A Reformulation of the MABK inequalities

Generating nonclassical correlations without fully aligning measurements

Abstract

We investigate the scenario where spatially separated parties perform measurements in randomly chosen bases on an -partite Greenberger-Horne-Zeilinger state. We show that without any alignment of the measurements, the observers will obtain correlations that violate a Bell inequality with a probability that rapidly approaches 1 as increases and that this probability is robust against noise. We also prove that restricting these randomly chosen measurements to a plane perpendicular to a common direction will always generate correlations that violate some Bell inequality. Specifically, if each observer chooses their two measurements to be locally orthogonal, then the observers will violate one of two Bell inequalities by an amount that increases exponentially with . These results are also robust against noise and perturbations of each observer’s reference direction from the common direction.

pacs:

I Introduction

Entangled quantum states can yield correlations between spatially separated measurements that are inconsistent with any locally causal theory (1); (2). These nonlocal (or nonclassical) correlations are a resource (3); (4) for a range of information processing tasks such as quantum key distribution (5); (6); (7); (8), teleportation (9); (10), certification and expansion of randomness (11); (12) and the reduction (13) of communication complexity (14).

These nonlocal correlations are correlations of measurement outcomes. As such, they are not solely a consequence of entanglement but also depend upon the choice of measurements.This point was emphasised by Bell in his seminal work (1), where he showed that the perfect correlations exhibited by the spin-singlet state do admit a simple locally causal model (see also Ref. (15) and references therein).

The demonstration of nonlocal correlations typically employs carefully chosen measurements whose implementation requires the spatially separated observers to share a complete reference frame (16); (17). To circumvent this requirement, observers who do not initially share a complete reference frame could share a particular state that is invariant under arbitrary rotations of the local reference frames (18); however, the state preparation involved is relatively complicated. Alternatively, they could use correlated quantum systems to establish a shared reference frame which can then be used to align measurements (19); however, this approach is resource-intensive as it requires coherently exchanging many entangled quantum systems.

Recently, it has been shown that such methods are not required to demonstrate violations of a Bell inequality (20). In particular, for spatially separated observers that share a Greenberger-Horne-Zeilinger (GHZ) state, it was found that most choices of measurements lead to nonlocal correlations between measurement outcomes (20). Therefore distant observers can randomly choose measurements that violate some Bell inequality with a probability that approaches 1 as increases. However, the successful detection of nonlocal correlations in this scenario requires checking the measurement statistics against a set of Bell inequalities that grows exponentially in .

In this paper, we show that for observers who share a GHZ state and are able to perfectly align a single measurement direction (but do not share a full reference frame), any choice of measurements satisfying a local constraint will generate nonlocal correlations. Furthermore, in contrast to the results of Ref. (20), verifying that these correlations are nonlocal only involves testing the measurement statistics against two Bell inequalities, thereby simplifying the verification process. Moreover, we show that as increases, the amount by which the observers violate one of the two Bell inequalities increases exponentially and, in the worst-case scenario, is a constant factor below the maximum violation permitted by quantum mechanics.

We also investigate the robustness of the above-mentioned results and those presented in Ref. (20) in the presence of uncorrelated local noise. We will demonstrate that the ability of observers to obtain correlations that violate some Bell inequality is robust against depolarizing or dephasing noise whether or not they share a direction. Finally, we show that even if the observers cannot perfectly align a measurement direction and so have randomly perturbed approximations to the common direction, they can still always obtain measurement statistics that violate one of two Bell inequalities.

This paper is structured as follows. We begin in Sec. II by illustrating these results in the simplest case, namely, when there are two observers who share a Bell state. We then outline the generalization to parties, discuss the relevant Bell inequalities and the methods of sampling random measurements that are used in this paper. In Sec. III, we consider the ideal case in which the parties share a state without any noise and can also share a reference direction perfectly. In Sec. IV.1, we relax the first assumption and show that the probability of violating a Bell inequality is robust against uncorrelated depolarizing or dephasing noise. In Sec. IV.2 we also show that the probability of violating a Bell inequality is robust against random perturbations in the shared direction. In Sec. V we discuss the implications of these results and offer some concluding remarks.

