aa r X i v : . [ qu a n t - ph ] A ug Non-separability does not relieve the problem ofBell’s theorem
Joe Henson ∗ August 24, 2018
Abstract
This paper addresses arguments that “separability” is an assumption ofBell’s theorem, and that abandoning this assumption in our interpretation ofquantum mechanics (a position sometimes referred to as “holism”) will allowus to restore a satisfying locality principle. Separability here means that allevents associated to the union of some set of disjoint regions are combinationsof events associated to each region taken separately.In this article, it is shown that: ( a ) localised events can be consistently de-fined without implying separability; ( b ) the definition of Bell’s locality conditiondoes not rely on separability in any way; ( c ) the proof of Bell’s theorem doesnot use separability as an assumption. If, inspired by considerations of non-separability, the assumptions of Bell’s theorem are weakened, what remains nolonger embodies the locality principle. Teller’s argument for “relational holism”and Howard’s arguments concerning separability are criticised in the light ofthese results. Howard’s claim that Einstein grounded his arguments on the in-completeness of QM with a separability assumption is also challenged. Instead,Einstein is better interpreted as referring merely to the existence of localisedevents. Finally, it is argued that Bell rejected the idea that separability is anassumption of his theorem. There are a plethora of views about Bell’s theorem [1]: what its essential mean-ing is, whether it is a problem, and if so, whether this problem can be solved (see e.g. [2, 3, 4]). Bell’s own position, set out most clearly in his final exposition of thesubject ([1], 1990, pp.232-248), was that the theorem showed a contradiction betweenquantum mechanics (and, arguably, experiment) and a reasonable formalisation of aban on superluminal causal influence. Broadly the same view has recently been cham-pioned by Norsen [5, 6, 7, 8]. As has already been noted in several places [9, 10, 11], ∗ [email protected] e.g. magnetic effects), we are led to add detail to our theories ( e.g. fields). Simi-larly, any basic principle can be employed in reformulations of present theories, andin the search for new theories. Finally, upholding a principle banning superlumi-nal influence would naturally explain the predicted and observed inability to sendsuperluminal signals.For all these reasons, the claim that locality can be maintained in modern physics,despite Bell’s theorem, is an interesting one. Some commentators have claimed thatthe problem can be solved by pointing out implicit assumptions of Bell’s theoremthat are independent of locality, once that concept is properly construed [13] ([3],Jarrett pp. 60-79). At the extreme, Wessels lists a veritable menagerie of conditions,amounting to (at least) seventeen in total, and shows that these imply a Bell’s theorem[14] ([3], Wessels pp. 80-96). However, if one wants to rely on one’s favourite derivationof Bell’s theorem for the purposes of this discussion, one needs to show that theassumptions one makes are equivalent to, or weaker than, what is used (explicitly orimplicitly) in the standard versions. After all, if I added the assumption that I livein London to a derivation of Bell’s theorem, that would not make it reasonable for agroup of angry realists to drive me out of town in the hope of saving locality .With this in mind we can turn to the supposed assumption that will be the maintopic of this paper: “separability” [16] [17] ([3], Teller pp. 208-223, Howard pp.224-253). As Howard puts it, “we might... all along have been testing not simply localhidden variable theories, but separable, local hidden variable theories” ([16], p.195).Among the definitions Howard makes we find that “separability says that spatiallyseparated systems possess separate real states” ([16] p. 173) and, in a later treatment,that separability “is a fundamental ontological principle” which... asserts that the contents of any two regions of spacetime separatedby a nonvanishing spatiotemporal interval constitute separable physicalsystems, in the sense that (1) each possesses its own, distinct physicalstate, and (2) the joint state of the two systems is wholly determined bythese separate states. Maudlin has coined the term “The Fallacy of the Unnecessary Adjective” to indicate this problemwith descriptions of Bell’s theorem [15], citing “realistic,” “hidden-variable” and “counterfactuallydefinite” as examples of unnecessary adjectives that have cropped up between “local” and “theories”. particularism, another version of separability: “[i]n applicationto relativistic theories, particularism takes the form of supposing the theory to applyexclusively to spacetime points and their non-relational properties.” Teller has madea case for dropping this supposed assumption, arguing that locality and the resultsof QM can both be saved in this way, and naming the converse principle “relationalholism” [17] ([3], Teller pp. 208-223).This type of reasoning has been taken up in many places, for example by Shimony([3], Shimony, pp. 25-37), and in Redhead’s influential book on quantum nonlocal-ity, in which “denial of separability” is claimed to “block the derivation of the Bellinequality” ([2], p.168). Developing on this theme, Healey has proposed an “inter-active” interpretation of QM that relies heavily on non-separability, and discussedholism more broadly [18, 19, 20, 21], and Morganti has built on Teller’s ideas [22].On the other hand, Laudisa and Maudlin [23, 4] have disputed some of Howard’sclaims, while Berkowitz [24] and Fine and Winsburg [25], followed by Fogel [26], havecriticised some parts of Howard’s argument while accepting others (see appendix B).Following Howard’s second definition, separability has commonly been taken tomean that, for a set of disjoint regions, any statement that can be made aboutevents in the union of the regions is in fact a logical combination of statements aboutevents in each separate region, or equivalently the events in the union “supervene”on those in each separate region in the terminology of Teller and Healey[27] . If thiswas indeed a necessary implicit assumption of Bell’s theorem, and independent oflocality, abandoning the principle would be an attractive way to avoid the problem ofBell’s theorem. Healey has made an argument that separability should be abandonedin the context of (classical) gauge theories [27]. If this is so, why not pay the samesmall price in order to save locality in quantum mechanics as well?Adding to the motivation to carefully consider such options, in recent years therehas been increased interest in causal principles and Bell’s theorem fueled mainly bythe rise of quantum information studies. The Ψ-epistemic position, for example, hasbeen much discussed and debated [29, 30, 31, 32]. It is natural in this approach tohope that dropping some deep assumption in Bell’s theorem will relieve the tensionbetween locality and QM, analogously to the way that special relativity relieves thetension between relativity of motion and invariance of the speed of light . This isa good reason to carefully scrutinise the EPR debate and the assumptions of Bell’stheorem from this perspective. In [30], which makes much use of references to Howard,Harrigan and Spekkens state the following: “A necessary component of any sensible Healy compares definitions of separability in which physical processes supervene on propertiesdefined at points and/or arbitrarily small neighbourhoods of those points ([27], p.46). See also [28]for a related discussion. The main conclusions of this article are independent of this distinction. This last point derives from comments made by Spekkens in a seminar [33]. . Similarly,Ghirardi and Grassi state that “usually, in discussing EPR-like situations one payslittle attention to separability, which is in a sense tacitly assumed as a prerequisiteof the locality principle” ([35] p.405).Ambiguity of terms has often been a problem in these discussions, allowing ar-guments to move from general principles to quite specific statements about Bell’stheorem in ways that are not obviously justified. In this paper, some mathematicalformalisation is sought as an antidote to this. To discuss the claims it is useful tocollect together some cursory definitions of the principles that play a part in thisdebate, to be enlarged upon later. The first three are adapted from definitions madeby Healey [27]. Locality (principle):
There is no superluminal influence.
Local action (principle): influence between events in separated regions must bemediated by some event(s) in the intervening regions.
Separability (principle): all events associated to a disjoint union of regions su-pervene on the events associated to each of the regions in the union.
Principle of common cause:
If there is a correlation between two events, theneither one of these events has influenced the other, or there exists some common causeof the correlation (associated to the past), such that, once this cause is conditionedon, the correlation disappears.Unless noted, these meanings will be used throughout the paper. This must bekept in mind especially for the slippery word “locality,” which is often given differentmeanings, but (albeit with some reluctance) has been defined here in such a way asto make common sayings like “Bell’s theorem exposes a tension between locality andQM” valid.Separability, it will be demonstrated below, has in many places been confoundedwith a much weaker principle that will be called the principle of localised events. Spekkens later endorsed an different view, however, arguing that Bell’s theorem does not relyon separability [34, 33]. As will become clear, “correlation” here takes on a specific meaning: a lack of statistical inde-pendence of the events given a probability distribution, i.e. in an obvious notation, P ( A ) P ( B ) = P ( A ∩ B ). This is distinct from definitions by which the actual occurrence of two events implies acorrelation between them, as discussed in the present context in [4]. In line with this, “event” hasthe meaning it does in the theory of stochastic processes and e.g. [28], that is, it is a propositionthat may or may not be true in any possible realisation (or “history”) of the process, and to whichis associated a probability (or probabilities). That is, the definition of correlation employed heredoes not refer to actual particular happenings such as the fact that a particular coin flip produced“heads”, but the theoretically given probabilities of such results. ocalised events (principle): All events can be associated to regions in spacetimein a consistent manner.In order to formalise some of these principles, a straightforward and minimal frame-work is introduced in this article, in section 2. Using these tools, the principle oflocalised events is formalised in section 2.1, and is shown not to imply separabilityeven when a sensible structure is imposed on the assignments of regions to events.It is then shown in section 2.2 that locality can indeed be defined without referenceto separability, and, in section 2.3, that Bell’s theorem can be derived without anyassumption of separability. In short, it is neither the case that separability is a nec-essary part of locality, nor that it is an implicit and necessary assumption of thetheorem. Furthermore, in section 2.5 it is argued that two attempts to weaken theassumptions of Bell’s theorem inspired by relational holism can quickly be seen to beproblematic. After this is established, in section 3 these conclusions are comparedto those of Teller and Howard (prefaced with some comments on a common ances-tor of their arguments from Jarrett), and several important flaws in the argumentsare pointed out. The article ends with some comments on what Einstein and Bellthemselves actually wrote about separability in the context of interpretations fromHoward, Harrigan and Spekkens. Appendix A treats a variant of Bell’s local causalitywhile in appendix B it is argued that Howard’s “separability of states” condition isnot a good formalisation of his separability principle.
