On the roles of Vorob'ev cyclicities and Berry's phase in the EPR paradox and Bell-tests
OOn the roles of Vorob’ev cyclicities and Berry’s phase in the EPR paradox and Bell-tests
David H. Oaknin ∗ and Karl Hess
2, † Rafael Ltd., IL-31021 Haifa, Israel Center for Advanced Study, University of Illinois, Urbana, Il 61801, USA
The well known inequalities of John S. Bell may be regarded, from a purely mathematical viewpoint, as adirect consequence of Vorob’ev-type topological-combinatorial cyclicities formed with functions on a commonprobability space. However, the interpretation of these cyclicities becomes more subtle when considerationsrelated to gauge symmetries and geometric-combinatorial phases are taken into account. These physics relatedconsiderations permit violations of all Bell-type inequalities within the realm of Einstein’s causal physical-mathematics.
Keywords: Bell’s Theorem | Vorob’ev cyclicities | Komolgorov’s probability | Einstein’s causality | Geometric (Berry) Phases
Many recent books on quantum theory and quantum in-formatics dedicate considerable sections to the Einstein-Podolsky-Rosen (EPR) [1] paradox and Bell’s inequalities[2, 3], which appear to represent the ultimate roadblock foran intuitive explanation of quantum phenomena. They seem toform the mathematical embodiment of Bohr’s enunciation thatthe atomic world cannot be explained by using the physicalconcepts of our macroscopic experiences and the correspond-ing mathematical language. In turn, removing this roadblockmight lead to an interpretation of the quantum formalism with-out forfeiting the fundamental physical principles that lay be-hind our views of the macroscopic world. This fact becomesparticularly obvious in reports of science writers, who com-monly present Bell’s inequality as a consequence of straight-forward algebra or logic and then struggle with the problemthat quantum theory and actual EPRB experiments seem tocontradict that logic. Their proposed way out of the conun-drum is the introduction of instantaneous influences at a dis-tance (that Einstein called “spooky”) and/or abandoning thenotion of physical reality that we ordinarily acknowledge inour macroscopic world.The non-sequitur of such radical measures has been pointedout in several thoughtful works e.g. in [4]. However, Wigner’s[5] set theoretical reasoning appeared difficult to overcomeeven to these authors and indeed may be overcome only by thedetailed mathematical physics given in the bulk of this paper.Bell-type inequalities constrain the correlations that mayappear in probabilistic models that fulfill certain generic fea-tures. Quantum phenomena appear to violate such constraintsand, thus, cannot be described or understood by any of thesegeneric models. This fact is interpreted as an experimentallyverifiable proof that quantum phenomena - and, in particu-lar, quantum entanglement - cannot be understood in terms ofKolmogorov’s classical probability concepts and fundamentalphysical principles such as Einstein’s notion of causality.
Elementary derivation of the key result for Bell tests
Bell tests involve a number of variations of EPR-type exper-iments and are thought to present the quintessential demon-strations of whether or not quantum systems may be described in terms of physical concepts taken from the macroscopicworld. In such experiments, a source emanates pairs of entan-gled particles, which propagate toward two distant detectionsystems that test their polarizations. Each detector may be po-sitioned in one of two available settings defined with respectto local lab frames. Upon detection each detector produces abinary response - either − +
1, so that the correlation be-tween the outcomes at the two measurement stations is givenby E ( ∆ ) = − cos ( ∆ ) , (1)where ∆ is the relative angle between the orientations ofthe two detectors. Eq.[1] represents the results of bothquantum theory and many experiments. However, this resultis considered to be inconsistent and impossible to obtainwith Einstein’s causality aided by Kolmogorov’s probabilitytheory.This supposed impossibility is very suspicious, because thecorrelation [1] can be accounted for on the basis of straight-forward symmetry arguments and smoothness constraints aswell as standard tools of information theory. In principle, wecould expect that the correlation between the outcomes of thetwo detection systems may depend on their separate orienta-tions, ∆ A and ∆ B , defined with respect to local lab frames.These orientations can be alternatively described in terms oftheir relative orientation ∆ = ∆ B − ∆ A and their global rigidorientation Γ = ( ∆ A + ∆ B ) /