Ii Preliminaries

In this section, we outline the simplest example, namely, when two spatially separated parties each perform two binary-outcome measurements on the Bell state . For this case, we define the probability of violating a Bell inequality and review the results presented in Ref. (20) for randomly chosen measurements without any shared reference frame. We then show that if both observers perform locally orthogonal random measurements in the -plane of the Bloch sphere, they will always obtain correlations that violate a Clauser-Horne-Shimony-Holt (CHSH) Bell inequality (2); (21). We then outline the general scenario for parties and discuss the relevant Bell inequalities. We conclude this section by explaining the various samplings of measurement bases employed in this paper.

ii.1 A two-party example

For the two-party case, a verifier prepares many copies of the maximally entangled Bell state

(1)

where , are the computational basis states, and distributes one qubit to each of two observers. Both observers choose two measurement bases. For each copy of the Bell state, the observers each randomly choose and perform one of their two measurements on their qubit. The observers then send the verifier a list of the measurement choice (a binary digit ) and the corresponding outcome for each qubit . The verifier uses the lists to determine if the measurement outcomes are inconsistent with a locally causal theory. The verifier does this by calculating the probabilities (as relative frequencies) that the outcomes satisfy given a specific choice of and . They then determine the correlation functions

(2)

and seek to determine if the correlation functions are consistent with a locally causal model. For two parties, the correlation functions are inconsistent with a locally causal theory if they violate the standard CHSH (2); (21) Bell inequality

(3)

However, the choice of the labels for the measurements (i.e., which measurement is labelled by ) is arbitrary, as is the labeling of the measurement outcomes . Therefore the correlation functions are also inconsistent with any locally causal theory if they violate inequality (3) after any relabeling of the and/or and/or the label . We will follow the terminologies of Refs. (22); (23) and refer to two inequalities that can be obtained from one another through such relabeling as being equivalent.

There are four equivalent inequalities that can be obtained from Eq. (3) by mapping to , , or , namely,

(4)

All permutations of the and map Eq. (3) to one of the four inequalities in Eq. (II.1), so the four inequalities in Eq. (II.1) are the complete set of Bell inequalities equivalent to the standard CHSH inequality. This set is referred to as the class of CHSH Bell inequalities. For two parties, the correlation functions are consistent with a locally causal theory if and only if they satisfy all inequalities in the class of CHSH Bell inequalities (24).

To see that quantum mechanics predicts violations of the CHSH inequalities, first note that for a quantum state and observables , where is the vector of Pauli matrices and

(5)

the correlation functions are

(6)

For the Bell state defined in Eq. (1), Eq. (6) becomes

(7)

If, for example, the measurements correspond to the vectors

(8)

then the corresponding correlation functions are

(9)

Substituting these into Eq. (II.1), one finds that , i.e., the CHSH inequality is violated.

Bell inequality violations with no aligned directions

From Eqs. (II.1) and (7), it is clear that the choices of measurements that generate CHSH-inequality-violating correlations must satisfy some constraints, i.e., the directions that correspond to the measurements must be aligned in particular ways. However, if the observers do not share a reference frame, directions satisfying such constraints can only be chosen probabilistically. If the measurement directions are sampled according to the normalized measure , then the observers will choose measurements that generate correlations inconsistent with any locally causal theory with probability

(10)

where is a function that returns 1 if the orientations generate correlations that violate any of the CHSH Bell inequalities and 0 otherwise. The probability with which the observers generate correlations inconsistent with any locally causal theory depends on the way they choose their measurements, which in turn depends on how much they can align their reference frames. For example, if the observers can completely align their measurement bases, they can simply choose the measurements (II.1), so , i.e., they always generate correlations inconsistent with any locally causal theory.

When the observers cannot align their measurements at all and randomly choose both of their measurements independently and isotropically, then the probability that they will choose measurements that generate correlations violating one of the CHSH inequalities is  (20). However, if the observers choose their measurements to be locally orthogonal, i.e.,

(11)

then the probability of generating correlations that violate a CHSH inequality increases to  (20).

