aa r X i v : . [ qu a n t - ph ] S e p Noname manuscript No. (will be inserted by the editor)
Does locality plus perfect correlation implydeterminism?
Michael J. W. Hall
Received: date / Accepted: date
Abstract
A 1964 paper by John Bell gave the first demonstration that quantummechanics is incompatible with local hidden variables. There is an ongoing andvigorous debate on whether he relied on an assumption of determinism, or instead,as he later claimed, derived determinism from assumptions of locality and perfectcorrelation. This paper aims to bring clarity to the debate via simple examplesand rigorous results. It is shown that the weak form of locality used in Bell’s 1964paper (parameter independence) is insufficient for such a derivation, whereas anindependent form called outcome independence is sufficient even when weak local-ity does not hold. It further follows that outcome independence, by itself, impliesthat standard quantum mechanics is incomplete. It is also shown that an appealby Bell to the Einstein-Rosen-Podolsky argument to support his claim fails, viaexamples that expose logical gaps in this argument. However, replacing the real-ity criterion underpinning the argument by a stronger criterion enables a rigorousderivation of both weak locality and determinism, as required for Bell’s 1964 paper.Consequences for quantum interpretations, locality, and classical common causesare briefly discussed, with reference to an example of local classical indeterminism.
Keywords locality · perfect correlation · determinism · outcome independence · Einstein-Podolsky-Rosen argument · classical common causes Bell’s theorem, that there are no local hidden variable models that can reproduceall quantum correlations [1, 2], is one of the most surprising results of twentiethcentury physics. It has important ramifications for both physics and metaphysics,in not only underlying the security of a number of device-independent communica-tion protocols such as quantum key distribution and random number generation,but also ruling out naive classical interpretations of quantum probability [2].
Department of Theoretical PhysicsResearch School of PhysicsAustralian National UniversityCanberra ACT 0200, Australia Michael J. W. Hall
The mathematics required to prove Bell’s theorem is remarkably straightfor-ward. However, there is one aspect of the assumptions used in Bell’s first expositionthat continues to generate differing opinions and vigorous debate [3]–[26]: do theassumptions of locality and perfect correlation made in his 1964 paper necessarilyimply that certain measurement outcomes must be predetermined, or does suchdeterminism represent a third assumption?It may be noted that the answer to this question is unimportant in one sense:later generalisations of Bell’s theorem do not rely on assuming perfect correla-tion nor determinism [2]. Nevertheless, the question remains of strong interest forseveral reasons. First, there is no general consensus on the answer. For example,while Wigner wrote that Bell postulated both deterministic hidden variables andlocality [3] (see also [4, 5]), Bell himself later claimed that determinism was in factinferred rather than assumed in his 1964 paper [6], and the debate has only inten-sified since then. Second, an appeal by Bell to the Einstein-Podolsky-Rosen (EPR)incompleteness argument [27] to support his claim puts the latter argument itselfinto question. Finally, the validity of Bell’s claim would suggest that there is nochoice between giving up determinism or giving up locality in interpreting quan-tum phenomena, i.e., that it would be compulsory to give up locality [9, 28]—aconclusion that is strongly contested in its own right (see, e.g., [29]–[36]).With the aim of making a clear and concise contribution to these issues, thispaper is guided by the following quote from an excellent early discussion on thesubject by van Fraassen:“I have made an effort to present the deduction . . . shorn of all superflu-ous mathematical technicalities and woolly interpretative commentary. (Areader as yet unfamiliar with the literature will be astounded to see theincredible metaphysical extravaganzas to which this subject has led.)” [7]In particular, attention will be focused on what can be proved in a simple yetrigorous manner, and what can be disproved via simple counterexamples, whilstavoiding woolly assertions and metaphysical extravaganzas.In section 2 the only explicit sense of locality supported by Bell’s 1964 pa-per is recalled, and shown to be too weak for deriving determinism for perfectlycorrelated measurement outcomes (Proposition 1). In contrast, an assumption ofoutcome independence (related to the existence of classical common causes) is suf-ficient to derive determinism, whether or not weak locality (or even experimentalfree will) is valid, generalising a result of van Fraassen [7] (Proposition 2). Outcomeindependence is also sufficient, by itself, to imply the incompleteness of standardquantum mechanics (Proposition 3).In section 3 the degree of support afforded to Bell’s claim by the EPR in-completeness argument is examined. Two gaps in the EPR logic are identified viasimple counterexamples, implying that the argument is itself incomplete (Propo-sition 4). However, a suitable strengthening of the EPR reality criterion closesthese gaps (Proposition 5), and further enables a rigorous derivation of both weaklocality and determinism as required for Bell’s 1964 paper (Proposition 6).Implications for quantum interpretations are noted in section 4, and the neces-sity or otherwise of nonlocality in quantum mechanics is discussed with referenceto a classical example of nondeterministic evolution. Conclusions are given in sec-tion 5. oes locality plus perfect correlation imply determinism? 3 without reference to determinism,if the latter is to be inferred rather than assumed. That is, a statistical formulationof locality is required.To obtain such a formulation, let x and y label possible measurements whichmay be made in two separate regions of spacetime, having respective outcomeslabeled by a and b , and let p ( a, b | x, y ) denote the joint probability density forthese outcomes. If λ denotes any further statistical variables of interest—arising,e.g., from a physical or mathematical model of the measurements—then it followsfrom the basic rules of probability that p ( a, b | x, y ) = X λ p ( a, b, λ | x, y ) = X λ p ( a, b | x, y, λ ) p ( λ | x, y ) . (1)Here summation is replaced by integration over any continuous range of λ . Thelocality of such a model, in the sense spelled out in the introduction to Bell’s 1964paper and quoted above, can now be rigorously formulated as Definition 1 (Weak locality) :
The probability distribution of the result of ameasurement in one region is unaffected by measurement operations in a distantregion, i.e., p ( a | x, y, λ ) = p ( a | x, λ ) , p ( b | x, y, λ ) = p ( b | y, λ ) , (2)for all x, y, λ . Michael J. W. Hall
