David H Oaknin
email: d1306av@gmail.com

The Bell theorem revisited: geometric phases in gauge theories

The Bell theorem stands as an insuperable roadblock in the path to a very desired intuitive solution of the EPR paradox and, hence, it lies at the core of the current lack of a clear interpretation of the quantum formalism. The theorem states through an experimentally testable inequality that the predictions of quantum mechanics for the Bell polarization states of two entangled particles cannot be reproduced by any statistical model of hidden variables that shares certain intuitive features. In this paper we show, however, that the proof of the Bell theorem involves a subtle, though crucial, assumption that is not required by fundamental physical principles and, hence, it is not necessarily fulfilled in the experimental setup that tests the inequality. Indeed, this assumption can neither be properly implemented within the standard framework of quantum mechanics. Namely, the proof of the theorem assumes that there exists a preferred absolute frame of reference, supposedly provided by the lab, which enables to compare the orientation of the polarization measurement devices for successive realizations of the experiment and, hence, to define jointly their response functions over the space of hypothetical hidden configurations for all their possible alternative settings. We notice, however, that only the relative orientation between the two measurement devices in every single realization of the experiment is a properly defined physical degree of freedom, while their global rigid orientation is a spurious gauge degree of freedom. Hence, the preferred frame of reference required by the proof of the Bell theorem does not necessarily exist. In fact, it cannot exist in models in which the gauge symmetry of the experimental setup under global rigid rotations of the two detectors is spontaneously broken by the hidden configurations of the pair of entangled particles and a non-zero geometric phase appears under some cyclic gauge symmetry transformations. Following this observation, we build an explicitly local model of hidden variables that reproduces the predictions of quantum mechanics for the Bell states.

I. INTRODUCTION

The Bell theorem is one of the fundamental theorems upon which relies the widespread belief that quantum mechanics is the ultimate mathematical framework within which the hypothetical final theory of the fundamental building blocks of Nature and their interactions should be formulated. The theorem states through an experimentally testable inequality (the Bell inequality) that statistical models of hidden variables that share certain intuitive features cannot reproduce the predictions of quantum mechanics for the entangled polarization states of two particles (the Bell states) [START_REF] Bell | On the Einstein-Podolsky-Rosen paradox[END_REF][START_REF] Fine | Hidden variables, joint probability, and the Bell inequalities[END_REF]. These predictions have been confirmed beyond doubt by very carefully designed experiments [START_REF] Hensen | Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres[END_REF][START_REF] Aspect | Experimental test of Bell's inequalities using timevarying analyzers[END_REF][START_REF] Tittel | Violation of Bell Inequalities by Photons More Than 10 km Apart[END_REF][START_REF] Weihs | Violation of Bell's inequality under strict Einstein locality conditions[END_REF][START_REF] Rowe | Experimental violation of a Bell's inequality with efficient detection[END_REF][START_REF] Giustina | Bell violation using entangled photons without the fair-sampling assumption[END_REF][START_REF] Christensen | Detection-Loophole-Free Test of Quantum Nonlocality, and Applications[END_REF][START_REF] Giustina | A significant-loophole-free test of Bell's theorem with entangled photons[END_REF][START_REF] Shalm | A strong loophole-free test of local realism[END_REF][START_REF] Wiseman | Quantum physics: Death by experiment for local realism[END_REF].

In these experiments a source emits pairs of particles whose polarizations are arranged in a Bell entangled state:

|Ψ Φ = 1 √ 2 | ↑ (A) | ↓ (B) -e iΦ | ↓ (A) | ↑ (B) , (1)
where {| ↑ , | ↓ } (A,B) are eigenstates of Pauli operators σ (A,B) Z along locally defined Zaxes for each one of the two particles. The two emitted particles travel off the source in opposite directions towards two widely separated detectors, which test their polarizations. The orientation of each one of the detectors can be freely and independently set along any arbitrary direction in the XY-plane perpendicular to the locally defined Z-axis.

Upon detection each particle causes a binary response of its detector, either +1 or -1.

Thus, each pair of entangled particles produces an outcome in the space of possible events P ≡ {(-1, -1), (-1, +1), (+1, -1), (+1, +1)}. We refer to each detected pair as a single realization of the experiment.

Quantum mechanics predicts that the statistical correlation between the binary outcomes of the two detectors in a long sequence of realizations of the experiment is given by:

E(∆, Φ) = -cos(∆ -Φ), ( 2
)
where ∆ is the relative angle between the orientations of the two detectors. In particular, when ∆ -Φ = 0 we get that E = -1, so that all outcomes in the sequence must be either (-1, +1) or (+1, -1).

The Bell theorem states that prediction (2) cannot be reproduced by any model of hidden variables that shares certain intuitive features. In particular, the CHSH version of the theorem states that for the said generic models of hidden variables the following inequality is fulfilled for any set of values (∆ 1 , ∆ 2 , δ) [START_REF] Clauser | Proposed experiment to test local hidden variables theories[END_REF]:

|E(∆ 1 ) + E(∆ 2 ) + E(∆ 1 -δ) -E(∆ 2 -δ)| ≤ 2. ( 3
)
On the other hand, according to quantum mechanics the magnitude in the left hand side of the inequality reaches a maximum value of 2 √ 2, known as Tsirelson's bound [START_REF] Cirelson | Quantum generalizations of Bell's inequality[END_REF], for certain values of ∆ 1 , ∆ 2 and δ -e.g, ∆ 1 = -∆ 2 = 1 2 δ = π 4 . As it was noted above, carefully designed experiments have confirmed that the CHSH inequality is violated according to the predictions of quantum mechanics and, therefore, have ruled out all the generic models of hidden variables constrained by the Bell inequality [START_REF] Hensen | Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres[END_REF].

In this paper we show, however, that the generic models of hidden variables constrained by the Bell theorem all share a subtle crucial feature that is not necessarily fulfilled in the actual experimental tests of the Bell inequality. Indeed, the considered feature cannot be derived from fundamental physical principles and may even be at odds with the fundamental principle of relativity. Moreover, this feature neither can be properly implemented within the standard framework of quantum mechanics. We follow this observation to explicitly build a local model of hidden variables that does not share the disputed feature and, thus, it is capable to reproduce the predictions of quantum mechanics for the Bell polarization states of two entangled particles.

Our model puts forward for consideration the possibility that quantum mechanics might not be the ultimate mathematical framework of fundamental physics. In fact, it is interesting to notice that the way how our model solves the apparent 'non-locality' associated to entanglement in the standard quantum formalism is very similar to the way how General Relativity solves the 'non-locality' of Newton's theory of gravitation: in our model quantum entanglement is the result of a curved metric in the space in which the hypothetical hidden variables live.

II. OUTLINE

Any local statistical model of hidden variables that aims to describe the Bell experiment consists of some space S of possible hidden configurations for the pair of entangled particles, labelled here as λ ∈ S, together with a well-defined (density of) probability ρ(λ) for each one of them to occur in every single realization. The model must also specify well-defined binary functions s

(A) Ω A (λ) = ±1, s (B)
Ω B (λ) = ±1 to describe the outcomes that would be obtained at detectors A and B when the pair of entangled particles occurs in the hidden configuration λ ∈ S and their polarizations are tested along directions Ω A and Ω B , respectively.