Bell inequality violations with one aligned direction

Consider another possible scenario, in which the observers can align one direction of their measurements, e.g., the direction of the Bloch sphere. Then each observer can choose two orthogonal measurements in the -plane, i.e., choose two angles and randomly according to a uniform distribution. The four corresponding measurements are

(12)

Substituting these into Eq. (7) and then Eq. (II.1) gives

(13)

Using standard trigonometric identities, we see that the CHSH inequalities in Eq. (II.1) are satisfied if and only if

(14)

where . But one of these inequalities is violated unless , so any choice of measurements (except for a set of measure zero) will violate one of two CHSH inequalities, i.e., . Therefore in order to choose measurements that generate correlations inconsistent with any locally causal theory, it is sufficient to perfectly align a single direction, namely the axis, and to check only two Bell inequalities.

ii.2 The general scenario

We now generalize the two-party case outlined in the previous section to parties and determine the extent to which the probability of generating correlations inconsistent with locally causal theories depends on the alignment of the measurements of the parties. To this end, we consider the scenario (abstracted from the physical implementation) wherein a verifier prepares a large number of copies of the -partite GHZ state (the GHZ state is chosen as it is a resource for obtaining maximum violations of some commonly used Bell inequalities (25); (26)),

(15)

where and denote states in which each of the qubits are prepared in the states and respectively. The verifier distributes 1 qubit from each copy to observers. As in the two-party case, each observer chooses two measurement bases, which corresponds to the observer choosing two directions , parametrized as in Eq. (5), in the Bloch sphere, where . Each observer measures their qubits, randomly choosing for each qubit. The observers then send a list of the measurement labels and outcomes for each copy back to the verifier, who will use the lists to determine if the measurement outcomes are inconsistent with a locally causal theory.

In contrast to the typical scenario where are fixed a priori to some optimal measurement bases that give the maximal violation of a specific labeling of a Bell inequality, we now consider a scenario where the measurement bases/directions are chosen randomly according to some distribution, but are fixed throughout the experiment. Formally, if we treat the measurement directions as random variables, we can define the probability, , that the verifier identifies that the correlation functions are incompatible with the class of Bell inequalities as

(16)

where d is the normalized measure associated with the sampling of measurement direction , and is a function that returns 1 if the measurements give rise to correlation functions that violate any of the Bell inequalities in the class and 0 otherwise.

Clearly, depends crucially on both the sampling of , which determines the probability of generating nonlocal correlations, and the class of Bell inequalities involved, which determines the probability of the verifier identifying nonlocal correlations as Bell-inequality-violating. For any given sampling of , is thus upper bounded by =, where refers to the complete set of Bell inequalities relevant to the scenario described above. The feasibility of demonstrating Bell inequality violation with randomly chosen measurement bases can then be quantified in terms of , which is the probability of randomly generating correlations that are incompatible with any locally causal theory. We now discuss the method of identifying nonlocal correlations using an appropriate class of Bell inequalities.

ii.3 The Bell inequalities

Bell inequalities are constraints on physically observable quantities that have to be satisfied by any locally causal theory (2). A relevant class of Bell inequalities for the scenario that we are considering is the set of Mermin-Ardehali-Belinskiǐ-Klyshko (MABK) inequalities (27); (28). A representative of this class is (29)

(17)

where is the vector of the measurement labels,

(18)

and the -partite correlation functions are the expectation values of the product of the measurement outcomes when the observer performs the -th measurement. Within quantum theory, the maximum possible value of is (28); (25).

As we show in Appendix A, this inequality can be rewritten as

(19)

where and

(20)

The equivalence class of MABK inequalities is the set of inequalities that can be obtained by permutating the measurement choices, , measurement outcomes, and labeling of the observers in the coefficients of inequality (19). However, as we prove in Appendix B, all such permuations can be obtained by permuting the measurement labels (i.e., for some set of ) and so each of the MABK inequalities can be obtained by one of the distinct permutations of measurement settings. In particular, the inequality

(21)

which will play an important role in the scenario where a single direction is shared, can be obtained from inequality (19) via the mapping for all .

When , the MABK inequalities reduce to the Bell-CHSH inequalities (21) and represent the complete set of Bell inequalities for this scenario (24). So for , the measurement outcomes are inconsistent with any locally causal theory if and only if they violate one of the MABK inequalities. For , there are also other equivalence classes of Bell inequalities (see, for example, Refs. (30); (25); (31)). An extensive set of such -partite Bell inequalities that include the MABK class as a subset is the well-known Werner-Wolf-Żukowski-Brukner (WWZB) inequalities (25); (29). These Bell inequalities can be put into the form of the following single nonlinear Bell inequality

(22)

Defining  mod 2, the above inequality is both necessary and sufficient for the set of -partite GHZ correlation functions [with measurements defined as in Eq. (5)]

(23)

to be describable within a locally causal theory. However, not all measurement statistics are captured by these full correlation functions. We can also compute the restricted correlation functions of the GHZ state,

(24)

, which involve the expectation value of the product of the measurement outcomes for a subset of the parties. As a result, one generally needs to check the measurement statistics against the complete set of Bell inequalities relevant to the particular experimental scenario to determine if these correlations are nonlocal. The characterization of the complete set of Bell inequalities is only known for and 3 (see Refs. (24); (30); (31) for details).