2. Because the pair of entangledparticles is invariant under rotations, the correlation betweenthe outcomes of the two detectors (polarizers) may (reason-ably speaking) only depend on the relative angle ∆ betweenthem and not on their global rigid orientation Γ , and the ex-perimental results indeed show so (with great accuracy). Thedependence of the correlation on the relative angle ∆ arisesnaturally from the simple fact that we compare and statisti-cally collect ’equal’ and ’not-equal’ experimental outcomeswhen evaluating the EPRB experiments; a procedure that in-volves the measurement results in both wings. This procedureis also basic to Wigner’s [5] approach. More detailed expla-nations will be given in the bulk of this paper.The probabilities for ’equal’ and ’not-equal’ outcomes atthe two detection systems may then be written, without any a r X i v : . [ phy s i c s . g e n - ph ] O c t loss of generality, as: p ( ’EQUAL’ ) = sin ( χ ( ∆ )) ≡ p ( χ ( ∆ )) , p ( ’NOT-EQUAL’ ) = cos ( χ ( ∆ )) ≡ p ( χ ( ∆ )) , (2)so that E ( ∆ ) = sin ( χ ( ∆ )) − cos ( χ ( ∆ )) = − cos ( · χ ( ∆ )) . (3)Since the correlation functions must fulfill the symmetry con-straints E ( ∆ + π ) = − E ( ∆ ) = − E ( − ∆ ) , (4)we postulate that the function χ ( ∆ ) fulfills χ ( − ∆ ) = − χ ( ∆ ) , (5) χ ( ∆ + π ) = (cid:16) π ± χ ( ∆ ) (cid:17) [ mod [ , π )] . (6)We will see in a section below that the plausible linear rela-tion: χ ( ∆ ) = ∆ , (7)may actually be derived from the above symmetry constraintsand well known tools of Informatics. Thus, the quantumcorrelations [1] are obtained as a variation of the Malus lawas suggested in [6].It is the purpose of this paper to show that Eq.[1] neithercontradicts Kolmogorov’s set theoretic probability theory norEinstein’s causality principle. The basis of our findings is thefollowing. Bell-type inequalities have actually been known tomathematicians in one form or another since the early work onprobability theory by Boole [7] and found their most generalformulation in the work of Vorob’ev [8]. Thus, it has beenknown that the constraints demanded by Bell type inequali-ties are a consequence of certain cyclicities of concatenatedrandom variables on a Kolmogorov probability space. The vi-olation of the inequalities implies that the joint probabilitiesassociated to the random variables cannot always be definedon a single probability space, unless the cyclicities may besomehow avoided. We shall show below that the cyclicitiesinvolved in the Bell and CHSH [9] theorems may be removedby well known yet subtle physics involving gauge symmetryconsiderations, geometric phases and other factors [10–13]. Bell-type inequalities and Vorob’ev cyclicities
Bell introduced function-pairs related to two measurementstations with certain instrument settings that are usuallydenoted by j , j (cid:48) = a , b , c , d . One function, A ( j , λ ) , describesthe measurement outcomes in station 1 and the function B ( j (cid:48) , λ ) describes those in station 2. Here, j and j (cid:48) arevariables representing the instrument settings. The variable λ is, according to Bell, corresponding to elements of physical reality and symbolizes the pair of entangled quantum entitiesthat are sent to the respective instruments from a commonsource. Therefore, λ appears as variable in both functions A , B and describes the information shared by the two stationsthrough the pair of entangled electrons or photons.The purpose of the function-pairs A , B is to describe inmathematical form the Einstein-Podolsky-Rosen (EPR) [1]Gedanken-experiment and its variations and it was Bell’sintention to link the domain and co-domain of his functions toactual, performable, experiments. In many later publications,it became evident that Bell’s functions needed to depend onall possible elements of physical reality that Einstein’s theoryof relativity has provided, which includes measurements withrigid rods and with clocks to describe dynamical effects.Thus, the interpretation of Bell’s λ has a long and checkeredhistory. However, for the following discussions, it is sufficientto regard λ as the mathematical