Are models of local hidden variables for the singlet polarization state necessarily constrained by the Bell inequality?
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Are models of local hidden variables for the singlet polarization state necessarilyconstrained by the Bell inequality ?
David H. Oaknin ∗ Rafael Ltd, IL-31021 Haifa, Israel
The Bell inequality is thought to be a common constraint shared by all models of local hiddenvariables that aim to describe the entangled states of two qubits. Since the inequality is violatedby the quantum mechanical description of these states, it purportedly allows distinguishing in anexperimentally testable way the predictions of quantum mechanics from those of models of localhidden variables and, ultimately, ruling the latter out. In this paper, we show, however, that themodels of local hidden variables constrained by the Bell inequality all share a subtle, though crucial,feature that is not required by fundamental physical principles and, hence, it might not be fulfilledin the actual experimental setup that tests the inequality. Indeed, the disputed feature neithercan be properly implemented within the standard framework of quantum mechanics and it is evenat odds with the fundamental principle of relativity. Namely, the proof of the inequality requiresthe existence of a preferred absolute frame of reference (supposedly provided by the lab) withrespect to which the hidden properties of the entangled particles and the orientations of each oneof the measurement devices that test them can be independently defined through a long sequenceof realizations of the experiment. We notice, however, that while the relative orientation betweenthe two measurement devices is a properly defined physical magnitude in every single realization ofthe experiment, their global rigid orientation with respect to a lab frame is a spurious gauge degreeof freedom. Following this observation, we were able to explicitly build a model of local hiddenvariables that does not share the disputed feature and, hence, it is able to reproduce the predictionsof quantum mechanics for the entangled states of two qubits. The Bell theorem is one of the pillars upon which re-lies the widespread belief that quantum mechanics is theultimate mathematical framework within which a hypo-thetical final theory of the fundamental building blocksof Nature and their interactions should be formulated.The theorem states through an experimentally testableinequality (the Bell inequality) that none theory of hid-den variables that shares certain intuitive features canreproduce the predictions of quantum mechanics for theBell states of two entangled qubits [1]. In fact, since thesepredictions have been experimentally confirmed beyondany reasonable doubt [2, 3] all said generic models of hid-den variables are currently ruled out.In a Bell experiment a source emits pairs of particleswhose polarizations are prepared in an entangled state: | Ψ Φ i = 1 √ (cid:16) | ↑i ( A ) | ↓i ( B ) − e − i Φ | ↓i ( A ) | ↑i ( B ) (cid:17) , (1)where {| ↑i , | ↓i} ( A,B ) are single-particle eigenstates ofPauli operators σ ( A,B ) Z along locally defined Z-axes,and two widely separated detectors oriented along in-dependently set directions within the correspondingXY-planes test them. Upon detection each particlecauses a binary response of its detector, either +1 or −
1. Thus, each detected pair of entangled particlesproduces an outcome in the space of possible events { ( − , − , ( − , +1) , (+1 , − , (+1 , +1) } . We refer toeach detected pair as a single realization of the exper-iment. The experiment consists of a long sequence ofrealizations along which each one of the detectors can beswitched between two possible settings, which we shall denote as Ω A and Ω ′ A for detector A and Ω B and Ω ′ B fordetector B, defined with respect to local lab frames.At the end of all these runs the outcomes recordedby the two detectors are compared and their statisticalcorrelations computed at each one of the available set-tings. Quantum mechanics predicts, and experimentaltests confirm, that these correlations are given by E (∆ − Φ) = − cos (∆ − Φ) , (2)where ∆ is the relative angle between the orientations ofthe two measuring devices and the phase Φ is defined by(1). Therefore, it can be readily check that | E (+ π/
4) + E ( − π/
4) + E ( − π/ − E ( − π/ | = 2 √ . (3)On the other hand, the CHSH version of the Bell inequal-ity states that for all models of hidden variables thatshare certain intuitive features the following inequality | E (∆ ) + E (∆ ) + E (∆ − δ ) − E (∆ − δ ) | ≤ , (4)must hold for any set of values (∆ , ∆ , δ ) and, in particu-lar, for ∆ = + π/