Nevertheless, for small , the problem of deciding if some given measurement statistics are compatible with a locally causal description can be solved numerically using linear programming 1. For larger values of , it may become infeasible to compute using linear programming. However, we can make use of the following inclusion relations:

(25)

to lower bound this probability, i.e.,

(26)

where etc. are the probabilities defined in Eq. (16) with being the respective classes of Bell inequalities.

ii.4 Sampling of measurement directions

Our sampling of measurement directions depends on the extent to which the observers are able to align their measurements within each physical scenario. For example, when all observers share a complete reference frame and can completely align their measurements, for any class of Bell inequalities , they can always pick in such a way that , assuming there exist such measurements. In this paper, we assume that the observers either cannot align their measurements at all or can only align them with respect to a single direction .

The following samplings of measurement directions will be applied to cater to the different extents in which the observers can align their measurements:

  1. Random isotropic measurements (RIM) – each party chooses both directions for independently and uniformly from the set of all possible directions;

  2. Random orthogonal measurements (ROM) – each local pair of measurement directions is chosen to be orthogonal but otherwise uniform, i.e., RIM with the additional constraint:

    (27)
  3. Planar random orthogonal measurements (PROM) – in addition to Eq. (27), all measurement directions are confined to a plane defined by some normal vector (which corresponds to the common direction the observers can align), i.e.,

    (28)

    for some shared by the parties.

Some of the results presented in Sec. III.1 for RIM and ROM have been discussed in Ref. (20) but are included here in more detail.

Iii Noiseless scenarios

When the experimenters do not align their measurements, one may expect that it is unlikely to find Bell-inequality-violating correlations by performing measurements in randomly chosen bases. Nonetheless, for the -partite GHZ state, the probability of choosing measurements that violate a Bell inequality rapidly approaches 1 as increases. In Sec. III.1 we briefly summarize the results for RIM and ROM presented in Ref. (20) and analyze the difference between testing the correlations against the WWZB inequalities and testing the correlations against the full set of Bell inequalities (obtained for small using linear programming).

Without any alignment of measurements, if the experimenters do not test the experimental statistics against a class of Bell inequalities that grows exponentially with , then the probability of identifying that the correlations generated by the randomly chosen measurements are inconsistent with any locally causal theory decreases with . However, if the observers can align the direction of their measurements, then we prove that for all they can always choose measurements that violate one of two Bell inequalities, namely, or from Eqns. (19)–(21), by an amount that grows exponentially with . We also numerically calculate the probability of violating four different classes of Bell inequalities and show that as the aligned direction is rotated away from the axis, the observers have to test their experimental statistics against more Bell inequalities in order to certify that the correlations generated by the randomly chosen measurements are inconsistent with any locally causal theory.

iii.1 No aligned directions - RIM and ROM

RIM ROM
2 7.080% 28.319% 28.319% 28.319% 10.326% 41.298% 41.298% 41.298%
3 1.328% 10.002% 13.313% 74.690% 2.324% 18.150% 26.604% 96.207%
4 0.972% 13.410% 23.407% 94.238% 1.714% 25.500% 59.034% 99.976%
5 0.644% 15.210% 25.675% 99.593% 1.108% 29.733% 52.798% 100.000%2
6 0.428% 16.879% 31.235% 99.965% 0.734% 34.442% 71.190% 100.000%3
8 0.183% 19.085% 37.509% - 0.312% 41.935% 80.420% -
10 0.077% 20.443% 42.254% - 0.133% 47.968% 86.926% -
15 0.009% 22.037% 50.515% - 0.017% 59.006% 95.204% -
Table 1: Probability of finding a Bell inequality violation from the -partite GHZ state for the scenario where each party is allowed to perform binary projective measurements in two randomly chosen measurement bases according to either RIM (left) or ROM (right). The number of parties is given in the leftmost column. Then, to the right, we have, respectively, the probability of violating , the MABK inequalities, the WWZB inequalities, and the complete set of Bell inequalities relevant to this scenario. Note that the probability of violation for each class of Bell inequalities is lower bounded by the corresponding entry to its left, as expected from Eq. (26).