The proof of the CHSH inequality (3) involves two well-defined possible orientations Ω A and Ω A for the polarization test of particle A and two well-defined possible orientations Ω B and Ω B for the polarization test of particle B, and assumes that the considered model of hidden variables assigns to each possible hidden configuration λ ∈ S a 4-tuple of binary values s 4 to describe the outcomes that would be obtained in each one of the two detectors in case that it would be set along each one of its two available orientations. Hence, it is straightforward to obtain that for any λ ∈ S,

(A) Ω A (λ), s (A) Ω A (λ), s (B) Ω B (λ), s (B) Ω B (λ) ∈ {-1, +1}
s (A) Ω A (λ) • s (B) Ω B (λ) + s (B) Ω B (λ) + s (A) Ω A (λ) • s (B) Ω B (λ) -s (B) Ω B (λ) = ±2, (4)
since the first term is non-zero only when s

Ω A (λ) • s (B) Ω B (λ) + s (B) Ω B (λ) + s (A) Ω A (λ) • s (B) Ω B (λ) -s (B) Ω B (λ) ≤ 2, (5)
while

dλ ρ(λ) s (A) Ω A (λ) • s (B) Ω B (λ) = E(∆ 1 ), (6)
dλ ρ(λ) s

(A) Ω A (λ) • s (B) Ω B (λ) = E(∆ 2 ), (7)
dλ ρ(λ) s

(A) Ω A (λ) • s (B) Ω B (λ) = E(∆ 1 -δ), (8)
dλ ρ(λ) s

(A) Ω A (λ) • s (B) Ω B (λ) = E(∆ 2 -δ). (9)
In this argument the orientations Ω A , Ω A , Ω B and Ω B seem to be fixed with respect to some external frame of reference supposedly provided by the labs. Nonetheless, the data collected in such an experimental setup could be alternatively analyzed within frames of reference aligned, for example, with the magnetic axis of the Sun or the rotational axis of the Galaxy, with respect to which the orientations of the detectors for different realizations of the experiment are not fixed anymore. Obviously, the conclusions of the analysis must remain the same, independently of the chosen lab frame. Indeed, the proof of the CHSH inequality actually requires only three well-defined angles, ∆ 1 ≡ ∠(Ω B , Ω A ), ∆ 2 ≡ ∠(Ω B , Ω A ) and δ ≡ ∠(Ω A , Ω A ), which correspond, respectively, to the relative orientations of Ω B , Ω B and Ω A with respect to Ω A , which serves as a reference direction. The reference direction Ω A serves also to define the hidden configuration λ ∈ S of the pair of entangled particles in every single realization of the experiment, since the description of a physical state must necessarily be done with respect to a reference frame. Otherwise, the orientation with respect to any external lab frame, either the optical table or the stars in the sky, of this reference direction Ω A at different single realizations of the Bell experiment is absolutely irrelevant: it is an spurious gauge degree of freedom, which can be set to zero (see Fig. 1).

The proof of the CHSH inequality, thus, seems straightforward and unovaidable. Nonetheless, the main claim of this paper is that this proof, as well as the proofs of all other versions of the Bell inequality, involve a subtle, though crucial, implicit assumption that cannot be derived from fundamental physical principles and, indeed, it might not be fulfilled in the actual experimental setup that tests the inequality. Namely, in each realization of a Bell experiment the polarization of each one of the two entangled particles is tested along a single direction. Hence, the relative orientation ∆ of the two measurement devices in each single realization of the experiment is a properly defined physical magnitude, which can be set to values ∆ 1 , ∆ 2 or any other desired value. On the other hand, the definition of the angle δ that appears in the proof of the CHSH inequality requires a comparison of the global rigid orientation of the measurement devices for different realizations of the Bell experiment and, thus, it requires the existence of an absolute preferred frame of reference with respect to which this global orientation could be defined. Otherwise, we could choose the orientation of, say, detector A as the reference direction for every single realization of the experiment and define the orientation of the other detector with respect to it, in which case the proof of the Bell theorem does not necessarily hold as we shall show later. Obviously, such an absolute preferred frame of reference would not be needed if the polarization of each one of the two entangled particles could be tested along two different directions at once in every single realization of the experiment, but this is certainly not the case. Similar concerns regarding the way how different settings of the detectors are compared within the framework of the Bell theorem and the crucial role that this comparison plays in the proof of the inequality are also raised by K. Hess in [START_REF] Hess | Kolmogorov's probability for 'entangled' data subsets of EPRB experiments: no violation of Einstein's separation principle[END_REF][START_REF] Hess | Bell's theorem and Instantaneous influences at a distance[END_REF], and much earlier in a different but related context in FIG. 1: The orientation of the reference direction Ω A with respect to the chosen lab frame is a spurious gauge degree of freedom. [START_REF] Hess | Bell's theorem and the problem of decidability between the views of Einstein and Bohr[END_REF][START_REF] Hess | Breakdown of Bell's theorem for certain objective local parameter spaces[END_REF][START_REF] Hess | From Boole to Leggett-Garg: Epistemology of Bell-Type Inequalities[END_REF].

The said preferred frame of reference needed to prove the Bell theorem is supposedly provided by the lab. However, the conditions that a reference frame must fulfill in order to qualify as a preferred absolute frame are far from obvious and, in any case, its existence is an overbold assumption whose fulfillment has never been explored neither theoretically or experimentally. In fact, the existence of an absolute preferred frame of reference would be clearly at odds with Galileo's principle of relativity. Moreover, it is straightforward to show that this assumption cannot be properly implemented within the standard framework of quantum mechanics either. The argument goes as follows. The Bell state (1) that describes the pair of entangled particles is defined in terms of the bases {| ↑ , | ↓ } (A,B) of eigenstates of the Pauli operators σ

(A,B) Z
along locally defined Z-axes for each one of the particles. Since these eiegenstates are defined up to a global phase, the phase Φ in expression (1) cannot be properly defined with respect to a lab frame. In order to properly define it we need to choose an arbitrary setting of the two detectors that test the polarizations of the pair of entangled particles as a reference. This reference setting defines parallel directions along the XY-planes at the sites where each one of the two particles are detected. Then, the phase Φ of the entangled state (1) can be properly defined with respect to this reference setting with the help of the measured correlations between the outcomes of the two detectors, E = -cos(Φ). Furthermore, we can use this reference setting to properly define a relative rotation ∆ of the orientations of the two measurement devices. On the other hand, since we must use an arbitrary setting of the detectors as a reference, their absolute orientation is an unphysical gauge degree of freedom (see Fig. 2). In summary, in order to describe the setting of the measurement devices in a Bell experiment within the standard framework of quantum mechanics we need to specify both Φ and ∆ with respect to some otherwise arbitrary reference setting of the detectors. Nonetheless, only their difference ∆ -Φ is independent of the chosen reference setting and, hence, the correlation between the outcomes of the two devices can only depend on this difference (2).

In the absence of an absolute preferred frame of reference the global rigid orientation of the two detectors is, as we have already noticed before, an spurious (unphysical) gauge degree of freedom and, hence, the proof of the CHSH inequality (as well as of all other versions of the Bell inequality) holds only for models in which the considered hidden configurations are symmetrically invariant under a rigid rotation of the two measuring devices. On the other hand, we shall show below that the proof of the inequality does not necessarily hold when this symmetry is (spontaneously) broken by the hidden configuration of the entangled particles, since then a non-zero geometric phase may appear under cyclic gauge transformations.

Indeed, the crucial role of the angle δ in the proof of the CHSH inequality is an obvious indication that in order to violate it the gauge symmetry under a rigid rotation of the two detectors must be spontaneously broken.

In fact, it is obvious from the correlation (2) that the entanglement of the two particles explicitly breaks the symmetry of the system under a rotation of the relative orientation of the two detectors. Since a reference direction is needed for this symmetry to get broken, the gauge symmetry under a rigid rotation of the two detectors must be also spontaneously broken. From this perspective the phase Φ that appears in the description of the source [START_REF] Bell | On the Einstein-Podolsky-Rosen paradox[END_REF] seems to play the role of a Goldstone mode associated to the spontaneously broken gauge symmetry, that is, the phase Φ appears instead of the spurious gauge degree of freedom δ that would describe the global rigid orientation of the two detectors. Under these circumstances, it is not possible to compare different settings of the detectors with respect to an external lab frame of reference: they can only be compared with respect to a frame in which they all share the same preferred direction, e.g. the reference frame set by the orientation of one of the detectors. This requirement can be explained as follows.