symbol that describes theelements of reality emanating from the source, some of thempossibly ’hidden’ to our current knowledge. Dynamicaleffects related to these elements of reality may then be“absorbed” in the method of their evaluation by measurementequipment and gauge considerations.Furthermore, Bell wished to connect his mathematicalconsiderations to quantum physics and to actual experimentsand, therefore, needed to involve probabilities. He thusassumed that some of his variables may be random variables.In Kolmogorov’s probability framework these are functionson a probability space. Mathematical work on probabilitythemes often starts with the words: “given a probabilityspace Ω ” and takes it for granted that such a probabilityspace exists. So did Bell and all the authors following hiswork. They concluded that the joint probabilities predicted byquantum mechanics for Bell experiments cannot be describedusing a single probability space.In fact, Vorob’ev had shown previously that certain ex-pectation values and corresponding probabilities for function-pair-outcomes cannot consistently exist on a single commonprobability space Ω , if a combinatorial-topological cyclicity isinvolved in the concatenation of random variables (functionson a probability space) and they exceed certain constraints.Vorob’ev’s generality of argument makes it necessary to in-volve combinatorial topology. The essence and principle ofhis reasoning can be made clear from the graphical represen-tation for the special case of Bell’s functions shown in Fig.1.The Bell inequality deals then only with the three function-pairs: A ( a , λ ) B ( b , λ ) ; A ( a , λ ) C ( c , λ ) ; B ( b , λ ) C ( c , λ ) , (8)Here we have used, with Bell, identical λ s for all of the threepairs, which is equivalent to the assumption that the cycli-cally connected functions may be defined on one commonprobability space. The Vorob’ev cyclicity that corresponds A( a , λ )B( b , λ ) C( c , λ ) FIG. 1: Vorob’ev cyclicity for Bell’s functions and inequality to these three function-pairs is that of the triangle shown inFig.1. Vorob’ev has emphasized that the arbitrary prescrip-tion of joint pair probability-distributions to the first two pairsdoes not permit complete freedom to choose the joint distribu-tion of the last pair. This fact puts a constraint on the possiblepair expectation values in form of Bell-type inequalities. Theform of the cyclicity determines the form of the inequalities.Similar considerations apply to the CHSH inequalities [9]as shown in Fig.2. The corresponding cyclicity is representedfor four pairs of Bell-type functions: A ( a , λ ) B ( b , λ ) ; A ( a , λ ) B ( b (cid:48) , λ ) ; A ( a (cid:48) , λ ) B ( b , λ ) (9)and A ( a (cid:48) , λ ) B ( b (cid:48) , λ ) . The cyclicity imposes again constraints if we restrict our-selves to one common probability space.Bell’s theorem is widely understood as a experimentallytestable statement that quantum mechanical joint probabilitiesfor the separate pair-wise measurement outcomes cannot bedefined on a single probability space and, hence, there cannotexist an underlying more fundamental description of quantumphenomena. We want to show here that these joint proba-bilities can be properly defined on a single probability spacethrough the use of the gauge symmetries of the problem. Itis our declared purpose to show that Vorob’ev’s cyclicitiesthat are inherent in Bell-type constraints can be eliminatedby a careful consideration of the involved gauge symmetries,geometric phases and other factors, so that the constraints canbe completely avoided.In order to show how to eliminate the cyclicities it isimportant to note that in Bell’s formulation the variables j , j (cid:48) that describe the settings of the instruments, as well asthe variables λ that describe the elements of reality of the A( a , λ )B( b , λ ) C( c , λ ) A( a , λ ) B( b , λ )B( b’, λ )A( a’ , λ ) FIG. 2: Symbolized Vorob’ev cyclicity for both Bell- and CHSH-type inequalities entangled pairs, are defined with respect to local lab frames.However, while the settings j , j (cid:48) can be defined with respectto local lab frames, it is not always true that the variables λ can be so defined when cyclicities and gauge symmetries areinvolved. Neither is it necessarily true that the response func-tions of the instruments can be defined in terms of variablesdefined