In this section a framework for dealing with causal principles will be explained, similarto that given in [36]. In essence the framework is that of stochastic processes, withsome extra structure to describe where the events occur in spacetime. The intentionis that the additional structure is only what is necessary for this task, and no more.In a sense, all that is being done here is to formalise some aspects of Bell’s idea of“local beables”. Attention will be payed to the issues discussed above while makingthese definitions. In particular, this will clarify the assumptions necessary to prove aBell theorem. Throughout this section, comments on the connections to the literatureare kept to a minimum, allowing the argument to stand for itself. Results will becompared to other treatments more fully in the following section.The first structure that is necessary is a spacetime M in which events will belocalised. We may as well take this to be Minkowski space, although any weaklycausal Lorentzian manifold ( i.e. one such that past and future causal relations areunambiguously defined as a partial order) will do. Subsets of M will be referred toas “regions”, and calligraphic script will be used to represent them. Next, we definea set of histories Ω, which may, for example, be a set of spacetime field configura-tions and/or particle trajectories. “Events” are subsets of Ω, for instance, in theprevious example, that a particle of some type passes through a region. As usualwith stochastic processes, the set of events is closed under an appropriate class oflogical operations. It forms a σ -algebra Σ( M ), meaning that that the set of events is5losed under complementation and countable unions of its members. The σ -algebraproperty allows us to consistently define a probability measure µ on the set of events.For countable history spaces this reduces to the more familiar Boolean algebra, andthe distinction will not be crucial for the main points below. This leaves open the question of how the events relate to the spacetime. Firstly, theprinciple of localised events given in the introduction requires that all events haveassigned to them some set of regions. But, one could argue, this on its own does notseem to embody the principle. It would surely violate the spirit of the principle if, forinstance, an event was associated to a region while its complement was not, or if thesame event could be assigned to many disjoint regions. It is necessary to be explicitabout these conditions, to see whether they imply anything about separability.To answer this, we introduce the following definition:
Localised Events: to every region in spacetime A is associated a subalgebra Σ( A )of Σ( M ), such that the following two conditions hold: \ i Σ( A i ) = Σ( \ i A i ) , (1)Σ( ∅ ) = {∅ , Ω } , (2)where {A i } is any countable collection of regions and ∅ is the empty set.Note that the intersection of a countable set of σ -algebras is also a σ -algebra. Wewill say that an event A is associated to a region A if A ∈ Σ( A ).The meaning of these expressions is straightforward. That a whole subalgebrais associated to each region means that, if some set of events is associated to aregion, then combinations (meaning logical combinations, if the events are thoughtof as propositions) of these events will be too. So for instance if “a blue ball passesthrough A ” and “a red ball passes through A ” correspond to events are associated to A , then “a blue ball and a red ball pass through A ” also is. But if we stopped there,we would still have the problem of one event being assigned to many disjoint regions.Equation (1) says that events associated to both A and B must be associated to theintersection of those two regions. This has the consequence that one event will notbe in the algebras associated to two disjoint regions, unless it is one of the “trivialevents” associated to the empty region, ∅ or Ω (although a non-trivial event can bein the algebra associated to one non-simply connected region).This last condition has a number of other interesting implications. Firstly, whenthis condition is assumed, A ⊂ B implies Σ( A ) ⊆ Σ( B ), or in other words an eventassociated to a region A is associated to any region that contains A , which is as itshould be. Secondly, every event A is associated to a unique “intrinsic” region thatis contained by all the other regions associated to A . Finally, because a union of a6et of regions is a superset of every region in the set, [ i Σ( A i ) ⊆ Σ( [ i A i ) . (3)In particular, for any two regions A and B , all the events that belong to A and B (andbecause they form a σ -algebra, combinations of these events) also belong to A ∪ B .Conditions (1) and (2) do not enter into the derivation of the Bell inequalitiesbelow. The only important thing about assuming localised events for that argumentis simply that all events are associated to some intrinsic region. However, theseconditions are relevant to the larger discussion in another way. Exhibiting themclearly shows that assuming the localised events principle need not imply separability,even when the association of regions to events is given a reasonable structure. In thisframework, the separability principle above becomes the following:
Separability: for a countable set of disjoint regions {A i } , G (cid:0)[ i Σ( A i ) (cid:1) = Σ( [ i A i ) , (4)where G ( X ) is the σ -algebra generated by the collection of events X . We will use theterm “non-separable event’ to refer to an event that is not a combination of eventsoccurring in a partition of some region to which it is associated.This condition agrees with the separability principle, and thus with the variousdefinitions in the literature. It is important to note that, from the localised eventscondition, we could only derive (3), which says that the algebra of events associatedto the union of many regions contains all the events belonging to the constituentregions and the combinations thereof. In contrast, separability says that, for disjointregions, these events are all that is to be found in the union of the regions.Unlike (3), it is easy to see that (4) does not follow from our definition of lo-calised events. Consider an example in which Σ( M ) = {∅ , Ω , X, ¯ X } , where the barindicates complimentation, and in which M contains a region X that itself containsmany disjoint regions. Let the algebra associated to X , and all regions containing it,equal Σ( M ), and all others equal the trivial algebra {∅ , Ω } . Clearly this meets theconditions for the existence of localised events, and just as clearly it violates separa-bility because X is non-separable. Such non-separable events can be added to anygiven stochastic process with localised events in an obvious way, showing that thisargument applies to a large and general class of models. The definition of localised events, although natural, does not say much about whatthese events are supposed to represent. Can we say that these events correspondto propositions about Bell’s local beables? Should we struggle to define intrinsic(which should rule out events associated to A corresponding to such statements as “a7oon-to-be-destroyed ball passes through region A ”) and qualitative (which shouldrule out events associated to A corresponding to statements like “the man passingthrough region A is the King of Sweden” unless they are equivalent to some qualitativeformulation) physical properties [19, 21]?The answer to this will be supplied in the form of more formal conditions. Inspirit, the answer is that the formal events of the stochastic process are intended torepresent the kinds of events that physics should be primarily concerned with accord-ing to Einstein and Bell, at least while spacetime remains fixed (see sections 3.3, 3.4and 3.5). That is, the events have two conceptual roles which are deeply intertwined.Firstly, operational events that are observable in a region (whatever we are comfort-able including here) must be associated to that region. Bell says “[t]he beables mustinclude the settings of switches and knobs on experimental equipment, the currentsin coils, and the readings of instruments” and he goes on to associate such settingswith appropriate regions in a straightforward fashion ([1], 1975, p. 52). Secondly, theevents can serve as causal factors for other events, operational or not, allowing causalexplanations. With these rules in place the events become recognisably physical,answering questions about “beables”. Questions of whether events refer to intrinsicor qualitative properties are, in essence, questions about whether the events are of atype that has the right to play these two roles; here, it will be assumed that they are.We represent the latter of these two ideas as “Bell locality”. To do this, we willneed to define a full specification of a region A . This is an event F that, when given,fixes the truth of every other event in A . More precisely, it is an event belonging to A such that, for every other event X ∈ Σ( A ), F ⊆ X or F ⊆ ¯ X [36]. The idea is thata full specification of the past of a region A includes all events that might possiblyinfluence events in A , and that any correlations between A and a spacelike region B should be a result of influences from these past events. Bell gave a such a principle,calling it “local causality” ([1], 1975, p. 54). Bell locality: for any two events A and B associated to spacelike-related regions A and B respectively, the following condition holds: µ ( A | λ ) µ ( B | λ ) = µ ( A ∩ B | λ ) ∀ λ ∈ Φ (cid:0) P ( A , B ) (cid:1) , (5)where Φ (cid:0) P ( A , B ) (cid:1) is the set of full specifications of the region P ( A , B ), which is asuitable past region for the pair {A , B} .Bell locality represents a conjunction of a principle of common cause (PCC) withrelativistic causal structure. Correlations between events should only be due to directcausal connections between the events, or due to some common cause, in which casethey should disappear when we condition probabilities on all events that might berelevant to such a cause. By adding relativistic causal structure, we ban the possibilityof direct causal connection between spacelike events and “all events that might berelevant” to the common cause become a full specification of the past region (see[37, 36] for more justification of this). 