In the proof of the CHSH inequality it is implicitly assumed, as we have already noticed above, that there exists a preferred frame of reference, which defines a set of coordinates λ ∈ S over the space S of all possible hidden configurations that can be used to describe the response function of each one of the two detectors in each one of its two available orientations (defined with respect to the said preferred frame). Above we denoted these response functions as s

(A) Ω A (λ), s (A) Ω A (λ), s (B) Ω B (λ), s (B) Ω B (λ).
Nonetheless, in general, we should allow for each one of the two detectors to define its proper set of coordinates over the space S. Thus, for a given setting of the detectors we shall denote as λ A and λ B the sets of coordinates associated to detector A and detector B, respectively, so that their responses would be given as s(λ A ) and s(λ B ) by some universal function s(•) of the locally defined coordinate of the hidden configuration. Since these two sets of coordinates parameterize the same space of hidden configurations S there must exist some invertible transformation that relates them:

λ B = -L(λ A ; ∆ -Φ), (10)
which may depend parametrically on the relative orientation ∆ -Φ between the two detectors. This transformation must fulfill the constraint

dλ A ρ(λ A ) = dλ B ρ(λ B ), (11)
in order to guarantee that the probability of every hidden configuration to occur remains invariant under a change of coordinates, while the (density of) probability ρ(•) is functionally invariant for both sets of coordinates. However, these constraints do not forbid the possibility that the set of coordinates accumulates a non-zero geometric phase α = 0 through certain cyclic gauge transformations:

(-L ∆2 ) • -L ∆2 -δ • -L ∆1 -δ • (-L ∆1 ) = I, (12)
In such a case there does not exist a single set of coordinates that can be used to define the response functions of each one of the two detectors in its two available orientations (defined with respect to an external frame), as required by the proof of the inequality (3). Therefore, in order to compare the four different experiments involved in the CHSH inequality we must choose the orientation of one of the detectors as a reference direction, as we do below in [START_REF] Clauser | Proposed experiment to test local hidden variables theories[END_REF], so that they all may be described within a common set of coordinates. The appearence of a non-zero geometric phase under a cyclic transformation is a well-known phenomena in physical models involving gauge symmetries [START_REF]Geometric Phases in Physics[END_REF] and, therefore, we should not rule out the possibility that it also occurs in models of hidden variables for the Bell states. The Bell theorem, however, cannot account for such models.

Following these observations we were able to explicitly build a local model of hidden variables that reproduces the predictions of quantum mechanics for the Bell polarization states. In our model the hidden configurations of the pair of entangled particles are described by a pointer, which sets an arbitrarily oriented preferred direction and, thus, spontaneously breaks the symmetry of the setup under rigid rotations of the two detectors. As we have just noticed, and we shall show later on in further detail, in order to compare different realizations of the experiment within the framework of such a model we must choose a common reference direction, which can be either the orientation of the hidden configuration of the pair of entangled particles or, alternatively, the orientation of one of the detectors, say, detector A.

Since the former may not be directly experimentally accessible, we are left only with the latter option. Thus, in such a model we only need to specify the binary values for s(λ A ), s(λ B ), s(λ B ), s(λ B ) and s(λ B ) for each possible hidden configuration λ A ∈ S of the pair of entangled particles, where

λ B = -L(λ A ; ∆ 1 ), λ B = -L(λ A ; ∆ 2 ), λ B = -L(λ A ; ∆ 1 -δ), λ B = -L(λ A ; ∆ 2 -δ).
It is then straightforward to notice that the magnitude

s(λ A ) • (s(λ B ) + s(λ B ) + s(λ B ) -s(λ B )) , (13)
which comes instead of (4), can take values out of the interval [-2, 2]. Hence, these models are not constrained by the CHSH inequality [START_REF] Hensen | Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres[END_REF]. A simplified version of these arguments is presented in Fig. 3 with the help of a toy model. In the game on the left the triangle is drawn on a plane surface and the reference unit vectors are contained within the plane, while in the game on the right the 'triangle' is defined on the surface of a sphere by segments of three great circles and the three reference unit vectors lay within the corresponding tangent planes. Two copies of a randomly oriented unit vector λ are generated at random at the center of one of the three segments of the triangle with density of probability ρ( λ), and detected, respectively, at the two detectors located at the ends of the segment. In the game on the left the vector λ is contained within the plane surface, while in the game on the right the vector λ is tangent to the sphere. The binary responses of the detectors are locally defined by parallely transporting the unit vector λ along the segment of the triangle to its end, and comparing its orientation to the orientation of the corresponding reference unit vector:

A( a, λ) = sign( a • λ), B( b, λ) = sign( b • λ) , C( c, λ) = sign( c • λ).
It is then straighforward to prove the Bell inequality for the game on the left, since for any settings a, b, c and any random vector λ the following equality

holds: A( a, λ) • B( b, λ) + A( a, λ) • C( c, λ) = 1 + B( b, λ) • C( c, λ).
Therefore, after integrating over the whole space of possible hidden configurations:

d λ ρ( λ) A( a, λ) • B( b, λ) + A( a, λ) • C( c, λ) ≤ d λ ρ( λ) A( a, λ) • B( b, λ) + A( a, λ) • C( c, λ) = d λ ρ( λ) A( a, λ) • B( b, λ) + A( a, λ) • B( b, λ) • B( b, λ) • C( c, λ) = 1 + d λ ρ( λ) B( b, λ) • C( c, λ),

Outcome

Setting

a = +1 = +1 a = +1 b = -1 a = -1 b = +1 a = -1 b = -1 A = +1 B = +1 p 1 0 0 1 -p 1 A = +1 B = -1 p 2 0 0 1 -p 2 A = -1 B = +1 p 3 0 0 1 -p 3 A = -1 B = -1 0 p 4 1 -p 4 0
Table 1. Conditional probabilities for a toy model with two binary inputs and two binary outcomes that cannot be reproduced by a realistic and local underlying theory [START_REF] Popescu | Quantum nonlocality as an axiom[END_REF].

These arguments can be stated in more abstract terms as follows. Quantum predictions for the Bell experiment are commonly described as a set of conditional probabilities This statement can be clearly illustrated with the help of the toy model described in Table 1 [START_REF] Popescu | Quantum nonlocality as an axiom[END_REF], where conditional probabilities for each one of the four possible results of an experiment with two binary outcomes a, b = ±1 (columns) are given for each one of four possible settings, defined by two independent binary inputs A, B = ±1 (rows). For these probabilities to be properly defined we require that p 1 , p 2 , p 3 , p 4 ∈ [0, 1]. It can be readily checked that for each set of input values (rows) the sum of the probabilities for all possible results of the experiment (columns) equals 1. These conditional probabilities, however, cannot be obtained within the framework of an underlying local model of hidden variables:

the conditional probabilities listed in the first three rows would imply a = b, that is, the outcomes of the two detectors in any of their four possible settings must have the same sign, which is obviously inconsistent with the conditional probabilities listed in the fourth row.

Nonetheless, it is straightforward to identify in this abstract reformulation of the Bell theorem the same unjustified implicit assumption that we have noticed above, namely, that there are two well-defined choices for the setting of each one of the detectors. We have noticed above that we can properly define and measure only the conditional probabilities p(a, b|D), where a = ±1 and b = ±1 are, as before, the outcomes at each one of the two detectors and D = 1, 2, 3, 4 defines four possible relative orientations between them. We did notice also that quantum mechanics as well makes theoretical predictions only for these conditional probabilities p(a, b|D). Under these looser constraints the Bell theorem does not necessarily hold.

Consider, for example, the toy model described in Table 2. The conditional probabilities are identical to those described in Table 1 for each one of the four possible results of the experiment, but the setting of the measurement devices is now described by a single A straightforward proof of the inequalities that constraint the correlations that can be obtained in any model of hidden variables with two binary inputs and two binary outcomes is presented in [START_REF] Revzen | Kolmogorov proof of the Clauser-Horne-Shimony-Holt inequalities[END_REF] using only Boolean logic. The analysis relies on the observation that any such model makes a prediction for the correlations AB , AB , A B and A B , and also for the correlations AA and BB that would be obtained in the hypothetical case that the polarization of each one of the two entangled particles could be tested along two different orientations at once. It can be immediately noticed that these constraints do not hold for the model of hidden variables discussed in this paper, for which the correlations AA and BB cannot be jointly bounded.

Outcome

Setting

a = +1 = +1 a = +1 b = -1 a = -1 b = +1 a = -1 b = -1 D = 1 p 1 0 0 1 -p 1 D = 2 p 2 0 0 1 -p 2 D = 3 p 3 0 0 1 -p 3 D = 4 0 p 4 1 -p 4 0
Table 2. Conditional probabilities for a toy model with a single input with four possible values and two binary outcomes. They can be reproduced by an underlying theory.