A natural strategy that the experimenters can adopt is to each randomly choose two independent measurement bases according to a uniform distribution on the surface of a sphere. As can be seen from Eq. (5), this corresponds to each observer randomly choosing 4 angles and for , where are chosen from a uniform distribution on the interval and from the interval with . When the observers restrict their measurements to be orthogonal to each other (i.e., when they sample according to ROM), then Eq. (27) fixes one of the angles.

The probability of violating 4 classes of Bell inequalities, namely, , the MABK inequalities, the WWZB inequalities and the complete set of Bell inequalities for two binary-outcome measurements at each site, are presented in Table 1. For (and only for ), the MABK, WWZB and full set of Bell inequalities are all identical to the set of CHSH inequalities, so the probability of violating each of these three classes coincide for both RIM and ROM.

For , these three classes of inequalities obey the strict inclusion relations given in Eq. (25), and we see that the probabilities of violating these three different classes follow the strict inequalities given in Eq. (26). For the MABK and WWZB inequalities, which contain a number of inequalities that is exponential in , the probability of violation generally increases with , except when increases from 2 to 3 and a few other cases for ROM. This increasing trend is even more pronounced for the complete set of Bell inequalities where is found to be strictly increasing (up to the limit of our analysis).

The WWZB inequalities are necessary and sufficient conditions for the full -partite correlation functions to be consistent with a locally causal theory. Given that the restricted correlation functions can be computed from the respective reduced density matrices of and are always separable, it may seem surprising that the WWZB inequalities fail to detect a significant fraction of the nonclassical correlations generated from the GHZ states. However, while the reduced density matrices of are separable and so can be modelled in a locally causal theory, there is an additional requirement: the locally causal models for the different reduced density matrices must be consistent, in that they must not contradict one another and must also reproduce the full correlation functions given in Eq. (23). The results given in Table 1 show that as increases, it becomes increasingly difficult to find a locally causal model that could simultaneously reproduce Eq. (23) and Eq. (24).

The results from Table 1 also imply that detecting nonclassical correlations with a probability that increases with requires a set of Bell inequalities containing a number of inequalities that is exponential in . If we only use one MABK inequality, e.g., , to detect these nonclassical correlations, then the probability of finding correlations that violate via ROM decreases exponentially as increases. Clearly, because each inequality in the same equivalence class can be obtained by adopting a different classical labeling, the probability of violating any one of the MABK inequalities is equal to . Therefore the probability of violating one of a set of MABK inequalities is upper bounded by . As decreases exponentially with , must increase exponentially with in order for the probability of violating one of a set of inequalities to either remain constant or increase.

As we will demonstrate in the next section, this is not the case if the experimenters can align one of their measurement directions. In particular, we will show that if observers can align a measurement direction, then there is a set of two inequalities such that the probability of violating either of these two inequalities is one for all .

iii.2 Partially aligned measurements - random measurements in the plane

Without any alignment of their measurements, observers need to check their experimental statistics against an exponentially large class of Bell inequalities to uncover nonclassical correlations with a probability that increases with the number of observers. However, there are physical situations in which it is relatively easy to align a single measurement direction, or such a direction is naturally defined by the system.

For example, if qubits are encoded in the polarization of single photons and transmitted over optical fibres, then the ordinary and extraordinary modes are stable but optical birefringence causes a phase shift between the two modes. If this phase shift is unknown, then the observers share a single ‘direction’ on the Bloch sphere but have an essentially random alignment of the other two directions. While experimental techniques are available to account for this phase shift and may exist for other situations in which there is a preferred direction, we show that if the observers are trying to violate a Bell inequality, then such techniques are unnecessary (the related question for quantum key distribution in this situation has also been investigated (35)).

Specifically, we show that if the reference direction, , is the -axis and the observers agree on a labeling convention for their measurements, they will always obtain correlation functions that violate either or if the measurements are sampled according to PROM (i.e., if the measurements are orthogonal and confined to the plane perpendicular to some normal vector shared by the parties).