4, ∆ = − π/ δ = + π/ {| ↑i , | ↓i} ( A,B ) of single-particle eigenstates of the Paulioperators σ ( A,B ) Z . Since these eiegenstates are defined upto a global phase, the phase Φ in (1) could not be prop-erly defined with respect to a lab frame of reference. Inorder to properly define this phase and, hence, the sourceof entangled particles we must choose an arbitrary refer-ence setting of the two measurement devices. The phaseΦ is then defined with respect to this reference setting ofthe detectors with the help of the measured correlationsbetween their outcomes, E = − cos(Φ). We can then usethis reference setting to properly define also a relativerotation ∆ of the orientations of the two apparatus. Itis interesting to notice at this point that the definitionsof the phase Φ and the angle ∆ do not rely at all on thequantum formalism and, therefore, we shall use the samedefinitions later on to build our model of hidden variablesto describe the experiment. It is also important to noticethat since we must use an otherwise arbitrary setting ofthe detectors as a reference in order to properly describethe experiment we cannot in any proper sense define theirglobal rigid orientation: it is an spurious gauge degree offreedom.Nevertheless, the proof of the CHSH inequality doesnot properly recognize this spurious gauge degree of free-dom. It proceeds as follows. Let us label as { λ } λ ∈S the space of all possible hidden configurations of the pairof entangled particles and let ρ ( λ ) be the (density of)probability of each one of them to occur in every singlerealization of the experiment. It is then assumed that itis possible to assign to each possible configuration λ ∈ S a 4-tuple of binary values (cid:16) s ( A )Ω A ( λ ) , s ( A )Ω ′ A ( λ ) , s ( B )Ω B ( λ ) , s ( B )Ω ′ B ( λ ) (cid:17) ∈ {− , +1 } (5)to describe the outcomes that would be obtained at each one of the measurement devices in case that their ori-entations were set along each one of the two availablesettings - Ω A , Ω ′ A and Ω B , Ω ′ B - defined with respect tolocal lab frames. Under this assumption, which we shallrefer to as the Bell assumption, it is straightforward toshow that for all possible configurations λ ∈ S , s ( A )Ω A ( λ ) · (cid:16) s ( B )Ω B ( λ ) + s ( B )Ω ′ B ( λ ) (cid:17) ++ s ( A )Ω ′ A ( λ ) · (cid:16) s ( B )Ω B ( λ ) − s ( B )Ω ′ B ( λ ) (cid:17) = ± , (6)since the first term equals either +2 or − s ( B )Ω B ( λ )and s ( B )Ω ′ B ( λ ) have the same sign and equals 0 when theyhave opposite signs, while the second term equals 0 when s ( B )Ω B ( λ ) and s ( B )Ω ′ B ( λ ) have the same sign and equals either+2 or − S and noticingthat each one of the four terms in the integrand producesone of the required correlations: − ≤ Z dλ ρ ( λ ) · h s ( A )Ω A ( λ ) · (cid:16) s ( B )Ω B ( λ ) + s ( B )Ω ′ B ( λ ) (cid:17) ++ s ( A )Ω ′ A ( λ ) · (cid:16) s ( B )Ω B ( λ ) − s ( B )Ω ′ B ( λ ) (cid:17)i ≤ +2 . The Bell assumption (5) intuitively seems a trivial fea-ture of any model of local hidden variables, which seem-ingly simply states that the response of each detector toeach possible hidden configuration λ ∈ S does not de-pend on the orientation chosen for the other detector.Indeed, this assumption would be indisputable if eachparticle of every single entangled pair could be tested atonce along the two available orientations of its detector.However, since each particle of every entangled pair canbe actually tested along only one possible orientation ofits detector it is crucial to identify the actual physicaldegrees of freedom of the experimental set-up. In fact,as we shall now show the assumption (5) is not requiredby fundamental physical principles and, therefore, mightnot be fulfilled in the actual experiments that test theBell’s inequality.In general, we should allow for each one of the twodetectors to define its proper set of coordinates over thespace S of possible hidden configurations. Thus, let usdenote as λ A and λ B the two sets of coordinates asso-ciated to detectors A and B, respectively. Since thesetwo sets of coordinates parameterize the same space ofhidden configurations S there must exist some invertibletransformation that relates them: λ B = −L ( λ A ; ∆ − Φ) , (7)which may depend parametrically on the relative angle∆ − Φ between the orientations of the two detectors. Thistransformation must fulfill the constraint dλ A ρ ( λ A ) = dλ B ρ ( λ B ) , (8)which simply states that the probability to occur of ev-ery hidden configuration must remain invariant under achange of coordinates, while the density of probability ρ ( l ) , l ∈ [ − π, π ), is functionally invariant for the twosets of coordinates. It can be readily shown [5, 6] thatwhen we define ρ ( l ) = 14 | sin( l ) | (9)the constraint (8) directly leads to the correlation (2).Furthermore, the transformation law (7) complies withthe trivial demand that a relative rotation of the measur-ing devices by an angle ∆ followed by a second relativerotation by an angle ∆ ′ results into a final rotation byan angle ∆ + ∆ ′ with respect to the original referencesetting. This can be readily shown as follows. Consider,for example, a setting in which the angular coordinatesof the