III. THE MODEL

We shall now build an explicitly local statistical model of hidden variables that reproduces the predictions of quantum mechanics for the Bell states (1) and, hence, it is not constrained by the Bell inequality (3). The fundamental ideas of the model were first discussed in [START_REF] Oaknin | Solving the EPR paradox: an explicit statistical local model of hidden variables for the singlet state[END_REF].

As we have already noticed above, the crux of the model is the spontaneous breaking of the gauge symmetry of the experimental setup under global rigid rotations of the orientation of the detectors. The symmetry is broken by the hidden configuration of the pair of entangled particles. Furthermore, we allow for a non-zero geometric phase [START_REF] Wiseman | Quantum physics: Death by experiment for local realism[END_REF] to accumulate through cyclic gauge transformations. Under these circumstances there does not exist an absolute preferred frame, other than the orientation of one of the detectors, to which we can refer in order to compare different realizations of the experiment (see Fig. 3).

The gauge symmetry is spontaneously broken because in the considered model the hidden configuration of the pair of entangled particles has a preferred direction randomly oriented over a unit circle S in the XY-plane. This orientation is carried by each one of the particles of the entangled pair. Each one of the two detectors defines over this circle S a frame of reference with its own set of associated coordinates, which we shall denote as λ A ∈

[-π, +π) for detector A and λ B ∈ [-π, +π) for detector B. Since the two sets of coordinates parameterize the same space S, they must be related by some transformation law:

λ B = -L(λ A ; ∆ -Φ), ( 14
)
where ∆ is the relative angle between the two detectors and Φ is the phase that characterizes the source of entangled particles as defined above. This transformation law states that a hidden configuration whose preferred direction is oriented along an angle λ A with respect to detector A, it is oriented along an angle λ B with respect to detector B.

The transformation law [START_REF] Cirelson | Quantum generalizations of Bell's inequality[END_REF] does not violate neither locality nor causality: it may well be a fundamental law of Nature. Indeed, the notions of locality and causality in special relativity stem from a similar relationship v = T (v; V ) beween the velocities v and v of a point particle with respect to two different inertial frames moving with relative velocity V . Moreover, ( 14) is only a generalization of the euclidean linear relationship that states that in a flat space given two detectors whose orientations form an angle ∆, then a pointer oriented along an angle ω with respect to one of them is oriented along an angle ω -∆ with respect to the other detector.

In order to reproduce the predictions of quantum mechanics we define the transformation law [START_REF] Cirelson | Quantum generalizations of Bell's inequality[END_REF] as follows:

• If ∆ ∈ [0, L(λ; ∆) =                                      q(λ -∆) • arccos -cos( ∆) -cos(λ) -1 , if -π ≤ λ < ∆ -π, q(λ -∆) • arccos + cos( ∆) + cos(λ) -1 , if ∆ -π ≤ λ < 0, q(λ -∆) • arccos + cos( ∆) -cos(λ) + 1 , if 0 ≤ λ < ∆, q(λ -∆) • arccos -cos( ∆) + cos(λ) + 1 , if ∆ ≤ λ < +π, (15)
• If ∆ ∈ [-π, 0), L(λ; ∆) =                                      q(λ -∆) • arccos -cos( ∆) + cos(λ) + 1 , if -π ≤ λ < ∆, q(λ -∆) • arccos + cos( ∆) -cos(λ) + 1 , if ∆ ≤ λ < 0, q(λ -∆) • arccos + cos( ∆) + cos(λ) -1 , if 0 ≤ λ < ∆ + π, q(λ -∆) • arccos -cos( ∆) -cos(λ) -1 , if ∆ + π ≤ λ < +π, (16)
where

q(λ -∆) = sign((λ -∆)mod([-π, π))),
∆ = ∆ -Φ and the function y = arccos(x) is defined in its main branch, such that y ∈ [0, π]

while x ∈ [-1, +1].
In Fig. 4 the transformation L(λ; ∆) is graphically shown for the particular case ∆ = π/3. It is straightforward to check that the transformation law ( 14) is strictly monotonic and fulfills the differential relationship

|d (cos(λ B ))| = dλ B |sin(λ B )| = dλ A |sin(λ A )| = |d (cos(λ A ))| , (17)
while the parameter ∆ plays the role of an the integration constant.

Locality is explicitly enforced in our model by requiring that the outcome of each one of the detectors in reponse to the hidden configuration of the pair of entangled particles depends only on its locally defined orientation, that is, s(λ A ) = ±1 for detector A and s(λ B ) = ±1 for detector B, where λ B and λ A are related by relationship [START_REF] Cirelson | Quantum generalizations of Bell's inequality[END_REF] and s(•) is the binary response function of the detectors, which for the sake of simplicity we define here as

s(l ) =    +1, if l ∈ [0, +π), -1, if l ∈ [-π, 0). (18)
In order to complete our statistical model we need to specify also the (density of) probability ρ(l ) of each hidden configuration l ∈ S over the space S to occur in every single realization of the pair of entangled particles. By symmetry considerations this density of probability must be functionally identical from the point of view of both detectors, independently of their relative orientation. Moreover, the condition of 'free-will' demands that the probability of each hidden configuration to occur in any single realization of the experiment cannot depend on the parameterizations of the space S associated to each one of the two detectors. This condition can be precisely stated as:

dλ A ρ(λ A ) = dλ B ρ(λ B ). (19)
It is straightforward to show from [START_REF] Hess | Bell's theorem and the problem of decidability between the views of Einstein and Bohr[END_REF] that this condition is fulfilled if and only if the probability density ρ(l ) is given by:

ρ(l ) = 1 4 |sin(l )| . (20)
We can now compute within the framework of this model the statistical correlations expected between the outcomes of the two detectors as a function of their relative orientation.

The binary outcomes of each one of the two detectors define a partition of the phase space S of all the possible hidden configurations into four coarse subsets, (s

(A) = +1; s (B) = +1) ⇐⇒ λ A ∈ [0, ∆ -Φ) (s (A) = +1; s (B) = -1) ⇐⇒ λ A ∈ [∆ -Φ, π) (s (A) = -1; s (B) = +1) ⇐⇒ λ A ∈ [∆ -Φ -π, 0) (s (A) = -1; s (B) = -1) ⇐⇒ λ A ∈ [-π, ∆ -Φ -π),
where we have assumed without any loss of generality that ∆ -Φ ∈ [0, π). Each one of these four coarse subsets happen a probability given by:

p (+1, +1) = ∆-Φ 0 ρ(λ A ) dλ A = 1 4 (1 -cos(∆ -Φ)) , p (+1, -1) = π ∆-Φ ρ(λ A ) dλ A = 1 4 (1 + cos(∆ -Φ)) , p (-1, +1) = 0 ∆-Φ-π ρ(λ A ) dλ A = 1 4 (1 + cos(∆ -Φ)) , p (-1, -1) = ∆-Φ-π -π ρ(λ A ) dλ A = 1 4 (1 -cos(∆ -Φ)) .
These conditional probabilities reproduce the predictions of quantum mechanics (2):

E(∆, Φ) = p (+1, +1) + p (-1, -1) -p (+1, -1) -p (-1, +1) = -cos(∆ -Φ).
Finally, we notice that in spite of the non-trivial transformation law ( 14) our model complies with the trivial demand that a relative rotation of the measurement apparatus by an angle ∆ followed by a second relative rotation by an angle ∆ results into a final rotation by an angle ∆+∆ . Consider, for example, an initial reference setting T 0 in which the outcomes of the two measurement apparatus are correlated by an amount E = -cos(Φ). The angular coordinates of the hidden configurations with respect to each one of the two measurement devices, λ A and λ B , would be related in this reference setting by the relationship:

λ B = -L(λ A ; -Φ). (21)
We now define a new measurement setting T 1 obtained from the initial setting T 0 by rotating the relative orientation of the two apparatus by an angle ∆. The angular coordinates λ A and λ B defined with respect to this new setting would be related by:

λ B = -L(λ A ; ∆ -Φ). (22)
A third measurement setting T 2 is obtained from the intermediate setting T 1 by rotating the relative orientation of the two apparatus by an additional angle ∆ . In the intermediate setting T 1 , which is now taken as reference to define the second rotation, the pair of particles appears to be in a polarization state characterized by a phase Φ = -∆ + Φ. Hence, the angular coordinates λ A and λ B defined with respect to the setting T 2 would be related by the transformation law:

λ B = -L(λ A ; ∆ -Φ ) = -L(λ A ; ∆ + ∆ -Φ). ( 23
)
By comparison of the transformation law [START_REF] Popescu | Quantum nonlocality as an axiom[END_REF] for the initial setting T 0 and the transformation law [START_REF] White | Preserving entanglement during weak measurement demonstrated with a violation of the Bell-Leggett-Garg inequality[END_REF] for the setting T 2 , we realize that the latter has been obtained from the initial setting by rotating the apparatus by an angle ∆ + ∆, as we had demanded.