This property is called weak locality here to distinguish it from other possiblenotions of locality, and is most easily justified when the regions are spacelike sepa-rated. It is also known as ‘hidden locality’ [7], ‘locality’ [37] and, more commonly,‘parameter independence’ [38]. Importantly, it is the only statistical formulationof locality that is explicitly supported by Bell’s 1964 paper.2.2 Determinism does not follow from weak locality and perfect correlationAs mentioned above, Bell claimed in 1981 that determinism was not assumed inhis 1964 paper, but was logically inferred [6]:“My own first paper on this subject . . . starts with a summary of the EPRargument from locality to deterministic hidden variables. But the commen-tators have almost universally reported that it begins with deterministichidden variables.”However, the only explicit argument for determinism in that first paper is suppliedby just two sentences, in the context of measurements on two perfectly correlatedspins σ and σ [1]:“Now we make the hypothesis, and it seems one at least worth considering,that if the two measurements are made at places remote from on another,the orientation of one magnet does not influence the result obtained withthe other. Since we can predict in advance the result of measuring any com-ponent of σ , by previously measuring the same component of σ , it followsthat the result of any such measurement must actually be predetermined.”Thus, it would appear that determinism is inferred as a logical consequence ofweak locality and perfect correlation. The ensuing debate in the literature arises,at least in part, from a simple observation that undermines this argument [15, 25]. Proposition 1 : weak locality + perfect correlation = ⇒ determinism . (3)That is, Bell’s claim certainly does not hold for the only form of locality explicitlysupported in his 1964 paper: weak locality is too weak.To demonstrate the above proposition, consider two particular measurements, x and y , that have a common range of possible values for their respective out-comes a and b . Perfect correlation of these outcomes then corresponds to p ( A = B | x , y ) = 1 , (4)where A and B denote the random variables corresponding to outcomes a and b . It follows that proposition can be demonstrated via any model of correlationssatisfying Eqs. (1), (2) and (4) such that the outcomes of measurements x and y are not predetermined by the model. The simplest such model is one for which λ takes a single fixed value, λ say, and only one measurement is possible in eachregion, x and y say, each with n possible outcomes. Then the joint measurementdistribution defined by p ( a, b | x , y ) = p ( a, b | x , y , λ ) := n − δ ab (5) oes locality plus perfect correlation imply determinism? 5 trivially satisfies Eqs. (1), (2) and (4) (with p ( λ | x , y ) = 1), and yet is clearlynot deterministic for any n > is deterministic (indeed, this is trivially the case for anygiven joint probability distribution p ( a, b | x , y ), perfectly correlated or otherwise):Bell’s prima facie claim is that determinism, rather than its mere possibility, isinferred from locality and perfect correlation alone. Counterexamples can alsobe easily constructed for continuous outcome ranges (e.g., with x and y cor-responding to perfectly correlated classical phase space measurements), and formultiple measurement settings (e.g., based on simple models of ‘no-signalling’ cor-relations [39, 40]).2.3 Determinism does follow from outcome independence and perfect correlationWe have from Proposition 1 that determinism can only be inferred from perfectcorrelation via some assumption other than, or possibly in addition to, weak lo-cality. For example, in his 1981 paper Bell interchangeably uses the terms ‘localcausality’, ‘local explicability’, and ‘action-at-a-distance’ for the condition p ( a, b | x, y, λ ) = p ( a | x, λ ) p ( b | y, λ ) (6)on models of correlations between distant regions (see also [41, 42]). This conditionis strictly stronger than weak locality in Eq. (2) (the latter follows by summingEq. (6) over each of a and b ), and is indeed sufficiently strong to derive determinismfor the case of perfect correlation [9, 12]. Thus local causality is a candidate for amissing additional assumption in Bell’s 1964 paper, albeit not actually supportedin any guise therein.However, local causality is a far stronger assumption than is necessary for thepurpose of deriving determinism from perfect correlations: physical models needin fact only satisfy the much weaker condition Definition 2 (Outcome independence) :
Any observed correlations betweenmeasurement outcomes arise from ignorance of some further variable(s) λ , i.e., p ( a, b | x, y, λ ) = p ( a | x, y, λ ) p ( b | x, yλ ) . (7)for all a, b, x, y and λ .This property has also been called, for example, ‘causality’ [7] and ‘complete-ness’ [37], and implies that any correlation between observed outcomes is duesolely to the average over λ in Eq. (1). In particular, λ can be interpreted as aclassical common cause for the correlations. Outcome independence is logicallyindependent of weak locality, and clearly weaker than local causality in Eq. (6).Indeed, simple substitution shows that local causality is equivalent to the combi-nation of weak locality and outcome independence [37].Importantly, in contrast to Proposition 1, one has Proposition 2 : outcome independence + perfect correlation = ⇒ determinism (8)with probability 1 (i.e., for a set of λ over which p ( λ | x , y ) sums or integrates tounity). Michael J. W. Hall
This result was first given for the case of discrete outcomes by van Fraassen [7] (aformally equivalent result for two-valued outcomes was also given in Theorem 2of[4]), but it does not appear to be well known. It is not discussed, for example, inany of the contributions [8]–[26] to the debate other than [8, 14]. It is extended hereto include the case of continuous outcomes, as proved at the end of this section.Significantly, Proposition 2 is independent of whether or not weak locality asper Eq. (2) is satisfied (although the latter is required for the further task of deriv-ing Bell inequalities). Thus locality, in the only sense that is explicitly discussedin Bell’s 1964 paper, is in fact irrelevant for the derivation of determinism fromperfectly correlated measurement outcomes! It is also worth noting the propositiondoes not rely on making an assumption of measurement independence or experi-mental free will (i.e., it is not assumed that p ( λ | x, y ) = p ( λ )), nor even that the‘common cause’ λ lies in the past.As further observed by van Fraassen [7], Eq. (8) has a simple corollary offundamental interest, also proved below. Proposition 3 : outcome independence = ⇒ incompleteness of standard QM . (9)Thus an assumption of outcome independence by itself is sufficiently strong toimply the incompleteness of the standard Hilbert space formulation of quantumtheory—regardless of whether or not weak locality or local causality or measure-ment independence holds. Noting the discussion following Eq. (7), this result canalso be informally stated as some quantum