8n the above, the definition of the past region has been left open deliberately. Itcould be the intersection of the past lightcones of A and B , the “mutual past”, ortheir union (with A and B removed so that they do not lie in the past region), the“joint past,” or the past of one of the regions but not the other . Below, we will seethat the choice does not matter much for the issues at hand.One of the conclusions becomes clear at this point: to define Bell locality, wehave not had to assume separability. We used nothing beyond the assumption thatevents happen in regions. Separability is not implicit here at all, and the existence ofnon-separable events in the past region makes this definition no less reasonable. Theywould be part of possible common causes in the past, and would be conditioned on aswell. After conditioning the probabilities, there is no remaining correlation between A and B . That is all Bell’s condition says.To expand on this important point, imagine a laboratory, and an event that isassociated to a region that extends for a light year around the lab in every directionat some moment in time (in the lab frame). Locality implies that such an eventshould not affect anything in the lab for at least a year. The point here is simplythat this is true whether the event is separable or not. The formal condition of Belllocality naturally incorporates this. Under these circumstances it is clearly mistakento argue, as some have, that the very existence of such an event would threaten ourindividuation of the lab system itself and thus any possible notion of locality (seesections 3.4 and 3.5).Another point in the above discussion needs to be emphasised as an assumption,although this does not have such a formal character. Operational consistency of localisation:
Operational events are associated withthe regions of spacetime in which they can be detected.Again, this condition does not imply separability. It does not even imply that theoperational events must be separable. But it is required to avoid absurd conclusions,as discussed at more length in section 2.5.Going forward to discuss the EPRB experiment, it is good to take stock of what weare assuming. We need to assume that some set of operational events are associatedto regions, and that it makes sense to condition on a full specification of the past ofthose regions; although this follows from the localised events condition, which seemsnatural to impose, nothing more from this condition is strictly necessary. We will alsoneed to assume Bell locality, operational consistency of localisation, and a “freedomof settings” assumption. We will not assume separability in any form. Conditions of this type, with these different past regions, are shown to be equivalent in a similarframework in [36] (for some debate on whether the proof is sound, see [38, 39]). However, the proofemployed separability, called rule ( iv ) there. .3 The EPRB experiment In this experiment, we have two spacelike regions A and B , representing two “wings”of our experimental apparatus. In these regions some operational events are defined.The events A s and B s correspond to whether one particular setting is chosen on someinstrument in regions A and B respectively (by tradition, some orientation of a Stern-Gerlach magnet rather than another). The events A o and B o refer to some outcomes(whether the spin is measured “up” according to the given orientation). We will alsorefer to the variable a o that can take the value A o or ¯ A o , and so on .Applying Bell locality directly to events in A and B , we have µ ( a o ∩ a s ∩ b o ∩ b s | λ ) = µ ( a o ∩ a s | λ ) µ ( b o ∩ b s | λ ) , (6)for all values of the variables and all full specifications of the past λ . Here we in-troduce the freedom of settings assumption, essentially that the choice of settings isindependent of past events: µ ( a s ∩ b s | λ ) = µ ( a s ∩ b s ) . (7)In this form, the condition makes the settings completely spontaneous in the sensethat they depend on nothing at all in the past .By summing over some variables, it follows from (6) and (7) respectively that µ ( a s ∩ b s | λ ) = µ ( a s | λ ) µ ( b s | λ )= µ ( a s ) µ ( b s ) . (8)From this, we can see that µ ( a o ∩ a s ∩ b o ∩ b s | λ ) = µ ( a o ∩ b o | a s ∩ b s ∩ λ ) µ ( a s ) µ ( b s ) , (9)and similar results are easily derived for each wing individually: µ ( a o ∩ a s | λ ) = µ ( a o | a s λ ) µ ( a s ) , (10) µ ( b o ∩ b s | λ ) = µ ( b o | b s λ ) µ ( b s ) . (11) The are a number of choices in how to present the experiment. Here I select the “big history spaceapproach” [28] that includes A s and B s as events with probabilities of their own, rather than havingto deal with the complication of treating them as parameters of a family of probability distributionswhich are nevertheless localised. These doubtful probabilities could be removed without harmingthe argument in its essentials, as usual. The experimental probabilities µ ( a o ∩ b o | a s ∩ b s ) do notdepend on the absolute probabilities of the