with respect to lab frames as assumed in Bell’s for-mulation. In general, the variables λ may be properly definedonly with respect to the setting of each instrument, and theresponse of the instrument would then be a function of thevariable λ defined only with respect to the instrument setting.More detailed explanations will be given below in section .To this regard, we must note that the original EPR argumentinvolves only instruments in parallel settings, for which theiroutcomes are fully (anti)correlated. Hence, the original EPRargument does not need to independently define the setting ofeach one of the instruments. Independently defined settingsfor each one of the instruments were introduced by Bell, notby Einstein and his collaborators. REMOVING THE CYCLICITIES AND BELL’SCONSTRAINTSDeriving the quantum result: Wigner’s counting
Wigner generalized Bell’s procedure by using set theoryand let the outcomes include any measurement result, giventhe instrument settings of both experimental wings. He thenused only a judgement of equal or not-equal measurement-outcomes and/or function values (such as spin “up” versus“down” or “horizontal” versus “vertical”, respectively) [5].As outlined in [6], Wigner’s counting ensures that only the rel-ative outcomes of the two wings are of importance. This factalso ensures that the physical variable responsible for the cor-relation between the outcomes of the measurements is the rel-ative angle ∆ between the directions of the polarizers (Stern-Gerlach magnets) and leads to the following equation that wasalready derived in the introduction: E ( ∆ ) = sin ( χ ( ∆ )) − cos ( χ ( ∆ )) = − cos ( · χ ( ∆ )) . In order to link this equation to the results of quantum the-ory and experiments, we still have to show the linearity of thefunction χ ( ∆ ) . This may be accomplished as follows: Be-cause the Fisher information for the random game discussedin the introduction is given by [14, 15] I F ( ∆ ) = ∑ i = , p i ( χ ( ∆ )) · (cid:18) ∂ p i ( χ ( ∆ )) ∂ ∆ (cid:19) = (cid:18) d χ ( ∆ ) d ∆ (cid:19) , (10)it attains a constant value consistent with the symmetry con-straints [5,6] for χ ( ∆ ) = ∆ . (11)In other words, the quantum correlation [3] corresponds tothe situation in which the correlation of the outcomes that aremeasured in the two stations for the single particle pairs car-ries the minimum possible information about the relative an-gle ∆ at which it was obtained.It is interesting to notice that the symmetry constraints [5,6]can also be obtained through similar considerations. Becausethe Shannon entropy for the random game is given by [16, 17] S [ χ ( ∆ )] = − ∑ i = , p i ( χ ( ∆ )) · log ( p i ( χ ( ∆ ))) , (12)we define the total entropy, with the help of some symmetryconsiderations, as Q [ χ ( ∆ )] ≡ (cid:90) π d ∆ S [ χ ( ∆ )] , (13)whose extrema obey the equation, δ Q [ χ ( ∆ )] δ χ ( ∆ ) = (cid:90) π d ∆ δ S [ χ ( ∆ )] δ χ ( ∆ ) = . (14)The last equation can be written as: (cid:90) π d ∆ sin ( χ ( ∆ )) log ( | tan ( χ ( ∆ )) | ) = , (15)which is fulfilled for all functions χ ( ∆ ) that obey the said sym-metry constraints.Similar arguments can be applied to other Bell states besidethe singlet state. In all cases there is a true physical angle,which corresponds to a properly defined relative orientationbetween the measuring devices, while its orthogonal combina-tion is a gauge degree of freedom that corresponds to a globalrigid orientation. This can be readily noticed by simply re-naming the ’up’ and ’down’ single-particle states at one of thetwo stations. This whole procedure goes against the grain of anyone whohas followed the work of Bell-CHSH, because ∆ contains theinstrument settings of both experimental wings, which appar-ently spells some kind of non-locality. However, as explainedabove, the computation of the correlation uses judgementsfor equal and not-equal measurement outcomes i.e. judge-ments relative to the other experimental wing. Such judge-ments are naturally based on global facts as opposed to onlylocal facts within the measurement stations. In fact, Bell-type constraints do not even rule out correlations of the form E ( ∆ ) = − + | ∆ ( mod [ − π , π )) | / π , which can be easily ob-tained in random games with macroscopic carriers, but areonly purported to rule out correlations depending on ∆ in theform predicted by quantum mechanics. A perspective on Vorobev’s cyclicities in terms of gaugesymmetries
