion of back ground noise would increase the bounds on the minimum detection efficiency. We find that bi-separable states can violate the inequality (8), as expected by the fact that two qubit CH inequality can be retrieved from the geometric tripartite extension of CH inequality (8) and violated by two-qubit pure entangled states.
The quantum violations of the geometric tripartite extension of CH inequality (8) are expected to experimentally realize with detectors, which differentiate singleand multi-photon detections with high efficiency and negligible dark count [44]. It is conjectured that the method to extend CH inequality can be further developed for arbitrary number of outcomes. This will be presented in a forthcoming work. [a] ≡ a mod 2. (A1) Two residues [a] and [b] satisfy triangle inequality, Also one may have an identity, Thus, one obtains the relation, Using Eqs. (4) and (A3), we have Using triangle inequality (A2), we also obtain Summing (A4) and (A5) we get Combining Eq. (A6) and Eq. (7), we obtain Or equivalently, in the form of Eq. (8), Thus we have two bounds, 0 for the lower bound and 1 for the upper bound.
The inequality (8) is a facet inequality as shown in the main text and the facet is specified by the set of linearly independent vertices. We present them in Table III and also those of another facet inequality with the upper bound 1 in Table IV.
Appendix C: Explicit inclusion of non-detection events

Clauser-Horne and Eberhard inequalities
Assume each observer gets 3 outcomes {+, −, u}, where 'u' is a undetected outcome. CH inequality is given by are probabilities of Alice's detecting the outcome '+' and P (A + , B + ) are joint probabilities of both detecting outcomes '+.' We may expand the singlesite probabilities in a form of the pair-site probabilities, CH inequality in Eq. (C1) then becomes This version of CH inequality is equivalent to Eberhard inquality [26].
Consider all the possible data of N events in each pair of local settings, assuming that arbitrarily large N is the same for the 4 pairs of local settings. Then, the CH inequality in terms of probabilities can be changed to the one with the coincident counts, , the number of coincident counts at both sites with outcome pair (a, b) for a given setting pair (i, j). Note that we also account the number of counts at one site and no count at the other . This is an Eberhard inequality.

Eberhard inequality for 3 qubits
We derive a 3-qubit Eberhard inequality from our inequality (8), based on the method in Sec. C 1. The inequality in Eq. (8) is recalled, . Similarly to Eq. (C1), we expand the local-and pairsite probabilities in a form of the triple-site. The singlesite term L is rewritten by where (X, Y, Z) is one of cyclic permutations of (A, B, C). The pair-site terms Q XY are rewritten as . Here we take Z k for each (X i , Y k ) such that k = 1 if i = j = 1 or i = j = 2, and k = 2 if i = 1 and j = 2 or  We replace P abc ijk with the coincident counts N abc ijk = N P abc ijk and obtain a 3-qubit Eberhard inequality. inequalities which can be reduced to Mermin inequalities for subsystems more than three In this subsection, we propose a method to derive an inequality for four qubits and also show that the extension to five qubits is also possible. Our derivation is based on the following identity, where X ij are events in the event space F (see Sec. II A), Y i4 = ⊕ 3 j=1 X ij , and Z i4 = (X i1 ∩ X i2 ) ⊕ (X i1 ∩ X i3 ) ⊕ (X i2 ∩ X i3 ). This applies to i = 1, 2, 3. Thus we have a set of 3 equations. In addition, we consider the 4-th equation, . We sum the 4 equations in Eqs. (D1) and (D2). Collecting the left hand sides together and similarly the right hand sides, we obtain a single equation.
Here, we have used Z 44 = ⊕ 3 i=1 Z i4 and Eq. (D3). In addition one can apply the procedure in Sec. II C to derive Mermin inequalities for four and five qubits by using inequality (D4).
We remark that the inequality presented in the section is derived by a vector of events X i for two and three qubits, and the one by a matrix of events X ij for four [Eq. (D4)] and five qubits. It is conjectured that one needs an n-th rank tensor of events X i1i2···in for the specific type of inequalities of 2n and 2n + 1 qubits. This is beyond the scope of this work.