For PROM in the plane, the observers share the axis. If the observer’s two measurements are and , then, because the labels and are arbitrary, they are free to relabel them as necessary so that forms a right-handed coordinate system for all (a similar result follows for left-handed coordinate systems). Under this labeling convention, randomly choosing and is equivalent to randomly choosing a single random angle , with and

(29)
Theorem III.1.

Any choice of orthogonal measurements in the plane on the -partite GHZ state will generate correlation functions that satisfy either

(30)

provided the observers obey the labeling convention described above.

Proof.

For the -partite GHZ state and the labeling convention in Eq. (29), the full correlation function, Eq. (23), becomes

(31)

where and as before, . Substituting this into the left-hand-side of inequality (19) gives

(32)

There are ways of choosing such that , so Eq. (32) can be rewritten as

(33)

Similarly, substituting Eq. (31) into the left-hand-side of inequality (21) gives

(34)

Because

(35)

either or . ∎

From Sec. II.3, the inequalities in Theorem III.1 are Bell inequalities with an upper bound of in any locally causal theory. Therefore for and for the observers will violate or . When , the observers will always violate or by a factor of at least . Moreover, the upper bound for both and in quantum mechanics is , so the violation of or is within a constant factor of the maximum violation possible in quantum mechanics.

We can also find the probability of the observers choosing measurements by PROM in the plane such that they will obtain a violation of a Bell inequality that is within a factor of the maximum violation possible in quantum mechanics.

Randomly choosing measurements by PROM in the plane is equivalent to randomly choosing in Eqs. (33) and (34). The probability of choosing such that

(36)

is the same as the probability of choosing such that , which is simply .

Therefore the probability of the observers choosing measurements by PROM in the plane such that they will obtain a violation of either or above some threshold is

(37)
MABK WWZB Complete set MABK WWZB Complete set
2 0.5411 0.7002 0.7002 0.7002 0.0033 0.0033 0.0033 0.0033
3 0.4129 0.4129 0.4580 0.9553 0.2189 0.2189 0.2406 0.4850
4 0.4729 0.4729 0.9130 0.9996 0.3219 0.3219 0.3721 0.8189
5 0.4832 0.4832 0.5867 0.9998 0.3741 0.3741 0.4059 0.8635
4 0.5035 0.5035 0.9155 0.9997 0.4129 0.4129 0.4544 0.9782
Table 2: Fraction of the surface area of a sphere that corresponds to normal vectors for which the probability of violating a class of Bell inequalities is nonzero () or unity (). The four classes of Bell inequalities considered are , the MABK inequalities, the WWZB inequalities and the complete set of Bell inequalities relevant to this scenario. Note that the value of both and exactly coincides for and the MABK inequalities when .
Figure 1: (a) Contour plot of the probability of violating or for when sampling measurements via PROM with a reference direction as in Eq. (38). (b) Contour plot of the probability of violating one of the WWZB inequalities for when sampling measurements via PROM with the reference direction as defined in Eq. (38). In both contour plots, in the center and increases radially to a maximum of and for smaller and generally decreases as increases along a line of fixed (i.e., along a radial line).

iii.3 Partially aligned measurements - random measurements in other planes

Theorem III.1 applies when the observers measure along two orthogonal directions in the plane, i.e., when the direction that the observers can align is the -axis (which corresponds to the computational basis used to define the GHZ state). When the common direction is at some angle to the basis in which the GHZ basis is defined, the probability of observers obtaining correlation functions that violate a Bell inequality can change significantly. For , we simulate as a function of and where

(38)

and when is and , the MABK inequalities, the WWZB inequalities or the complete set of Bell inequalities for this scenario.

Given a normal vector, , shared by parties, we want the probability that allows the parties to violate a given class of Bell inequalities with probability 1 or with some nonzero probability. Consequently, we define the ratio of the set of normal vectors that give violations of the class of Bell inequalities with probability to the set of all normal vectors (i.e., the set of points on the surface of a unit sphere with ) by

(39)

where

(40)

Similarly, we define the ratio of the set of normal vectors that give violations of the class of Bell inequalities with probability to the set of all normal vectors by

(41)

where

(42)

gives the fraction of the set of unit normal directions such that and observers who share can always obtain correlations that violate a Bell inequality when they sample measurements using PROM. Likewise, gives the fraction of the set of normal directions such that and observers who share can always obtain correlations that violate a Bell inequality with nonzero probability when they sample measurements using PROM. The values of and for are given in Table 2 and the probability of violating and the WWZB inequalities for is plotted in Fig. 1.