In order to complete the description of the Bell experiment we define two new settings T 3 and T 4 , which are obtained, respectively, from T 1 and T 2 by cancelling the phase Φ in the reference setting T 0 . Hence, in these settings the angular coordinates of the hidden configurations with respect to the two measurement apparatus are related by the relationships:

λ B = -L(λ A ; ∆). ( 24
)
and

λ B = -L(λ A ; ∆ + ∆), (25)
respectively. Thus, we could intuitively think about the four settings of the detectors involved in a Bell experiment as corresponding to two possible values for the relative angle ∆ and two possible values for the phase Φ, while they all four share the orientation of one the two detectors, say detector A, taken as reference.

Finally, let us notice that when we substitute the coherent source of pairs of entangled particles (1) by the incoherent classical source (where all the mixed coherent sources are defined with respect to the same arbitrary setting of the two detectors):

μ = 2π dΦ |Ψ Φ Ψ Φ | = | ↑ ↑ | (A) ⊗ | ↓ ↓ | (B) + | ↓ ↓ | (A) ⊗ | ↑ ↑ | (B) , (26)
the broken rotational symmetries are statistically restored and the outcomes of the two measurement devices become uncorrelated for all settings. Only then, when the rotational symmetries are restored, we can safely define separately the orientations of each one of the measurement devices with respect to some external reference frame and, thus, describe the phase space of its possible settings with the help of these two angles (Ω A , Ω B ).

IV. A PROPOSAL FOR AN TEST

The statistical model of hidden configurations described in the previous section reproduces the quantum mechanical prediction for the correlation (2) between the binary outcomes of projective polarization measurements performed on each one of the two particles of every entangled pair, as a function of the angular parameter ∆ -Φ that characterizes the experimental setting. However, with the help of additional weak polarization mesurements the predictions of this statistical model can still be experimentally distinguished from those of the standard framework of quantum mechanics.

Let us consider as before a source of pairs of entangled particles prepared in a Bell state (1) and a pair of measuring devices that test their polarizations through projective measurements at a relative angle ∆ -Φ = π/4, so that the correlation between their binary

outcomes is E A 1 ,B 2 = E(π/4) = -1/ √ 2.
For reasons that will be immediately clear we denote this correlation as E A 1 ,B 2 . This correlation is only very slightly modified if we perform on particle B a very weak polarization measurement before the projective polarization test [START_REF] White | Preserving entanglement during weak measurement demonstrated with a violation of the Bell-Leggett-Garg inequality[END_REF][START_REF] Dressel | Avoiding loopholes with hybrid Bell-Leggett-Garg inequalities[END_REF]. If we design the weak measurement on particle B so that it is oriented along a relative angle ∆ -Φ = -π/4 with respect to the projective polarization measurement on particle A, the correlation between their outcomes in a long sequence of repetitions will be given by E

A 1 ,B 1 = E(-π/4) = -1/ √ 2.
We can now ask ourselves what would be the correlation E B1,B2 between the outcomes of the weak measurement performed on particle B and the projective measurement performed on the same particle later on. According to quantum mechanics their correlation should be

E QM B1,B2 = cos(π/2) = 0, (27)
while in the statistical model presented in the previous section their correlation would be

[33] E SM B1,B2 = 4 π/2 π/4 ρ(λ) dλ - π/4 0 ρ(λ) dλ = = π/2 π/4 |sin(λ)| dλ - π/4 0 |sin(λ)| dλ = = -cos(0) + cos(π/4) -cos(π/2) + cos(π/4) = = √ 2 -1 0.41 = E QM B1,B2 . (28)
The Bell theorem is one of the pillars upon which relies the widely accepted belief that quantum mechanics is the ultimate mathematical framework within which the hypothetical final theory of the fundamental building blocks of Nature and their interactions must be formulated. The theorem proves through an experimentally testable inequality (the Bell inequality) that the predictions of quantum mechanics for the Bell polarization states of two entangled particles cannot be reproduced by any underlying theory of hidden variables that shares certain intuitive features.

In this paper we have shown, however, that these intuitive features include a subtle, though crucial, assumption that is not required by fundamental physical principles and, hence, it is not necessarily fulfilled in the actual experimental setup that tests the inequality.

In fact, the disputed assumption cannot be implemented within the framework of standard quantum mechanics either.

Namely, the proof of the Bell theorem requires the existence of a preferred frame of reference, supposedly provided by a lab, with respect to which the orientations of each one of the two measurement devices can be independently defined for every single realization of the experiment. This preferred frame is required in order to compare the orientations of the detectors in a sequence of repetitions of the experiment, since in every realization each particle's polarization can be tested along a single orientation.

Notwithstanding, the existence of a preferred frame of reference is at odds with Galileo's fundamental principle of relativity and, indeed, it cannot exist when the hidden configurations of the pair of entangled particles spontaneously break the rotational symmetry of the experimental setup under rigid rotations of the two detectors and a non-zero geometric phase accumulates through cyclic gauge transformations. In such a case, in order to compare different realizations of the experiment, we must pick the orientation of one of the detectors as a common reference direction, with respect to which the relative orientation of the second detector is defined. Under these conditions the Bell theorem does not necessarily hold, see [START_REF] Clauser | Proposed experiment to test local hidden variables theories[END_REF], Fig. 2 and Fig. 3.

Following these ideas we explicitly built a model of hidden variables for the Bell states of two entangled particles that reproduces the predictions of quantum mechanics. Further details of the model are discussed in [START_REF] Oaknin | Solving the EPR paradox: an explicit statistical local model of hidden variables for the singlet state[END_REF]. In two additional accompanying papers we have used these same ideas to explicit local models of hidden variables for the GHZ state of three entangled particles [START_REF] Oaknin | Solving the Greenberger-Horne-Zeilinger paradox: an explicit local model of[END_REF] and also for the qutrit [START_REF] Oaknin | Bypassing the Kochen-Specker theorem: and explicit non-contextual model of hidden variables for the qutrit[END_REF].

The derivation of a model of local hidden variables for the entangled states of two or more qubits means that entanglement, the quintessential quantum phenomenon, can be fully described without the quantum formalism. Indeed, the model shows that entanglement can be described in terms of classical statistical concepts, with the help of the well-understood classical notions of curved spaces and gauge degrees of freedom. Thus, the model proves that there are not mysterious fundamental differences between classical and quantum correlations.

Furthermore, the model of hidden variables presented here opens the window to the possible existence of an unexplored physical reality that might underlay the laws of quantum mechanics [START_REF] Ball | Exorcising Einstein's spooks[END_REF] and, thus, it might lead to a whole new area of research in physics in quest for the fundamental laws of this underlying reality. The existence of such a reality was first suggested 85 years ago by Einstein, Podolsky and Rosen through their famous EPR paradox [START_REF] Einstein | Can quantum mechanical description of physical reality be considered complete ?[END_REF][START_REF] Bohm | Quantum Theory 1951[END_REF], but following Bell's arguments it had been thought that an underlying reality was incompatible with quantum mechanics [START_REF] Bell | On the problem of hidden variables in quantum mechanics[END_REF][START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF].