hidden configurations with respect to each one ofthe two measurement devices, λ A and λ B , are related bythe transformation λ B = −L ( λ A ; ∆) , (10)Thus, with respect to this setting the source is describedby a phase Φ = − ∆. Hence, by adding a relative angle∆ ′ to the relative orientation of the detectors we obtaina new setting in which the sets of coordinates associatedto the two detectors are related by the transformation λ ′ B = − L ( λ A ; ∆ ′ − Φ) = − L ( λ A ; ∆ ′ + ∆) . (11)By comparing with the transformation law (10) we realizethat the final setting corresponds to a relative angle of∆ + ∆ ′ with respect to the original reference setting, aswe had demanded.Finally, since the global rigid orientation of the twodevices is an spurious gauge degree of freedom, the set ofcoordinates over the space of hidden configurations mayaccumulate a non-zero geometric phase through a cyclictransformation:( −L ∆ ) ◦ ( −L ∆ − δ ) ◦ ( −L ∆ − δ ) ◦ ( −L ∆ ) = L α = I . (12)The appearance of geometric phases in physical modelsinvolving gauge symmetries is a well-known phenomenon[7] and, therefore, we should not rule out the possi-bility that it also occurs in models of hidden variablesthat describe quantum phenomena. Under such circum-stances there does not exist a common set of coordinatesin which we can jointly define binary responses for thetwo detectors in each one of their two available orienta-tions. Hence, the Bell’s assumption (5) does not holdand, therefore, such models are not constrained by theinequality (6). In the presence of a non-zero geometric phase we mustchoose the orientation of one of the detectors as a com-mon reference direction in order to compare the four ex-periments involved in the CHSH inequality. Thus, in-stead of (6) we should have written: s ( λ A ) · [ s ( λ B ) + s ( λ ′ B ) + s ( λ ′′ B ) − s ( λ ′′′ B )] , (13)where we have assumed that the two detectors share thesame universal binary response function s ( l ) , l ∈ [ − π, π ),and we have now defined λ B = −L ( λ A ; ∆ ) , (14) λ ′ B = −L ( λ A ; ∆ ) , (15) λ ′′ B = −L ( λ A ; ∆ − δ ) , (16) λ ′′′ B = −L ( λ A ; ∆ − δ ) . (17)It is obvious that the new expression (13) is no longerconstrained to equal either +2 or − A of detector A the polar-ization properties of its particle along any other directionΩ ′ A would not be binary. Of course, in the description re-ferred to orientation Ω ′ A the polarization properties alongthis direction must be binary, while the properties alongΩ A would not be necessarily so. In other words, thehidden polarization properties of the entangled particlesmay not be scalar magnitudes under a rotation of thedetector that test them, much like the components of anelectromagnetic field are not scalar magnitudes under thetransformation that connects two observers related by aboost.Finally, it is worth to stress that eq. (7), which relatesthe sets of coordinates defined by each one of the twodetectors in a Bell experiment, does not introduce anynon-local interaction between them. In order to clarifythis issue let us consider a source that produces pairs ofparallel macroscopic arrows randomly oriented along alocally defined XY plane. The arrows are then parallelytransported in opposite directions along the Z axis totwo distant detectors, each one of them consisting of anarrow that can also be arbitrarily oriented in the XY plane. For every pair of arrows the following constraintis fulfilled: θ A = θ B − Θ , (18)where θ A is the relative angle between the orientationof detector A and its incoming arrow, θ B is the relativeangle between the orientation of detector B and its in-coming arrow and Θ is the relative angle between theorientations of the two detectors. The constraint (18) isdictated by the euclidean structure of the macroscopicspace. Thus, it is fulfilled no matter who decides how toorient the detectors or whenever the decisions are takenand, obviously, it does not introduce any non-local in-teraction between the detectors. Eq. (7) is nothing buta non-linear generalization of the euclidean relationship(18), and it simply means that the entangled particlesmay carry with them a non-euclidean metric.In summary, we have shown that the Bell theoremholds only for a very particular class of models of lo-cal hidden variables that share a subtle, though crucial,feature. This feature, nonetheless, is not required byfundamental physical principles and it is not necessarilyfulfilled in the actual experimental setup that tests theinequality. Indeed, following this observation we have presented in [5, 6] an explicitly local statistical model ofhidden variables that does not share the said feature andreproduces the predictions of quantum mechanics for theBell states. ∗ [email protected][1] J.S. Bell, Physics , 195-200 (1964).[2] B. Hensen et al , Nature , 682 (2015).[3] H. Wiseman, Nature , 649 (2015).[4] J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt,Phys. Rev. Lett.23