correlations have no classical commoncause . Further, although not recognised by van Fraassen, it is independent of theEPR incompleteness argument [27] (discussed in section 3), as is demonstrated bythe proof below.In particular, Proposition 3 follows via consideration of any quantum wavefunction that predicts perfect correlation between two measurements. Perhaps thesimplest example is one given by Einstein at the 1927 Solvay Congress and laterstreamlined [43, 44]: a superposition of a single particle in two regions, R1 and R2say, with a wave function of the form | ψ i = 1 √ | R i + | R i ) . (10)Hence, if x is a measurement detecting whether the particle is in R1, and y is a measurement of whether the particle is in R2, it follows that the outcomes(suitably labeled) are perfectly correlated. It follows from Proposition 2 that anyoutcome-independent model of the detection statistics for this example must bedeterministic. But quantum theory predicts a nondeterministic (50%) probabilityfor either outcome. Hence, the wave function description is incomplete under anassumption of outcome independence, as claimed.Finally, to demonstrate Proposition 2, note that perfect correlation for theoutcomes of measurements x and y implies P λ p ( A = B | x , y , λ ) p ( λ | x , y )= 1 via Eqs. (1) and (4). Hence P λ [ 1 − p ( A = B | x , y , λ ) ] p ( λ | x , y ) = 0, andso, noting each factor is nonnegative, p ( A = B | x , y , λ ) = 1 (11) oes locality plus perfect correlation imply determinism? 7 with probability 1. Now, for the case of discrete outcomes we have p ( A = B | x , y , λ ) = X a p ( a, a | x , y , λ ) . (12)It follows via Eq. (11) and outcome independence as per Eq. (7) that X a p ( A = a | x , y , λ ) [1 − p ( B = a | x , y , λ )] = 1 − . Thus, since each factor is nonnegative, p ( A = a | x , y , λ ) [1 − p ( B = a | x , y , λ )] = 0 (13)with probability 1. But p ( A = a | x , y , λ ) > a = a ′ say, and therefore p ( B = a ′ | x , y , λ ) = 1, implying p ( B = b | x , y , λ ) = δ a ′ b . Thisvanishes for b = a ′ , and so p ( A = a | x , y , λ ) = 0 for a = a ′ from Eq. (13), yielding p ( A = a | x , y , λ ) = δ a ′ a . Thus the measurement outcomes are deterministic, asclaimed in Proposition 2.The case of continuous outcomes is more subtle, noting that Eq. (12) doesnot generalise to p ( A = B | x , y , λ ) = R da p ( a, a | x , y , λ ): the right hand sideis not a probability, and indeed it diverges for perfectly correlated cases such as p ( a, b | x , y , λ ) = δ ( a ) δ ( b ). Instead, one has p ( A = B | x , y , λ ) := lim ǫ → p ( | A − B | < ǫ | x , y , λ )= Z da p ( a | x , y , λ ) lim ǫ → Z { b : | b − a | <ǫ } db p ( b | x , y , λ ) , (14)where the last line follows assuming outcome independence as per Eq. (7). Equa-tion (13) therefore generalises to p ( a | x , y , λ ) " − lim ǫ → Z { b : | b − a | <ǫ } db p ( b | x , y , λ ) = 0 (15)with probability 1. Essentially the same argument as for the discrete case may nowbe applied. First, p ( a | x , y , λ ) > a = a ′ say, implying that p ( b | x , y , λ )is fully supported on { b : | b − a ′ | < ǫ } with probability 1, for all ǫ >
0, i.e., p ( b | x , y , λ ) = δ ( b − a ′ ). But this vanishes for b = a ′ , implying from Eq. (15)that p ( a | x , y , λ ) = 0 for a = a ′ , and hence that p ( a | x , y , λ ) = δ ( a − a ′ ).Thus, again, the measurement outcomes are determininistic with probability 1,and Proposition 2 follows. Bell’s 1964 two-sentence derivation of determinism from locality and perfect cor-relation, as quoted in section 2.2 above, is certainly incomplete. In particular, asevidenced by Proposition 1, it is not supported by the only explicit form of localitydiscussed in that paper, and must therefore rely on some other unstated assump-tion. While either of local causality or outcome independence would do the job, asimplied by Proposition 2, it is natural to ask, given that Bell claims his derivationis a version of the EPR argument, whether his unstated assumption can in fact bededuced from the original EPR paper. This is the subject of this section.
Michael J. W. Hall
Definition 3 (EPR reality criterion) :
If, without in any way disturbing asystem, we can predict with certainty (i.e., with probability equal to unity) thevalue of a physical quantity, then there exists an element of physical reality corre-sponding to this physical quantity.This condition is then applied to cases of perfect correlation as per Eq. (4) (andin particular to perfect position and momentum correlations), using the followinglogic (see also section 2 of [45]):1. Assume measurement x is made in a first region, with result a (assumption).2. The outcome of a measurement y in a distant second region can then bepredicted as b = a with certainty (perfect correlation).3. This prediction can be obtained without disturbing the distant region in anyway (assumption).4. Hence, the value of b is an element of physical reality, prior to any actualmeasurement of y in the distant region (EPR reality criterion).The above steps are all that EPR explicitly use to derive reality/determinism forany given perfect correlation. However, when one tries to make the above stepsmathematically rigorous, a gap shows up in the above logic. Indeed, as will beshown via counterexamples below, the EPR reality criterion is in fact insufficientfor concluding the reality of b via these steps: a further assumption is required.Moreover, EPR go on to consider what can be said if there are two or moreperfectly correlated pairs of measurements, and add the following further step:5. If x and y are a second pair of perfectly correlated measurements, for thefirst and second regions respectively, then applying the same steps as aboveimplies that the outcomes of both y and y are real and predetermined.EPR rely on this last step to conclude that the incompleteness of quantum me-chanics follows from their reality criterion [27]. However, as is also shown via acounterexample below, Step 5 does not strictly follow as a logical consequence ofSteps 1–4: there is a second missing assumption. It follows that Proposition 4 :
The logic of the EPR argument for the incompleteness of quan-tum mechanics is itself incomplete, i.e.,
EPR reality criterion = ⇒ incompleteness of QM . (16)Thus, comparing with Proposition 3, it is seen that the EPR reality criterion isnot as strong as outcome independence in this regard. Not all is lost, however. Inparticular, it is not difficult to formulate a stronger form of the reality criterionthat allows the EPR argument to go through.3.2 Counterexamples to the EPR logicThe EPR logic, as summarised above, has two important gaps which preventSteps 4 and 5, respectively, from going through without additional assumptions,and lead to Proposition 4. These gaps relate to symmetry and joint measurements,and are best illuminated via counterexamples. oes locality plus perfect correlation imply determinism? 9 The first gap in the EPR logic is relatively minor, and arises from the inherentasymmetry of the argument: it requires only that a measurement made in the firstregion does not disturb the system in the second region in any way, as per Step 3(e.g., “at the time of measurement . . . no real change can take place in the secondsystem in consequence of anything that may be done to the first system” [27]).This does not rule out, however, the possibility that making a measurement in thesecond region can disturb the system in the first region. Note this is consistentwith no direct interaction between the two systems if it is the measurement devicethat is responsible for the disturbance.One class of counterexamples that exploits this gap is to allow (faster-than-light) signalling from the second region to the first region, but not vice versa:
Example 1 : Suppose that each region contains a single spin- particle in a totallymixed state, and that if a device measures spin in the y direction of the secondparticle, with result b , it sends a signal (e.g., superluminally or along its backwardlightcone) that puts the first particle into a − b eigenstate of spin in the y direction.Hence, if a measurement of spin in the x direction is made in the first region, thestatistics of the singlet state are reproduced, i.e., p ( a, b | x , y ) = (1 − ab x · y ).Note for this example, choosing y = ± x , that the outcome in the second re-gion can be predicted with probability unity from the result of a measurementin the first region, and that there is no disturbance of the second region by anymeasurement carried out in the first region, thus fulfilling the conditions for theEPR reality criterion. Nevertheless, contrary to Step 4 of the EPR logic, there isno pre-existing element of reality for the outcome of a measurement in the secondregion: a totally random and unpredictable result b = ± y . Thus, use of the EPR reality criterion fails for this example, implyingthat the EPR logic is incomplete without some further assumption. Note that ifthe spin measurements are replaced by linear polarisations, for a two-photon state,then the two initial states can be replaced by eigenstates of circular polarisation.An interesting second class of counterexamples exploiting the asymmetry gapis temporal in nature, and applies to the case of a single particle: Example 2 : Suppose, a single spin- particle is initally described by a maximallymixed state, and that the second region lies within the past lightcone of the firstregion. Suppose further that a measurement of spin in some direction, if madein the second region, acts to leave the particle in an eigenstate of spin in thatdirection. Hence, if the same spin direction is subsequently measured in the firstregion, the results are perfectly correlated.Thus, the outcome in the second region can be predicted from the result of ameasurement made in the first region and, assuming no retrocausality, it is impos-sible for any measurement in the first region to disturb the second region in anyway, as required by the EPR reality criterion. Yet there is no pre-existing elementof reality for the outcome in the second region, contrary to Step 4 of the EPRargument.Classical examples can also be obtained by replacing the particles in the aboveexamples by classical models thereof [1]. One way to close the asymmetry gap is simply to symmetrise the EPR logic, by further assuming that measurement y ismade in the second region, without disturbing the first region in any way (whichhas the additional advantage that one can also obtain the reality of the outcomeof x , via Steps 1–4 with the roles of x and y reversed). It turns out, however,that this assumption is too strong in that it prevents closure of a second gap. To expose the second (and most important) gap in the EPR argument (see alsosection 2 of [45]), observe that the logic starts with the assumption in Step 1that x is measured. Without this assumption the value of b cannot be predictedwith certainty, as required for Steps 2 and 4. It follows that, even ignoring (or insome way closing) the asymmetry gap, one cannot proceed directly from Step 4to Step 5. In particular, for a second pair of perfectly correlated measurements x and y , one can only conclude that the outcomes of both y and y are real andpredetermined if both x and x are measured without disturbing the predictedperfect correlations. Conversely, if they cannot both be so measured, then Step 5does not logically follow (without some additional assumption). Note that thisissue is only exacerbated if the asymmetry gap is closed as per the method of theprevious subsection, as Step 5 would then require that both y and y are alsomeasured, without disturbing the correlations.In his response to the EPR paper, Bohr famously gave an explicit quantumcounterexample exploiting this joint measurement gap, by showing how perfectmomentum and position correlations between two quantum particles cannot bothbe maintained as soon as either the position or momentum of one particle is mea-sured [46] (note that the fourth and fifth pages, describing the counterexample,are printed in reverse order). The singlet state measurements considered by Bellsupply a similar counterexample (since a Stern-Gerlach magnet cannot simulta-neously have two measurement orientations), and one can also construct classicalcounterexamples [47].EPR allude to the joint measurement gap in the penultimate paragraph oftheir paper, when they reject a possible replacement of their reality criterion by amore restrictive criterion for the reality of the momentum P and position Q of aparticle in the second region [27]:“Indeed, one would not arrive at our conclusion if one insisted that two ormore physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. . . . This makesthe reality of P and Q depend upon the process of measurement carriedout on the first system, which does not disturb the second system in anyway. No reasonable definition of reality could be expected to permit this.”This quote suggests a possible method for closing the joint measurement gap: bymaking the ‘reasonable’ assumption that the reality of a physical quantity in agiven region is independent of any measurement that does not in any way disturbthat region (note that this assumption would represent a necessary condition foran element of reality, in contrast to the sufficient condition given by the EPRreality criterion). Thus, if Steps 1–4 are taken to be valid, then the reality of both y and y must still hold even if neither of the non-disturbing measurements of x and x are actually made. However, this assumption is too weak to also close the oes locality plus perfect correlation imply determinism? 11 asymmetry gap, which prevents Step 4 from being used to actually conclude thereality of y (or y ) even in the case that x (or x ) is measured. In particular,the assumption can only be applied once the reality of the outcome of y (or y )has actually been established.3.3 Closing the gaps: a stronger reality criterionIt is clear from the counterexamples of the previous subsection that the EPRreality criterion is not strong enough to derive the incompleteness of quantummechanics, as per Proposition 4. It is further clear that the common issue arisingfrom the gaps identified by these counterexamples is the need for stronger notionsof non-disturbance and reality. A natural solution to this issue is to minimallystrengthen the EPR reality criterion as follows. Definition 4 (Local reality criterion) :