Finally, we wish to notice that our model of hidden variables is built upon fundamental physical concepts shared by the formalism of General Relativity and, thus, it might eventually lead to a unified description of quantum phenomena and gravitation.

  have the same sign, while the second term is non-zero only when they have opposite signs. The CHSH inequality (3) is then obtained by averaging (4) over the whole space S of all possible hidden configurations, since dλ ρ(λ) s (A)

FIG. 2 :FIG. 3 :

 23 FIG. 2: Two descriptions of the experimental setup required for testing the Bell inequality. In the description above the lab frame is taken to be fixed, while in the description below the orientation of detector A is taken to be fixed. The relative angle between the two detectors is set at four possible values ∆ 1 , ∆ 2 , ∆ 1 -δ and ∆ 2 -δ. When considering models in which the hypothetical hidden configurations of the pairs of entangled particles spontaneously break the symmetry under rigid rotations of the orientations of the two measurement devices, only the latter choice allows to properly compare the four different settings.

  and therefore, E A,B ( a, b) + E A,C ( a, c) ≤ 1 + E B,C ( b, c). This proof, nonetheless, does not hold for the random game on the right hand side, since the orientation of a vector λ parallelly transported along the closed contour of the triangle ABC gets rotated by a geometric phase α due to the curvature of the sphere. In fact, in the game on the right the three bipartite correlations are constrained by the inequality E A,B ( a, b) + E A,C ( a, c) ≤ 1 + E B,C ( b, Rα c), where Rα c denotes the rotation of vector c by an angle α.

p

  (a, b|A, B), where a = ±1 and b = ±1 are the two possible outcomes at each one of the two detectors and A = ±1 and B = ±1 describe two possible choices for the setting of each one of the two detectors. It is then proven that these conditional probabilities cannot be obtained in terms of a local model of hidden variables, defined by its configuration space λ ∈ S, its density of probability ρ(λ) and its local response functions a = f (λ, A), b = f (λ, B) [2].

parameter D = 1 , 2 , 3 , 4 .

 1234 Each input value corresponds to a given relative orientation of the two devices. The new model simply states that when the devices are set at D = 1, 2, 3 their outcomes must have the same sign, and when they are set at D = 4 their outcomes must have opposite signs. Obviously, this latter model is not necessarily in contradiction with an underlying local model of hidden variables.

FIG. 4 :

 4 FIG. 4: Plot of the transformation law λ → λ = L(λ; ∆) for ∆ = π/3 (solid line), compared to the corresponding linear transformation (dotted line).

VI. ACKNOWLEDGEMENTS

This manuscript has been released as a pre-print at https://arxiv.org/abs/1912.06349, D.H. Oaknin [34]. Its author D.H. Oaknin was employed by the company RAFAEL Advanced Defense Systems.


domain: phys sdu math

text:
David H Oaknin
email: d1306av@gmail.com

Solving the Greenberger-Horne-Zeilinger paradox: an explicit model of local hidden variables for the GHZ quantum state

The Greenberger-Horne-Zeilinger (GHZ) version of the Einstein-Podolsky-Rosen (EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of any model of local hidden variables in which the observers are free to choose their own experimental settings. In this paper, we show, however, that the GHZ argument implicitly relies on a subtle, though crucial, assumption that is not required by fundamental physical principles, and which had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles, as well as the orientations of the measurement devices that test them, can be defined. We further note that such an absolute frame of reference may not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping the disputed requirement we are able to build a model of local hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions. Our model, thus, completes -in the EPR sense -the quantum description of the GHZ state.

Introduction

The inability to accommodate the seemingly trivial notions of locality and physical realism within the current interpretation of the quantum mechanical wavefunction is at the core of a long lasting debate about the foundations of the quantum theory and the role played by measurements, whose origins go back to the formulation of the renowned Einstein-Podolsky-Rosen (EPR) paradox eighty five years ago 1 . Solving these key issues would require to develop a description of quantum phenomena in terms of a statistical model of local hidden variables. Nonetheless, according to the current wisdom, such a description is not possible in so far as we insist on keeping the notion, also seemingly trivial, that the observers' choice of their measurement settings is not constrained by the actual hidden configuration of the observed system (free-will). Indeed, several fundamental theorems state that generic models of hidden variables that share certain intuitive features cannot fully reproduce the predictions of quantum mechanics [2][3][4][5][6][7][START_REF] Greenberger | Bell's Theorem, Quantum Theory, and Conceptions of the Universe[END_REF][START_REF] Cabello | [END_REF][10][11] . Moreover, carefully designed experimental tests have consistently confirmed the predictions of the quantum theory and, thus, have ruled out all these generic models of hidden variables [12][13][14][15][16][17][18][19][20][21] .

The best known among these theorems is the Bell theorem [2][3][4][5] , which proves that such generic models of hidden variables cannot reproduce the statistical correlations predicted by quantum mechanics for the outcomes of long sequences of strong polarization measurements performed along certain relative directions on pairs of entangled qubits.

The Greenberger-Horne-Zeilinger version of the Bell theorem 8 is an even more conclusive proof of the limitations of these generic models of hidden variables, since it proves that such models cannot reproduce even single outcomes of strong spin polarization measurements performed along certain relative directions on three or more entangled qubits prepared in the so-called GHZ state,

GHZ⟩ ≡ ↑ ↑ ..... ↑ ⟩ + ↓ ↓ ..... ↓ ⟩ √ 2 . (1)
However, in a series of recent papers [22][23][24] we have shown that the proof of the Bell theorem crucially relies on a subtle assumption that is not required by fundamental physical principles. Namely, we noted that the proof implicitly assumes the existence of an absolute angular frame of reference with respect to which we can define the polarization properties of the hypothetical hidden configurations of the pairs of entangled particles as well as the orientations of the measurement devices that test them. Furthermore, we showed that such an absolute frame of reference may not exist if the hidden configurations spontaneously break the gauge rotational symmetry along an otherwise arbitrary direction.

An illustrative example of the absence of an absolute frame is described in Fig. 3 in reference 22 . Let first consider a Bell-type game played between three parties located at the vertices of a triangle drawn on a plane. At the start of the game, each party sets at his/her vertex a reference unit vector contained within the plane. A long sequence of unit vectors randomly oriented within the plane are then produced at the center of each one of the three edges of the triangle and sent to the two parties located at their respective ends. Upon receiving a sampled random vector, each party compares its orientation to the local reference unit vector and produces a binary outcome, either +1 or -1, according to a deterministic response function. In this game, the affine structure of the euclidean plane allows comparing at once the relative orientations of the reference unit vectors at the three vertices, as well as the sampled random unit vectors, and, thus, it defines an "absolute frame of reference". It is then straightforward to derive the Bell inequality for the pairwise correlations between the binary outcomes of the parties. However, it can be readily seen that such an "absolute frame of reference" does not exist if we consider a similar Bell-type game played between parties located on the surface of a sphere: a tangent vector parallel-transported over a closed-loop drawn on the sphere may accumulate a non-zero geometric rotation phase due to a holonomy. Therefore, even though any two parties can calibrate and agree on a common frame of reference to describe the relative orientations of their reference unit vectors, as well as the orientation of the random vectors shared between them, there does not exist a common frame of reference upon which all the three parties can agree at once. In order to compare (and maybe constraint) the pairwise correlations that can be attained in the latter game, it is necessary to set the reference unit vector of one of the parties as a fixed common frame by taking advantage of the gauge degrees of freedom involved in the problem.

Following this insight, we built a statistical model of local hidden variables that fully reproduces the predictions of quantum mechanics for the Bell states of two entangled qubits, while complying with all the required symmetry demands and the hypothesis of 'free-will' [22][23][24] . Thus, our model completes the description of these quantum states in the sense advocated by Einstein, Podolsky and Rosen 1 . The hidden configuration of the pair of entangled qubits is represented in this model by a randomly oriented vector, whose orientation is described with respect to two reference unit vectors representing the settings of two detectors. The holonomy appears when tracking the orientation of the hidden vectors for the three or four possible settings of the detectors considered in the setup of the Bell experiment. This holonomy is allowed because any hidden configuration of the entangled qubits can be tested for only one of the possible settings of the two detectors.

In this paper we develop these ideas to build a model of local hidden variables for the GHZ state of three entangled qubits. The paper is organized as follows. In Section 2 we review the argument put forward by Greenberger, Horne and Zeilinger as a proof of the impossibility to reproduce the quantum mechanical predictions for the GHZ state within the framework of any model of local hidden variables. In Section 3 we introduce a simple explicit model of hidden variables that overcomes this argument. In Section 4 we extend this model and discuss it in detail. Our conclusions are summarized in section 5.