settings. This may not be so realistic; in a more general treatment, we might allow settings to correlateto some set of past events in the model which nonetheless were independent of every other eventof relevance, or even allow limited correlations using some “careful epsilonics” as Bell puts it ([1]1977 p.102). Another way to represent the same situation would be to replace the requirement that λ be a “full specification” with the requirement that it has “sufficient completeness for a certainaccuracy”, as Bell explains ([1] 1977 p.104). See [6, 40] for more on this. In any case, the mainpoint of this article does not turn on the issues. µ ( a o ∩ b o | a s ∩ b s ∩ λ ) = µ ( a o | a s ∩ λ ) µ ( b o | b s ∩ λ ) . (12)This last condition is commonly referred to as “factorisability”. The CHSH inequal-ities can be derived from it in a straightforward manner [41]. This remaining partof the proof of Bell’s theorem is essentially a series of elementary mathematical ma-nipulations, and, at least in the published literature, it is uncontroversial that thesesteps make no use of separability.What have we learned from this brief rehearsal of the well-known result? Previ-ously we saw that it is not necessary to assume separability in order to make senseof Bell locality, and that none of our assumptions imply separability. Now we haveseen that Bell inequalities can be derived in the normal manner without assumingseparability. That is, separability does not enter into the argument at all .A reader versed in the lore of Bell’s theorem might, at this point, wonder whathas become of another condition sometimes labelled “separability,” Howard’s “sep-arability of states” [16] ([3], Howard pp. 224-253). How do such conditions relateto the definition given here, and if they differ, is this a hidden assumption of Bell’stheorem? These issues are discussed in appendix B and section 3.3 respectively. Before going on, it is important for the following discussion to note that Bell localityand freedom of settings together imply a prohibition on superluminal signalling, whichwill be referred to as “no-signalling” for short: µ ( a o | a s ∩ b s ) = µ ( a o | a s ) , (13) µ ( b o | a s ∩ b s ) = µ ( a o | b s ) . (14)It is well-known that, as predicted by QM, this condition holds in the EPRBexperiment and in general, unlike the set of assumptions used to derive Bell’s theorem.It is relevant for later discussion to note a few basic consequences of this.To start with, if one claims that a prohibition on superluminal influence is equiv-alent to a prohibition on superluminal signalling, there is no need to analyse Bell’s Healey once convincingly argued that a non-separable event in a region
A∪B could be influencedby events to its past without violating “relativistic locality” [21]. However, the conclusion that “thereis no sound reason” to say that his interactive interpretation [18] of the EPRB experiment “violatesany defensible relativistic locality condition” ([21], p. 369) is unwarranted. The argument aboveshows that it must violate one: the very condition that causes all the fuss about quantum non-locality. It should be noted that Healey has since put forward a different interpretation, however[42]. Note also that the exact shape of the past region seemed not to enter, except insomuch as theregion arguably has to be a certain shape to sufficiently specify the past (see appendix A). As longas freedom of settings (7) refers to the same λ as Bell locality (6) the proof will go through. e.g. [1], 1990, pp. 232-248) and has been defended by e.g. Maudlin [4] and Norsen[8]. This means that, if we are worried about Bell’s theorem at all, then we areworrying about a definition of superluminal influence that goes beyond superluminalsignalling. It would be one thing to take on the arguments of Bell directly, and todiscuss in detail the difference between the two conditions and the motivation forBell’s locality. However, it is another to simply re-assert the claim that no-signallingsuffices. Attempts of this type have already been answered in the earlier treatments.Also, no-signalling can be thought of as a consistency check for the idea thatBell locality is a good representative of locality. Bell locality implies no-signalling,given the freedom of settings assumption. That assumption serves here to define anecessary feature of a genuine signal (that is, that the cause of the signal, the setting,is not correlated with the outcome only because of the influence of some past event).This is necessary, because it is a foundational assumption that operational signallingis a form of influence, which we will not question here.Both of these points will be relied upon in what follows to deal with some argu-ments that the assumptions of Bell’s theorem could be weakened while preserving lo-cality. If the argument is just a round-about way of reducing locality to no-signalling,it will be dismissed (or more properly referred back to other writings on the subjectincluding Bell’s own). If it leaves us with a candidate for locality that does not implyno-signalling by itself, it will be rejected as too weak.