If, without in any way disturbing orbeing disturbed by a system, we can predict with certainty (i.e., with probabilityequal to unity) the value of a physical quantity, then there exists, prior to theprediction , an element of physical reality corresponding to this physical quantity.The italicised phrases indicate additions to the original EPR reality criterion inDefinition 3, and close the asymmetry and joint measurement gaps respectively.Note that they correspond to a weakening of the method of closing the asymmetrygap considered in section 3.2.1 (a direct measurement of y is not required), andto a strengthening of the method of closing the joint measurement gap consideredin section 3.2.2 (the reality of the outcome does not require the prediction to haveactually been established by some suitable measurement).It is shown here that this stronger criterion of local reality restores the desiredlogical completeness of the EPR argument, i.e., in contrast to Proposition 4: Proposition 5 local reality criterion = ⇒ incompleteness of standard QM , (17)It is further shown that one can rigorously obtain: Proposition 6 local reality criterion + perfect correlation = ⇒ weak locality + determinism , (18)as required by Bell for his 1964 paper. Hence it is natural to identify the unstatedassumption in that paper with the local reality criterion. Notet that Proposition 6is also valid if the local reality criterion is replaced by local causality as per Eq. (6).Thus this criterion is also closely related to the assumption used to derive generalBell inequalities [2, 6, 41, 42].The proof of Proposition 5 is straightforward. First, the EPR logic for pairsof perfectly correlated measurements x , y and x , y is modified as follows (withprimes and italics indicating modified steps):1. Assume measurement x is made in a first region, with result a (assumption).2. The outcome of a measurement y in a distant second region can then bepredicted as b = a with certainty (perfect correlation). ′ . This prediction can be obtained without disturbing or being disturbed by thedistant region in any way (assumption).4 ′ . Hence, the value of b is an element of physical reality, prior to any actualmeasurement of y in the distant region, and prior to obtaining the predictionvia an actual measurement of x (local reality criterion).5 ′ . If x and y are a second pair of perfectly correlated measurements, for thefirst and second regions respectively, then applying the same steps as aboveimplies that the outcomes of both y and y are real and predetermined priorto any actual measurement of x or x .Note that Step 3 ′ rules out the counterexamples in section 3.2.1, and Step 4 ′ rulesout those in section 3.2.2.The above logic is sufficient to obtain Proposition 5, using either the perfectlycorrelated momentum and position wave function considered by EPR or the sin-glet state considered by Bell. It also sufficient for the determinism component ofProposition 6. However, the weak locality component of the latter requires a littlemore work, including several uses of the following Lemma. Lemma 1 :
If the conditional distribution of a random variable j is deterministicfor given prior information k (i.e., p ( j | k ) = 0 for all but one value of j ), then itremains deterministic when conditioned on any further information l compatiblewith k (i.e., p ( j | k, l ) = 0 for all but one value of j ).Proof : We have p ( j | k ) = 0 for all j = j , for some j , and p ( l | k ) >
0. Hence0 ≤ p ( j, l | k ) ≤ p ( j | k ) = 0 for j = j , and so, using the standard rules of probability, p ( j | k, l ) = p ( j, l | k ) /p ( l | k ) = 0 for all j = j .To prove Proposition 6, note first from Step 4 ′ above that the outcome of y is predetermined prior to any measurement of x or y . Let λ ′ denote anyadditional variables needed to determine this outcome, i.e., p ( b | a, x , y , λ ′ ) = 0for all but the predetermined value of b . Further, from Step 3 ′ , actually making ameasurement of x in the first region cannot affect this predetermined outcome,i.e., x and a are redundant in determining its value, and hence p ( b | a, x , y , λ ′ )= p ( b | y , λ ′ ) = 0 for all but one value of b .Second, suppose that in fact some other measurement, x say, rather than x ,is actually made in the first region, without disturbing the second region in anyway. Then, since we have p ( b | y , λ ′ ) = 0 for all but one value of b , and x must becompatible with y and λ ′ if it is actually made, it follows from the Lemma that p ( b | x, y , λ ′ ) = p ( b | y , λ ′ ) = 0 (19)for all but one value of b and any such measurement. Similarly, reversing the rolesof the first and second regions, and letting λ ′′ denote any additional informationneeded to determine the outcome of x , it also follows that p ( a | x , y, λ ′′ ) = p ( a | x , λ ′′ ) = 0 (20)for all but one value of a and any measurement y made in the second region thatdoes not disturb the first region.Third, suppose there is some set of perfectly correlated pairs of measurements, { ( x s , y s ) : s ∈ S } , for some index set S , where the measurement in each region oes locality plus perfect correlation imply determinism? 13 does not disturb and is not disturbed in any way by the measurement in the otherregion. Using an obvious notation, the above two equations then generalise to p ( b | x s , y t , λ ′ t ) = p ( b | y t , λ ′ t ) = 0 , p ( a | x s , y t , λ ′′ s ) = p ( a | y t , λ ′′ s ) = 0for all but one value of a and b and all s, t ∈ S . Hence, letting λ denote the set ofpairs { ( λ ′ s , λ ′′ s ) : s ∈ S } , application of the above Lemma immediately gives p ( a | x s , y t , λ ) = p ( a | x s , λ ) = 0 , p ( b | x s , y t , λ ) = p ( b | y t , λ ) = 0 (21)for all but one value of a and b and all s, t ∈ S . Thus, the outcomes of all pairs( x s , y t ) are deterministic, and satisfy weak locality as per Eq. (2), thereby provingProposition (6). prior to beingable to make a prediction with probability unity. For example, in the many worldsinterpretation such a prediction cannot be made until a measurement establishesthe relevant branch (or memory sequence) of the observer making the predic-tion [50], while in the Copenhagen interpretation Bohr emphasises there is “ aninfluence on the very conditions which define the possible types of predictions re-garding the future behavior of the system . . . [where] these conditions constitute aninherent element of the description of any phenomenon to which the term ‘physicalreality’ can be properly attached” [46] (his italics). Another approach is to assertthe reality of wave function collapse, thus implying a disturbance of one region bymeasurement in another, as in spontaneous collapse interpretations [52, 53] (andin