The authors of this present work have more recently pro-posed that relativity and gauge symmetry considerations per-mit and actually demand steps that remove the Vorob’evcyclicity for a correct theoretical analysis of actual EPRB ex-periments [10–13]. These considerations follow the observa-tion that the absolute direction of the polarizers or magnets ina Bell experiment is a redundant gauge variable, while the rel-ative orientation of polarizers or magnets is the only true phys-ical variable. A most important consequence of this observa-tion is that we may and even must remove the cyclicity in theBell-type inequalities as shown in Fig.3. Here we have fixedthe instrument setting in one wing as a reference direction andjust chosen the instrument settings in the other wing to obtainthe correct Bell-CHSH angle-differences. As we shall showin what follows this is the only choice for which we can avoidwith certainty assigning a double identity to any of the possi-ble hidden events (which is the reason why it is usually saidthat some joint probabilities cannot be described on a singlespace). As one can see, the cyclicity is removed and so is theconstraint by Bell-CHSH inequalities and the equivalent in-equalities given by Wigner’s procedure. The probability dis-tribution of the elements of reality emanating from the sourceis not changed by this procedure as it must not be. However,the evaluation of the relative outcomes by the measurementinstruments may change and give different numbers for the’equal’ and ’not-equal’ outcomes, depending on the relativeinstrument settings. Under certain circumstances that we shallnow detail, this procedure is essential in order to make it pos-sible to describe all the involved pair-wise random games ona single probability space.We assume that the space of random events available atevery repetition of the experiment form an unbiased samplewithin the whole space of events, so that we can consider forthe sake of simplicity that the latter is always available .Thus, let ( Ω , Σ , µ ) be a probability space, where Ω is a non-empty sample space, Σ is the σ -algebra of all its subsets and µ is a (probability) measure defined on it, and let ξ : Ω → A( a , λ )B( b , λ ) B( b’ , λ )B( b’’ , λ ) A( a , λ )B( b , λ ) B( b’ , λ )B( b’’ , λ )B( b’’’ , λ ) FIG. 3: Removed Vorob’ev cyclicity for Bell- and CHSH-type in-equalities [ − , + ] ⊂ R be a random variable defined on it that takesvalues on the real interval [ − , + ] .Furthermore, let { F } ∆ ∈ Z be a (continuous or discrete)group of isomorphic parameterizations of the probabilityspace labelled over an additive group Z . That is, F ∆ : Ω → Ω are bijective applications from the sample space onto the sam-ple space that preserve the probability measure: ( ∀ S ⊆ Ω )( µ ( F ∆ [ S ]) = µ ( S )) . (16)In particular, F = I : Ω → Ω denotes the identity transforma-tion.Thus, two observers related by a relative ’displacement’ ∆ would describe the same random event, respectively, as S ⊆ Ω and F ∆ [ S ] ⊆ Ω . Hence, the correlation between their descrip-tions of the random variable ξ would be given by: E ( ∆ ) = (cid:90) Ω d µ ( w ) ξ ( w ) · ξ ( F ∆ ( w )) . (17)Bell-type inequalities constrain the two-parties correlations { E ( ∆ i ) } i = , ,..., n that can exist between parties for which, seeFig.2, ∑ i = , ,..., n ∆ i = , (18)assuming that this cyclicity constraint requires that F n ◦ ... ... ◦ F ◦ F = F . (19)However, we notice that whenever gauge symmetries are in-volved a non-zero geometric (Berry) phase α ∈ Z may appearthrough some finite cyclic sequences [18]: F n ◦ ... ... ◦ F ◦ F = F α (cid:54) = F . (20)In other words, we consider the case in which cyclic se-quences may be associated by a re-definition of the identityof symmetric events [10–13], so that the parties cannot all be described within a single probability space. In such a casethe same symmetry consideration can and must be used in or-der to remove the cyclicities, as shown in Fig.3. Obviously,this freedom cannot be allowed when all parties can test therandom events at once, since it would imply that the eventscould have a “double-identity” for at least one of the involvedparties. On the other hand, this freedom must be consideredin cases in which every random event can be tested only bya strict subset of the involved parties. In this case, the free-dom [20] is equivalent to stating that the identities of singleparties cannot be properly defined, but only their relative ’dis-placements’. In physical terms we shall say that the identityof the parties is a gauge (non-physical) degree of freedom.This gauge freedom is tantamount to relaxing the cyclicityconstraint [18] and, therefore, it allows us to avoid the con-straints that would appear otherwise. Remark on the concept of contextuality