The GHZ paradox

The Greenberger-Horne-Zeilinger spin polarization state of three entangled qubits is described by the quantum wavefunction:

Π⟩ Φ = 1 √ 2 ↑⟩ (A) ⊗ ↑⟩ (B) ⊗ ↑⟩ (C) + e iΦ ↓⟩ (A) ⊗ ↓⟩ (B) ⊗ ↓⟩ (C) ,
where { ↑⟩, ↓⟩} denotes a basis of single particle spin polarization eigenstates along its locally defined Z-axis. In this state all the three outcomes in every single event of a long sequence of strong spin polarization measurements performed on each one of the three particles along their corresponding Z-axes must be consistently equal, either

S (A) Z (n) = S (B) Z (n) = S (C) Z (n) = +1, or S (A) Z (n) = S (B) Z (n) = S (C) Z (n) = -1,
for all n ∈ {1,...,N}, with each one of the two possibilities happening with a probability of 1/2.

In fact, in the GHZ state (2) the expected average values of long sequences of strong spin polarization measurements performed along any arbitrary directions

Ω (A) α , Ω (B) β , Ω (C) γ
in the XY-planes orthogonal to the local Z-axes are equal to zero: of the measurement devices that test them.

⟨S (A) Ω α (n)⟩ n∈N = ⟨S (B) Ω β (n)⟩ n∈N = ⟨S (C) Ω γ (n)⟩ n∈N = 0, (2) 2
as well as their two-particles correlations:

⟨S (A) Ω α (n) ⋅ S (B) Ω β (n)⟩ n∈N = ⟨S (B) Ω β (n) ⋅ S (C) Ω γ (n)⟩ n∈N = ⟨S (C) Ω γ (n) ⋅ S (A) Ω α (n)⟩ n∈N = 0. (3)
Notwithstanding, the three-particles correlation is non-zero, in general, and given by:

⟨S (A) Ω α (n) ⋅ S (B) Ω β (n) ⋅ S (C) Ω γ (n)⟩ n∈N = cos ∆ Ω (A) α + ∆ Ω (B) β + ∆ Ω (C) γ + Φ , where ∆ Ω (A) α , ∆ Ω (B) β and ∆ Ω (C) γ
describe the relative orientations of each one of the measurement devices with respect to some implicit local reference directions labelled as X-axes, see Fig. 1.

In particular,for Φ = 0 the following four relationships follow:

S (A) X (n) ⋅ S (B) X (n) ⋅ S (C) X (n) = +1, n = 1,....,N S (A) X (m) ⋅ S (B) Y (m) ⋅ S (C) Y (m) = -1, m = 1,....,M S (A) Y (k) ⋅ S (B) X (k) ⋅ S (C) Y (k) = -1, k = 1,....,K S (A) Y (l) ⋅ S (B) Y (l) ⋅ S (C) X (l) = -1, l = 1,....,L, (4)
for any four sequences of strong measurements performed along directions (X,X,X) , (X,Y,Y ), (Y,X,Y ) and(Y,Y,X).

These four relationships (4) lie at the core of the Greenberger-Horne-Zeilinger paradox [START_REF] Greenberger | Bell's Theorem, Quantum Theory, and Conceptions of the Universe[END_REF] . On one hand, these relationships imply that we can gain certainty about the polarization properties of any of these three particles without in any sense disturbing them. Thus, according to the notion introduced by Einstein, Podolsky and Rosen 1 , these polarization properties are elements of reality whose values must be set at the time when the three entangled particles are produced. On the other hand, this notion seems to be inconsistent: by multiplying the last three equations in (4) and assuming that all polarization components must take values either +1 or -1, we would obtain that

S (A) X (n) ⋅ S (B) X (n) ⋅ S (C) X (n) = -1, n = 1,....,N (5)
which is in contradiction with the first one.

This argument is widely considered as the most clear-cut evidence against the possibility of giving the wavefunction (2) an statistical interpretation within the framework of a local model of hidden configurations, in which the observers are free to choose the setting of their measurements. Figure 2. By symmetry considerations the orientation of one of the measurement devices, say A, can always be defined as a local X-axis for every one of the repetitions of the experiment. Moreover, the orientation of a second measurement device, say B, can also be always defined as a local X-axis since any rotation in it can be accounted for through the definition of the phase Φ that characterizes the source of the photons. In fact, as explained in paragraph 3, this is strictly necessary in order to properly define the quantum state (2). Thus, the experimental setting of the three measurement devices is described by a single angle ∆, while the expected correlation between their outcomes depends only on the linear combination ∆ + Φ.

The paradox revisited

The above argument crucially relies on the implicitly assumed existence of an absolute angular frame of reference, with respect to which the polarization properties of the hidden configurations of the triplets of entangled particles, as well as the orientations of the measurement devices that test them, can be defined. In such an absolute frame of reference all the polarization components of each possible hidden configuration must take a binary value, either +1 or -1, and relationships (4) immediately follow. However, as we already did notice i previous works [22][23][24] , the existence of such an absolute angular frame of reference is not required by fundamental physical principles. Indeed, the polarization properties of the hidden configurations may only be properly defined with respect to the reference directions set by the orientation of the corresponding measurement devices. Furthermore, the orientation of the measurement devices can only be properly defined with respect to each other, while their global orientation is actually a spurious gauge degree of freedom. Namely, the correlation between the binary outcomes of the three measurement devices,

⟨S (A) Ω α (n) ⋅ S (B) Ω β (n) ⋅ S (C) Ω γ (n)⟩ n∈N = cos(∆ + Φ), (6)
is described by a single physical degree of freedom, the angle ∆ + Φ, where the angle ∆ is measured with respect to a set of locally defined X-axes at the sites of each one of the three measurement devices, for which the correlation is given by:

⟨S (A) X (n) ⋅ S (B) X (n) ⋅ S (C) X (n)⟩ n∈N = cos(Φ), (7)
and, in particular, for Φ = 0 is given by

⟨S (A) X (n) ⋅ S (B) X (n) ⋅ S (C) X (n)⟩ n∈N = +1. (8)
Actually, condition (7) defines the notion of parallel directions at the sites of each one of the three measurement devices. All sets of axes for which this condition is fulfilled are physically indistinguishable through measurements performed on triplets of entangled particles in the GHZ state and, hence, all such sets of axes are gauge equivalent. It is worth noticing at this point that the single-particle eigenstates ↑⟩ (A,B,C) , ↓⟩ (A,B,C) of the locally defined operators σ

(A,B,C) Z
are defined only up to a phase (as any normalized eigenvector of any operator) and, hence, the phase Φ in (2) is not, in principle, properly defined yet. In order to properly define this phase, it is necessary to set a reference setting (7) of the three measurement devices and experimentally obtain the threesome correlation. With respect to this reference setting of parallel directions it is possible to properly define a subsequent rotation of any one of the devices by an angle ∆, see Fig. 2.

Furthermore, also the polarization properties of the hidden configurations of the triplets of entangled particles may be properly defined only with respect to the local reference directions set by the three measurement devices. That is, the actual number described in 25 .

Before closing this section, it is worth noticing that eq. (16-17) is a coordinates transformation that relates the orientations of the hidden configurations of the triplets of entangled qubits with respect to the reference directions set by two different detectors oriented at an angle ∆ relative to each other. Hence, eq. (16-17) does not introduce any non-local interaction between the detectors. In order to clarify this issue consider a source that produces pairs of macroscopic arrows parallel to each other and randomly oriented within a locally defined XY plane. The twin arrows are then parallel-transported in opposite directions along the Z axis towards two distant detectors, each one of them consisting of an arrow that can also be arbitrarily oriented within the local XY plane. For every pair of twin arrows the following relationship must hold:

ω ′ = ω -∆, (31)
where ω and ω ′ are the relative angles between the orientations of the incoming arrows and their corresponding detectors, and ∆ is the relative angle between the two detectors. This relationship (31) does obviously not introduce any non-local interaction between the detectors, since it is dictated by the euclidean structure of the macroscopic space, and, therefore, it is fulfilled no matter who decides how to orient the detectors or when these decisions are taken. Eq. (16-17) is nothing but a non-linear generalization of the euclidean relationship (31), and it simply means that the entangled particles might carry with them a non-euclidean metric. In this sense, it is useful to think about eq. (16-17) as somehow similar to the Lorentz transformation that relates, for example, the frequencies ν and ν ′ of a signal emitted by a source towards two detectors moving with relative velocity V ,

ν ′ = L(ν;V ). (32)
Obviously, this non-linear relationship neither violates causality, since it is dictated by the Minkowski metric of space-time from which the very notion of causality stems. Finally, let us remind again that the non-linearity of the coordinates transformation eq. (16-17) introduces an holonomy (30) only in the un-physical gauge degrees of freedom, while the physical degrees of freedom that describe the settings of the detectors comply with all the usual symmetry demands.