some variants of the Copenhagen interpretation, albeit explicitly rejected byBohr [46]: “Of course there is in a case like that just considered no question of amechanical disturbance of the system under investigation”—see also [56]).In contrast, interpretations which either supplement or replace standard quan-tum theory with a deterministic description of outcomes, such as the deBroglie-Bohm [57, 58], many-interacting-worlds [59, 60] and superdeterministic [61, 62] in-terpretations, treat standard quantum mechanics as incomplete from the outset, and hence are untroubled by Propositions 3 and 5. Note that any such interpre-tation automatically satisfies outcome independence (deterministic models neces-sarily satisfy Eq. (7)), and may or may not satisfy the local reality criterion. Forexample, for the case of perfect position and momentum correlations consideredby EPR, both the deBroglie-Bohm and many-interacting-worlds interpretationssatisfy this criterion for position measurements (the positions of all particles are‘real’), but not for momentum measurements (such measurements on one par-ticle necessarily disturb the other particle, directly via a nonlocal influence inthe deBroglie-Bohm interpretation [58], and indirectly via interactions with otherworlds in the many-interacting-worlds approach [60]). In contrast, the local real-ity criterion is always satisfied in superdeterministic interpretations (as is localcausality), as these give a measurement-dependent explanation of quantum corre-lations in terms of local classical common causes, which not only determine themeasurement outcomes but the measurement selections themselves [61, 62].4.2 Is outcome independence a locality property?As noted in the Introduction, Bell’s claim that determinism follows from localityand perfect correlations would further suggest, in the light of his 1964 inequality [1],that quantum mechanics is necessarily nonlocal. While debate on this question isalso extensive (see, e.g., [9, 15, 28]–[36]), it largely centres around different possibledefinitions of ‘locality’ [15] and so, in the spirit of avoiding ‘woolly interpretivecommentary’ as per the Introduction, will not be reviewed here. Nevertheless,some basic points are made below that help clarify some aspects, including anexample of local classical indeterminism with perfect correlations.First, standard quantum mechanics is certainly local in the sense of weak lo-cality: operations carried out in one region cannot affect the statistics in a distantregion [63, 64]. Any claim of nonlocality must therefore rely on a different sense (oron an interpretation that goes beyond the standard quantum description). Second,the local reality criterion in section 3.3 clearly involves notions of both locality andreality, and hence its failure cannot unambiguously rule out locality. Third, theviolation of Bell inequalities by quantum correlations do not necessarily rule outlocality in the strong sense of local causality as per Eq. (6), unless one furtherassumes measurement independence [65]. Fourth, even under the assumption ofmeasurement independence, local causality is equivalent to the combination ofweak locality and outcome independence, and hence its failure corresponds to anunambiguous signature of nonlocality if and only if the failure of outcome inde-pendence is also such a signature. But is this the case?The answer to the above question is, prima facie , negative. Outcome inde-pendence can be interpreted purely in terms of classical common causes [7] orcompleteness [37], and hence any violation can be taken to correspond to, for ex-ample, the existence of nonclassical common causes, rather than to some type ofnonlocality. Further, while Shimony refers to the failure of outcome independenceas an ‘uncontrollable nonlocality’ [28], the nature of this nonlocality is only inthe vague and unexplained sense of ‘passion at a distance’ (amusingly criticisedby Mermin as ‘fashion at a distance’ [30]). These points do not, however, strictlyrule out the possibility that some overlooked and unambiguous nonlocal effect isassociated with the failure of outcome independence. In this context it is of inter- oes locality plus perfect correlation imply determinism? 15 est to note the existence of classical examples that further curtail this possibility,based on systems in classical mechanics and electrodynamics that are inherentlynondeterministic [66, 67, 68, 69].For example, consider two classical particles of mass m moving in one dimensionunder a potential of the form V ( x , x ) = ( − γ | x − x | / , | x − x | ≤ d, − γ d / , | x − x | > d, (22)where d is some fixed separation distance beyond which the particles do not inter-act. If the particles are initially at rest at the origin, i.e., ˙ x (0) = ˙ x (0) = 0, thenconservation of energy in the centre of mass frame corresponds to m ˙ x − γ | x | / =0 for separations x = x − x less than d , and free motion for separations greaterthan d . Remarkably, this system has the non-unique set of solutions: x R ( t ) = − x L ( t ) = , ≤ t ≤ T, γ m ( t − T ) , T < t ≤ T + τ, d + q γm d / ( t − T − τ ) , t > T + τ, (23)where x L and x R label the left-moving and right-moving particles, T ≥ τ := 2( dm /γ ) / . Thus the particles remain at the origin for time T ,then repel each other for a fixed duration τ , and are noninteracting thereafter. Theevolution is therefore nondeterministic, since the ‘pause time’ T is not uniquelyfixed despite all forces and initial conditions being specified and well-defined.Suppose now that two detectors are placed at equal distances greater than d/ t = 0. It follows that there is a perfectcorrelation between their readouts at all times, irrespective of the value of T (thereis also a perfect correlation between the particle positions and momenta, up untildetection). We thus have a classical example of perfect correlations between twononinteracting regions, in a nondeterministic context.It follows via Propostion 6 that the local reality criterion in section 3.3 fails forthis example (as does the EPR reality criterion). Further, while outcome indepen-dence cannot be directly assessed (the example is nonstatistical), it neverthelessfails in the broad sense of being incompatible with Proposition 2 for this example.It is clear that the these failures are not due to any nonlocal effect, but to the lackof a classical cause for the pause time T , where this lack may itself be regarded asform of incompleteness.The above example of classical indeterminism corresponds to reinterpreting thesingle-particle model in [68] as describing the relative displacement of two identicalparticles initially at rest, and switching off the inter-particle interaction above aspecified separation distance to ensure locality. Similar examples can be obtainedvia the single-particle models in [67], while an inherently local and relativisticexample is provided by a weakly bound system of a classical electromagnetic fieldand two identical classical particles in two-dimensional spacetime [66, 69]. It wouldbe of interest to determine whether the model of indeterministic classical motionsuggested by Gisin can provide further examples [70] (although it is not clear howto satisfactorily obtain perfect correlations via conservation laws in this model). Bell’s “insistence that the determinism was inferred rather than assumed” [6],via the combination of locality and perfect correlation in his 1964 paper, is notsupported by the only form of locality considered in that paper (weak localityis simply too weak), nor by an appeal to the EPR paper (the incompletenessargument therein is itself incomplete), as per Propositions 1 and 4. It is interestingin this regard that Bell did not back up his claim by spelling out any details of howdeterminism might be rigorously derived. Indeed, when revisiting the example ofperfect spin correlations in section 3 of his 1981 paper [6], he merely states that “ weseem obliged to admit that the results on both sides are determined in advance” (myitalics). Likewise, in earlier papers he states only, for example, that “