The notion of contextuality , frequently included in discus-sions about quantum foundations, actually corresponds in thecase of Bell-type tests to different choices for the gauge-fixingcondition in which the physical system is described. Hence,the contextuality of quantum phenomena may be removedby appropriately choosing a common gauge-fixing condition,namely, the common orientation of the measurement devicein one of the two stations as shown in Fig.3.
Remark on the relationship to experiments
It is important to realize that the removal of the Vorob’evcyclicity as discussed above is not necessarily associated withan actual geometrical rearrangement of the sources and mea-suring instruments involved in the EPRB experiments. Thisremoval may be accomplished as a part of the theoretical de-scription and analysis of the experiment by taking advantageof the involved gauge symmetries and the fact that the polar-ization of each particle of every pair of entangled photons orelectrons can be tested only along a single orientation. In fact,when a non-zero Berry phase [20] appears through the con-sidered cyclic arrangement it is a must to remove the cyclicityin order to describe all the involved pair-wise experiments to-gether.Thus, our analysis applies to both, experiments in whichthe detectors at the two experimental wings can be actually ro-tated [18] and to the experiment of Giustina et al. [19], whichuses electro optical modulators (EOMs) that may actually belocated at any place between source and detectors in both ex-perimental wings. The analysis applies as well to hypotheticalexperiments, not yet performed, with different sources emit-ting the entangled pairs and detectors arranged geometricallyin a Vorob’ev cyclicity. Current optical fiber technology doesappear to permit the construction of such experiments.
CONCLUSION
Bell-type inequalities for random games involving at leastthree functions defined on a single probability space have beenknown to mathematicians since the early works of Boole onprobability theory [7], and found their more general formula-tion in the work of Vorob’ev [8]. These inequalities constrainthe correlations between pairs of random variables that canappear in this kind of games, and it is known that they areassociated to certain cyclicities in the way how the variablesare concatenated. The violation of the inequalities is under-stood to imply that the variables cannot be jointly defined ona single probability space.In particular, the violation of this kind of inequalities bythe correlations predicted by Quantum Mechanics has beenunderstood as the ultimate proof of the impossibility to de-scribe quantum phenomena in terms of any underlying morefundamental theory based on the same fundamental physicalprinciples derived from our macroscopic experience, and thusit represents the ultimate roadblock towards an intuitive inter-pretation of the quantum formalism.In this paper we have shown, however, how subtle phys-ically motivated considerations related to the gauge symme-tries involved in the considered games may allow to removethe cyclicities and, hence, lift the constraints derived from theinequalities, paving the way to an explanation of the quan-tum formalism within the framework of standard Einstein’scausality and Kolmogorov’s probability theory [10–13].As a byproduct of our symmetry argumentation we havepresented an elegant way of obtaining the correlations pre-dicted by quantum mechanics for the Bell experiment (2) fromsymmetry principles using standard tools of information the-ory.Acknowledgement: We appreciate important comments byHans De Raedt, Louis Marchildon and Andrei Khrennikov. ∗ Electronic address: [email protected] † Electronic address: [email protected] [1] Einstein, A., Podolsky, B. & Rosen, N. (1935) Can quantummechanical description of physical reality be considered com-plete?
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