Discussion

We have shown in this paper that the argument behind the renowned GHZ paradox crucially relies on an implicit assumption that is not required by fundamental physical principles. Hence, the argument can be overcome by giving up this unnecessary requirement. Namely, the argument put forward by Greenbereger, Horne and Zeilinger thirty years ago 8 implicitly assumes that there exists an absolute angular frame of reference with respect to which we can define the polarization properties of the hypothetical hidden configurations of the entangled qubits, as well as the orientations of the measurement devices that test them. However, we have remarked in this paper that in order to properly define the phase Φ that characterizes the state (2) of the triplets of entangled qubits it is necessary to fix a reference setting of the measurement devices that test their polarizations. Thus, only a subsequent relative rotation ∆ of one of the devices with respect to the fixed reference setting is a properly defined physical magnitude, see Fig. 2. Moreover, we have noticed that the polarization properties of each one of the qubits may be properly defined only with respect to the orientation of the measurement device that tests it. With these observations in mind we have built an explicit model of local hidden variables that complies with the 'free-will' assumption and reproduces the predictions of quantum mechanics for the GHZ state of three entangled qubits. The model, thus, completes -in the EPR sensethe quantum description of the GHZ state. This model closely resembles the model of hidden variables for the Bell polarization states of two entangled qubits that we recently described [22][23][24] .

Acknowledgments

I wish to thank an anonymous referee for useful comments regarding the statistical model described in this paper. I state that there is no conflict of interests.

FFigure 1 .

 1 Figure 1. The GHZ argument implicitly requires the existence of an absolute frame of reference with respect to which it is possible to describe the polarization properties (±1,±1,±1) of the hypothetical hidden configurations of the triplets of photons as well as the orientations Ω (A) α , Ω (B) β , Ω (C) γ

value s (A) Ω (Ω α ,ω) of the polarization component of, say, particle A along some direction Ω may be, in general, a function of the reference direction Ω α set by the measurement apparatus of observer A (and, of course, also of the coordinate ω ∈ S that labels the hidden configuration in which the system of three entangled particles occurs). This dependence does not conflict with the principles of locality and physical realism, which only demand that the value of the polarization components of particle A cannot depend on the orientations of the reference directions Ω β , Ω γ along which observers B and C choose to test their particles. Therefore, we must not restrict our models within the constraint that all polarization components of either one of the particles must take a binary value, either +1 or -1: only the polarization component of each one of the particles along the reference direction set by the orientation of the corresponding measurement device must take a binary value. That is, on all possible hidden configurations of the triplet we must have:

but the polarization components along any other directions must not necessarily take either one of these two values. Indeed, the only experimental access that we can have to the spin polarization components along these other directions is through weak measurements, whose outcome can have absolute values larger and smaller than one and may even be complex 26 . In fact, weak values of physical observables are complex numbers independently of the linear dimension of the Hilbert space of the described quantum system. Therefore, it is crucial to realize that in order to obtain a meaningful description of the system we must be careful to compare magnitudes defined with respect to the same reference directions. For example, we can state that with respect to a set of parallel reference directions X (A) , X (B) and X (C) defined by condition (8), the polarization components of the particles along the orthogonal directions Y (A) , Y (B) , Y (C) take values either +i or -i, according to the relationship:

X (X,ω) = ±1, see Fig. 3. Then, the four constraints (4) become trivially identical,

In other words, the argument put forward by Greenberger, Horne and Zeilinger as a proof of the impossibility to reproduce the predictions of quantum mechanics for the GHZ state within the framework of a model of local hidden variables can be overcome by realizing that there does not necessarily exist an absolute frame of reference with respect to which the hidden polarization properties of the entangled particles can be defined and, in consequence, allowing their actual values depend on the reference direction with respect to which they are described.

A statistical model for the GHZ state

In this section we build and discuss in detail an explicit statistical model of local hidden variables for the GHZ state of three entangled qubits. The model complies with the 'free-will' assumption and reproduces the quantum mechanical predictions for the average values and correlations of long sequences of strong spin polarization measurements performed on the three qubits along any three arbitrary directions.

Our statistical model consists of infinitely many possible hidden configurations continuously distributed over the unit circle S 1 , a well-defined density of probability for each one of these configurations to occur and locally defined binary response functions that specify the outcomes that each one of these hidden configurations would produce in each one of the three measurement devices as a function of their orientations.

We define two sub-populations within the space of all possible hidden configurations, which we label as η = ±1, each one occurring with probability of 1 2. These two sub-populations correspond, respectively, to the two possible outcomes of the measurement performed on one of the particles, say, particle A. That is,

5/10 The device that measures the polarization of particle B along an arbitrary direction orthogonal to its locally defined Z-axis fixes a reference frame of angular coordinates ω B ∈ [-π,π) over the circle S 1 . We assume that the density of probability for each one of the hidden configurations to occur is given by

and define the outcome of the measurement on particle B as:

with:

Similarly, the device that measures the polarization of particle C along some other arbitrary direction Ω orthogonal to its locally defined Z-axis sets its own frame of angular coordinates ω C ∈ [-π,+π) over the circle S 1 . By symmetry considerations we demand that the outcome of this measurement is described by the same response function:

). The two sets of angular coordinates ω B and ω C are related by the relationship:

where

and

6/10

with

and the function y = arc-cos(x) is defined in its main branch, such that y ∈ [0,π] while x ∈ [-1,+1]. The parameter ∆ ≡ ∆ + Φ in this transformation law denotes the orientation of the measurement setting, as defined in (6,7).

It is straightforward to check that the probability density ( 13) remains invariant under the coordinates transformation (18,19), since

and

In fact, these equalities state in precise terms that the probability of each hidden configuration to occur does not depend on the orientation of the reference direction chosen by the observers to describe their particles or, in other words, that our model complies with the requirements of 'free-will'.

We can now define a partition of the circle S 1 into four disjoint regions,

as follows:

where we have assumed without any loss of generality that 0 ≤ ∆ ≤ π.

In each one of these four segments, the two measurements are fully correlated or anti-correlated:

Ω (ω C ,η)

Ω (ω C ,η)

• If η = -1,

Ω (ω C ,η)

Ω (ω C ,η)

It is straighforward to notice that

where µ(⋅) denotes the normalized measure over the circle according to the probability density distribution (13). Hence,

• Over the sub-population of states with η = +1, ⟨S

• Over the sub-population of states with η = -1,

Therefore, over the whole population the two measurements are completely uncorrelated,

since each one of the two sub-populations η = +1 and η = -1 happens with probability 1 2. Furthermore, the three-particles correlation is given by:

which reproduces the quantum mechanical prediction (6) for the GHZ state.

Let us remark that in the model that we have described here, like in the quantum formalism, the orientation of two of the three measurement devices sets a reference frame with respect to which the orientation of the third device is described, see Fig. 2. Hence, it does not make sense to compare two different orientations for the reference setting. The ultimate reason is the fact that the set of angular coordinates over the circle S 1 acquires a non-zero geometric phase α ≠ 0,π through certain cyclic transformations (18,19) due to an holonomy:

Such a geometric phase is allowed precisely because the orientation of the reference setting is, as we have noticed above, a gauge degree of freedom and each triplet of entangled qubits can be tested at only one setting of the detectors.

The possible appearance of a geometric phase in closed loops (in some parameter space) of gauge transformations is well-known, also in classical physics. A particularly beautiful example is the gauge theory of swimming at low Reynolds