This stronglysuggests that the outcomes of such measurements, along arbitrary directions, areactually determined in advance” [71]; or asks “
Is it not more reasonable to assume that the result was somehow predetermined all along?” [72] (my italics).It follows in any case, from Proposition 1, that an extra assumption of somesort, either additional to or in place of weak locality, is required to derive deter-minism from perfect correlation (see also [15, 25]). Further, as first noted by vanFraassen [7], the assumption of outcome independence is sufficient for this purpose(whether or not weak locality or local causality or even measurement independenceholds) and, moreover, is even sufficient to derive the incompleteness of quantummechanics, as reviewed in Propositions 2 and 3.It similarly follows that the EPR incompleteness argument requires an addi-tional assumption of some sort, to avoid the asymmetry and joint measurementgaps identified in section 3.2. Strengthening the EPR reality criterion to the local reality criterion in Definition 4 is sufficient for this purpose, as per Proposition 5.Moreover, the strengthened criterion is also sufficient, when combined with perfectcorrelation, to imply both weak locality and determinism as per Proposition 6, asrequired for deriving Bell’s original inequality. Given Bell’s appeal to the EPRargument to support his claim, the local reality criterion is therefore a naturalcandidate for the unstated assumption in his 1964 paper.As noted in section 4, the completeness of quantum mechanics can be main-tained by all interpretations that rely exclusively on the standard Hilbert spacedescription. Locality can also be maintained, in the absence of any evidence thatthe failure of outcome independence corresponds to an unambiguous signature ofnonlocality. Indeed, if we follow van Fraassen in interpreting outcome indepen-dence as corresponding to classical common causes, it follows that its failure instandard quantum mechanics instead corresponds to the existence of nonclassical common causes. In this regard it is of interest to note that some quantitative dis-tinctions between classical, quantum, and nonclassical common causes have beenrecently identified [73].Finally, it is well known that the experimental violation of Bell inequalitiesrequires that models of quantum correlations give up at least one of weak locality,outcome independence or measurement independence. But stronger conclusionscan be obtained by finding experimental tests that relax at least one of theseassumptions. For example, a relaxed Bell inequality has recently been given [74]and experimentally tested [75] for models satisfying measurement independence oes locality plus perfect correlation imply determinism? 17 but which relax weak locality and outcome independence to the weaker condition p ( a, b | x, y, λ ) = w A → Bλ p ( a | x, λ ) p ( b | a, y, λ ) + w B → Aλ p ( a | b, x, λ ) p ( b | y, λ ) (24)for all x, y, λ , where w A → Bλ + w B → Aλ = 1. Thus, the measurement x in a first regioncan influence the statistics of b in a second region via its outcome a (with someprobability w A → Bλ ), while the measurement y in the second region can influencethe statistics of a in the first region via its outcome b (with probability w B → Aλ ).Violations of a relaxed Bell inequality for such models, by some quantum systems,show that some quantum correlations cannot be described even if weak localityand outcome independence are relaxed to this extent [74, 75]. In contrast, thesimple one-particle example used to prove Proposition 3 suggests the possibilityof semi-device independent tests of models that are only constrained by outcomeand measurement independence (i.e., allowing weak locality to be fully relaxed),based on modifying the example to allow approximate perfect correlations. Thiswill be investigated elsewhere. References
1. J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics , 195–200 (1964)2. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev.Mod. Phys. , 419–478 (2014)3. E.P. Wigner, On hidden variables and quantum mechanical probabilities, Am. J. Phys. ,1005–1009 (1970)4. P. Suppes and M. Zanotti, On the determinism of hidden variable theories with strict corre-lation and conditional statistical independence of observables, in: P. Suppes (ed.), Logic andProbability in Quantum Mechanics, pp. 445–455, D. Reidel Publishing Company, DordrechtHolland (1976); Stanford postprint5. W. Demopoulos, Boolean representations of physical magnitudes and locality, Synthese ,101–119 (1979)6. J.S. Bell, Bertlmann’s socks and the nature of reality, Journal de Physique Colloques ,41–62 (1981)7. B. C. van Fraassen, The Charybdis of realism: epistemological implications of Bell’s in-equality, Synthese , 25–38 (1982)8. G. Grasshof, S. Portmann and A. W¨uthrich, Minimal assumption derivation of a Bell-typeinequality, Brit. J. Phil. Sci. , 663–680 (2005)9. T. Norsen, Bell locality and the nonlocal character of nature, Found. Phys. Lett. , 633–655 (2006)10. G. Blaylock, The EPR paradox, Bell’s inequality, and the question of locality Am. J. Phys. , 111–120 (2010)11. T. Maudlin, What Bell proved: a reply to Blaylock, Am. J. Phys. , 121–125 (2010)12. G.C. Ghirardi, On a recent proof of nonlocality without inequalities, Found. Phys. ,1309–1317 (2011)13. R.B. Griffiths, EPR, Bell, and quantum locality, Am. J. Phys. , 954–965 (2011)14. A. W¨uthrich, Local acausality, Found. Phys. , 594–609 (2014)15. H.M. Wiseman, The two Bell’s theorems of John Bell, J. Phys. A , 424001 (2014)16. T. Maudlin, What Bell did, J. Phys. A , 424010 (2014)17. R.F. Werner, Comment on ‘What Bell did’, J. Phys. A , 424011 (2014)18. T. Maudlin, Reply to Comment on ‘What Bell did’, J. Phys. A , 424012 (2014)19. R.F. Werner, What Maudlin replied to, arXiv:1411.2120 (2014)20. T. Norsen, Are there really two different Bell’s theorems?, Int. J. Quantum Found. ,65–84 (2015)21. H.M. Wiseman and E.G. Rieffel, Reply to Norsen’s paper “Are there really two differentBell’s