Separability has been shown to play no part in the derivation of Bell’s inequalities.However, considerations involving non-separability might give grounds to weaken theconditions set out above. The aim would be to perform such a weakening while pre-serving a realisation of the locality principle. Expanding on Teller’s work, Morgantihas made arguments broadly of this form [22]. We will examine similar ideas in thissection.Howard writes that “[i]f two systems are not separable, then there can be nointeraction between them, because they are not really two systems at all” ([16] p. 173,Howard’s italics). Similarly, Redhead writes that the particle in one wing of the EPRBexperiment “does not possess independent properties of its own” ([2], p.107). Thesecomments give some inspiration for our task. But they also beg the question of whatin fact we are measuring and where it is, if statements of this sort are supposed toease our minds on the problem of Bell’s theorem.To make this rather vague idea explicit, we need to translate it into a criticism ofone of the assumptions made above. Perhaps it is wrong to assume that an event A associated to A can only be influenced by events belonging to regions that lie entirely in the causal past of A . Perhaps it is only necessary, for non-separable events, that12 ome points of the region lie to the past of A . This motivates altering Bell localityto condition not only on a full specification of the past, but also on any non-separableevents whose intrinsic regions overlap the past region.However, if we take the past region P for Bell locality to be the union of the pastsof A and B , it is indeed possible that a full specification of the region could includenon-separable events not contained in the past of A alone. But the proof of Bell’stheorem still stands, as we have seen. Perhaps then, not only can events be influencedby non-separable events associated to regions only partially in their past, but also,events can influence non-separable events associated to regions only partially in theirfuture? This does not work either: with this modification Bell locality no longer banssuperluminal signals. Consider two spacelike regions A and B as above, and a region C , some points of which lie to the future of A , and some points of which lie to the pastof B . If we accept the proposed modification of Bell locality, any events in A , evensettings, can be correlated to events in C and thus to B , without any restrictions. Sothis proposed weakening clearly amounts to denial of the locality principle.There is another way to challenge to the above framework, which might also beseen as a version of an argument that is present in the literature . The idea is toweaken operational consistency of localisation. For the EPRB setup, perhaps theoutcome A o in the A wing could be said to belong to A ∪ B , and not to A as before.This seems to be what is suggested by saying that such events have no separateexistence in the separate wings. Note that, for this to undermine the proof of Bell’stheorem, no operational event correlated to this outcome that happens subsequentlycould be assigned to a region using operational consistency of localisation, either. Forinstance, if someone in the lab might remark “look, the spin is up” as a result of thatoutcome, and we assign this event to A (or any region spacelike to B ), we can still Morganti agrees with this proposal:At most, the evidence requires one to put into doubt what, following Jones, one maycall causal separability, that is, the requirement that an event A can be the causeof another event B only if A has a part entirely in the past light cone of B thatentirely causes B . Indeed, if the emergent property of the whole system is such thatit is affected in its entirety by a measurement localized where one particle is and, asa consequence of this, determines a new categorical property of another particle at adifferent location, it is clear that causal separability fails. But this, as explained, isnot in itself a violation of locality.([22], p. 1033).The reference is to [43]. Morganti clearly states his belief that “locality,” which canbe taken to share the meaning it is given here, can be preserved while making this move. The following quote from Morganti illustrates this:In other words, a measurement on an entangled system, commonly understood as anevent E localized where one of the particles is and determining another event E localized where the other particle is in fact an event E located everywhere the totalsystem is (in particular, at the locations of both component particles) that determinesevents E and E localized at different places and yet in physical and spatiotemporalcontinuity with their cause, E . This means that one has a process that is entirelylocal at each stage.