The case when is exceptional because almost any rotation of the reference direction [except when in Eq. (38)] reduces the probability of violating a Bell inequality to below 1. This occurs because there are dense sets of measurements that produce arbitrarily small violations of and when and these measurements do not produce a violation of or (or any other Bell inequality) when the reference direction is rotated an arbitrarily small amount from the axis.

However, for , there are no such sets of measurements and so, as our results show, the reference direction can be rotated from the axis by some “small” angle (i.e., ) in any direction without reducing .

Numerically, we found that for any rotated reference vector , considering the full class of MABK inequalities provides no advantage over just considering and , except when , in which case the MABK inequalities form the complete set of Bell inequalities. Considering the full set of WWZB inequalities does increase the value of , but not very substantially. However, testing against the full set of WWZB inequalities can substantially increase the value of , i.e., the area of normal vectors for which , as can be seen in Table 2. Testing against the full set of WWZB inequalities also reveals a strong dependence on the parity of , which occurs due to the term in Eq. (23).

For , the dependence on the azimuthal angle, , is small when testing against and . In particular, for all and . That is, for any , the reference direction can be rotated from the axis by at least in any direction without affecting the probability of generating correlations that violate one of two Bell inequalities, namely, and . As increases, this threshold value of the polar angle appears to increase slowly.

Iv Noisy scenarios

So far, we have made use of various idealisations. We now examine what happens when some of these assumptions are relaxed. In Sec. IV.1, we determine how depolarizing and dephasing noise upon the GHZ state reduce the probability of violating a Bell inequality when observers do not align any measurement directions or only align a single measurement direction. In Sec. IV.2 we analyze how the probability of violating a Bell inequality is affected by random perturbations in each observer’s alignment of the common direction, i.e., when the observers cannot align their measurements perfectly.

iv.1 Decoherence

In order to study the ability of observers to violate a Bell inequality in the presence of noise, we consider depolarizing and dephasing noise. For simplicity, we assume that each qubit is transmitted over equally noisy, uncorrelated channels, so the noise for all qubits is described by a single parameter , where corresponds to zero noise and corresponds to maximal noise (i.e., complete depolarizing or dephasing). We begin by outlining the correlation tensor formalism, which is a convenient method of examining the effect of uncorrelated noise. We then give a brief introduction to depolarizing and dephasing noise before presenting our results from numerical simulations on the probability of violating a Bell inequality in the presence of noise.

Correlation tensor formalism

An arbitrary -qubit state can be expanded in any basis of Hermitian operators acting on the Hilbert space . In particular, the -fold tensor products of local Pauli operators

(43)

is one such basis; here is a string of indices, , and is the identity operator acting on .

Together with the orthogonality relation,

(44)

we can then represent by

(45)

where is the correlation tensor (36). The description in terms of the correlation tensor is thus equivalent to the description in terms of the density operator. In what follows, we will follow Ref. (36) and describe the effect of noise on a quantum state using the correlation tensor, which allows us to define the effects of uncorrelated noise on each qubit. For the GHZ state, we have

(46)

All of these terms are unless is a tensor product of either (1) Pauli and Pauli matrices or (2) Pauli and identity matrices for some .

Depolarizing noise

Depolarizing noise maps local Pauli operators as (37)

(47)

Full correlation functions correspond to all observers performing non-trivial projective measurements, while restricted correlation functions correspond to some subset of observers performing the measurement (i.e., ignoring the outcomes from some observers). Therefore the effects of depolarization on the correlation functions are

(48)

and

(49)

for arbitrary subsets of observers, .

Note that the effect on the full correlation functions is identical to the effect of mixing the GHZ state with the maximally mixed state according to

(50)

when .

Dephasing noise

We also consider dephasing noise, which is appropriate when there is some preferred basis in the system which is particularly stable. Dephasing noise in the computational basis suppresses off-diagonal terms for each qubit, i.e., it maps local Pauli operators as (37)

(51)

Clearly, from Eqs. (45) and (46) all diagonal terms of are unaffected and, because off-diagonal terms of the correlation tensor are zero unless the term corresponds to tensor products of only and matrices, all off-diagonal terms are uniformly reduced by a factor of . Therefore dephasing takes the GHZ state to