theorems?”, Int. J. Quantum Found. , 85–99 (2015)22. E.J. Gillis, On the Analysis of Bell’s 1964 Paper by Wiseman, Cavalcanti, and Rieffel, Int.J. Quantum Found. , 199–214 (2015)8 Michael J. W. Hall23. H.M. Wiseman, E.G. Rieffel, and E.C.G. Cavalcanti, Reply to Gillis’s “On the Analysis ofBells 1964 Paper by Wiseman, Cavalcanti, and Rieffel”, Int. J. Quantum Found. , 143–154(2016)24. H.R. Brown and C.G. Timpson, Bell on Bell’s theorem: the changing face of nonlocality,in: M. Bell and S. Gao (eds), Quantum Nonlocality and Reality—50 Years of Bell’s Theorem,pp. 91–123, Cambridge University Press (2016); arXiv:1501.0352125. R. Tumulka, The assumptions of Bell’s proof, in: M. Bell and S. Gao (eds), QuantumNonlocality and Reality—50 Years of Bell’s Theorem, pp. 79–90, Cambridge University Press(2016); arXiv:1501.0416826. T. Norsen, Quantum solipsism and non-locality, in: M. Bell and S. Gao (eds), QuantumNonlocality and Reality—50 Years of Bell’s Theorem, pp. 204–237, Cambridge UniversityPress (2016); eprint27. A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physicalreality be considered complete?, Phys. Rev. , 777–780 (1935)28. A. Shimony, Controllable and uncontrollable non-locality, in: S. Kamefuchi et al. (eds.),Foundations of Quantum Mechanics in the Light of New Technology, pp. 225–230, PhysicalSociety of Japan, Tokyo Japan (1984)29. D. Albert and B. Loewer, Interpreting the many-worlds interpretation, Synthese , 195–213 (1988)30. N.D. Mermin, What do these correlations know about reality? Nonlocality and the absurd,Found. Phys. , 571–587 (1999)31. A. Kent, Locality and reality revisited, in: T. Placek and J. Butterfield (eds.) QuantumLocality and Modality, pp. 163-171. Kluwer, Dordrecht (2002)32. N. Gisin, Non-realism: deep thought or a soft option?,Found. Phys. , 80–85 (2012)33. M.J.W. Hall, Comment on ‘Non-realism: deep thought or a soft option?’ by N. Gisin,arXiv:0909.001534. M. ˙Zukowski and C. Brukner, Quantum nonlocality—it ain’t necessarily so. . . , J. Phys. A , 424009 (2014)35. C.A. Fuchs, N.D. Mermin and R. Schack, An introduction to QBism with an applicationto the locality of quantum mechanics, Am. J. Phys. , 749–754 (2014)36. R.B. Griffiths, Nonlocality claims are inconsistent with Hilbert space quantum mechanics,Phys. Rev. A , 022117 (2020)37. J. Jarrett, On the physical significance of the locality conditions in the Bell arguments,Noˆus , 569–589 (1984)38. A. Shimony, Events and processes in the quantum world, in: R. Penrose and C.J. Isham(ed.), Quantum Concepts in Space and Time, pp. 182–203, Clarendon Press, Oxford UK(1986)39. P. Rastall, Locality, Bell’s theorem and quantum mechanics, Found. Phys. , 963–972(1985)40. R. Spekkens, Evidence for the epistemic view of quantum states: a toy theory, Phys. Rev.A , 032110 (2007)41. J.F. Clauser and M.A. Horne, Experimental consequences of local objective theories, Phys.Rev. D , 526–535 (1974)42. J.S. Bell, The theory of local beables, in: J.S. Bell, Speakable and Unspeakable in QuantumMechanics, pp. 52–62, Cambridge University Press, Cambridge UK (1987)43. T. Norsen, Einstein’s boxes, Am. J. Phys. , 164–176 (2005); quant-ph/040401644. G. Bacciagaluppi and A. Valentini, Quantum Theory at the Crossroads: Reconsidering the1927 Solvay Conference, pp. 432–449, Cambridge University Press, Cambridge UK (2009);quant-ph/0609184, pp. 476–49645. J.F. Clauser and A. Shimony, Bell’s theorem: Experimental tests and implications, Rep.Prog. Phys. , 696–702 (1935)47. S.D. Bartlett, T. Rudolph, R.W. Spekkens, Reconstruction of Gaussian quantum mechan-ics from Liouville mechanics with an epistemic restriction, Phys. Rev. A , 012103 (2012)48. N. Bohr, Atomic Physics and Human Knowledge, pp. 32–101, New York, Wiley (1958)49. H. Everett III, Rev. Mod. Phys. , 454 (1957)50. H. Everett III, The theory of the universal wavefunction, in: B.S. DeWitt and N. Gra-ham (eds.), The Many-Worlds Interpretation of Quantum Mechanics, pp. 3–140, PrincetonUniversity Press, New Jersey USA (1973)oes locality plus perfect correlation imply determinism? 1951. R.B. Griffiths, Consistent histories and the interpretation of quantum mechanics, J. Stat.Phys. , 219-272 (1984)52. G.C. Ghirardi, A. Rimini and T. Weber, Unified dynamics for microscopic and macroscopicsystems, Phys. Rev. D , 470–491 (1986)53. P. Pearle, Combining stochastic dynamical state-vector reduction with spontaneous local-ization, Phys. Rev. A , 2277–2289 (1989)54. C. Rovelli, Relational quantum mechanics, Int. J. Theoret. Phys. , 1637-1678 (1996)55. C.A. Fuchs, C.M. Caves and R. Shack, Phys. Rev. A , 022305 (2002).56. D. Howard, Who invented the “Copenhagen Interpretation”? A study in mythology, Phil.Sci. , 669–682 (2004)57. D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variablesI, Phys. Rev. , 166–179 (1952)58. D. Bohm, A suggested interpretation of the quantum theory in terms of “hidden” variablesII, Phys. Rev. , 180–193 (1952)59. M.J.W. Hall, D.-A. Deckert and H.M. Wiseman, Quantum phenomena modeled by inter-actions between many classical worlds, Phys. Rev. X , 041013 (2014)60. M. Ghadimi, M.J.W. Hall and H.M. Wiseman, Nonlocality in Bells theorem, in Bohm’stheory, and in many interacting worlds theorising, Entropy , 567 (2018)61. C.H. Brans, Bell’s theorem does not eliminate fully causal hidden variables, Int. J. Theoret.Phys. , 219–226 (1988)62. G. S. Ciepielewski, E. Okon and D. Sudarsky, On superdeterministic rejections of settingsindependence, arXiv:2008.00631 (2020)63. G.C. Ghirardi, A. Rimini, and T. Weber, Lett. Nuov. Cim. , 293–298 (1980)64. M.J.W. Hall, Imprecise measurements and non-locality in quantum mechanics, Phys. Lett.A , 89–91 (1987)65. M.J.W. Hall, Relaxed Bell inequalities and Kochen-Specker theorems, Phys. Rev. A ,022102 (2011)66. B.P. Kosyakov, Holography and the origin of anomalies, Phys.Lett. B , 349–356 (2000)67. S.P. Bhat and D.S. Bernstein, Example of indeterminacy in classical dynamics, Int. J.Theoret. Phys. , 545–550 (1997)68. J.D. Norton, Causality as folk science, Philosophers Imprint , 1–22 (2003)69. B.P. Kosyakov, Is classical reality completely deterministic?, Found. Phys. , 76–88(2008)70. N. Gisin, Indeterminism in physics, classical chaos and Bohmian mechanics: are real num-bers really real?, Erkenntnis, doi: 10.1007/s10670-019-00165-8 (2019)71. J.S. Bell, Introduction to the hidden-variable question, in: J.S. Bell, Speakable and Un-speakable in Quantum Mechanics, pp. 29–39, Cambridge University Press, Cambridge UK(1987)72. J.S. Bell, Einstein-Podolsky-Rosen experiments, in: J.S. Bell, Speakable and Unspeakablein Quantum Mechanics, pp. 81–92, Cambridge University Press, Cambridge UK (1987)73. M. Gachechiladze, N. Miklin and R. Chaves, Quantifying causal influences in the presenceof a quantum common cause, arXiv:2007.01221 (2020)74. R. Chaves, R. Kueng, J.B. Brask, and D. Gross, Unifying framework for relaxations of thecausal assumptions in Bells theorem, Phys. Rev. Lett. , 140403 (2015)75. M. Ringbauer, C. Giarmatzi, R. Chaves, F. Costa, A.G. White and A. Fedrizzi, Experi-mental test of nonlocal causality, Science Adv.2