([22], pp. 1031-1032). This seems consistent with the version made explicit here, and the quotesfrom Howard and Redhead given earlier in this section also tend in this direction. this event instead of A o – even though the quantumentanglement has been eliminated by the time the person speaks.Consider a situation in which an experimenter in the A wing sees a light flash. Thecorrelation between this and the flashing of another distant light cannot be explainedby some common cause in the past, according to our assumptions. The suggestionabove is that one could say to the worried experimenter “don’t worry, when you sawthe flashing light, that actually corresponded to an event in the whole experimentalregion, not an event in your lab. So you see, it’s all local... Actually, your reaction tothe flashing light didn’t happen in your head either, but in the larger region.” If weopen the door to such arguments, a more elegant way to ensure locality would be toassociate every event to one point, despite the convictions of the metaphysically na¨ıveon the matter. More specifically, it is in no way more unreasonable to apply this kindof reasoning to a hypothetical case of superluminal signalling than it is to apply it tooutcomes in the EPRB experiment. This, as has been repeatedly emphasised, clearlyentails a violation of locality.In both of these scenarios, it might seem that we can escape by falling back onan assumption of prohibition on superluminal signalling. Is not that what causesthe unpalatable conclusions? But if we rely on this, we may has well have avoidedanalysis of Bell’s theorem by rejecting all locality assumptions except no-signallingin the first place. Our policy, as set out in the previous section, is to dismiss sucharguments in the context of analysing Bell’s theorem.Thus, we see that the various pieces of wordage on separability to be found inthe literature, attractive though they might seem, start to look considerably lessattractive in the light of the framework used above. Once they are cast into anexplicit form by pointing to a specific place at which the assumptions of Bell’s theoremshould be weakened, they can be overturned with elementary arguments. There maybe more weakenings of the assumptions of Bell’s theorem that can be motivated bysome words about non-separability. To my knowledge, no other such option has beenput forward with sufficient clarity to justify adding it to the list. It is a stronglymotivated conjecture that any other attempt of this sort could be just as easily ruledout. In the previous section, we saw that denying separability does not save us from thederivation of Bell inequalities. The discussion there was, hopefully, fairly straightfor-ward. The greater difficulty comes in comparing this conclusion to previous writingsno the subject, in order to see how misconceptions have arisen.Since a number of the arguments surrounding non-separability and holism aremotivated in large part by Jarrett’s analysis of the EPRB experiment [13] ([3], Jarrettpp. 60-79), and use terminology related to it, it is worth taking a slight detour toexamine these ideas, ahead of criticisms of Teller and Howard.14 .1 Jarrett
The relevant part of Jarrett’s analysis starts with the application of Bell causality tothe EPRB set-up, (6) above. Assuming freedom of settings, he resolved this conditioninto two which together imply (6). The first he calls “completeness”, which Shimony([3], Shimony pp. 25-37) renamed “outcome independence”: µ ( a o | a s ∩ b s ∩ b o ∩ λ ) = µ ( a o | a s ∩ b s ∩ λ ) , (15) µ ( b o | a s ∩ b s ∩ a o ∩ λ ) = µ ( b o | a s ∩ b s ∩ λ ) , (16)while the second he called “locality” (“parameter independence” for Shimony) whichwill be referred to here as Jarrett locality: µ ( a o | a s ∩ b s ∩ λ ) = µ ( a o | a s ∩ λ ) , (17) µ ( b o | a s ∩ b s ∩ λ ) = µ ( a o | b s ∩ λ ) . (18)Note that this relies on treating outcomes and settings differently, unlike Bell’s localcausality in which they are all “beables”. The only difference between them in theprevious discussion was, naturally, in the freedom of settings condition, (7).This mathematical derivation is used as the pushing-off point for a certain view ofBell’s theorem, given in a generic form here. Jarrett emphasises the claim that Jarrettlocality is equivalent to no-signalling, stating “LOCALITY ⇔ NO SUPERLUMINALSIGNALS” in a box in one paper ([3], Jarrett pp. 69). He holds that Bell localityis saying something stronger than this, to do with principles of common cause. Buthe also refuses to view a failure of Bell locality as a failure of locality, relying on hisown definition instead (in his argument he is chiefly concerned with potential conflictbetween QM and special relativity). Howard, as we will see, goes a little further inhis wording, identifying Jarrett locality with a satisfactory realisation of the localityprinciple.Jarrett’s analysis has been criticised on many grounds [4, 8]. Maudlin points outthat a prohibition on signalling does not imply Jarrett locality, and so Jarrett localityshould not be confounded with no-signalling as in eqn. (13). It is not hard to seethat Jarrett locality, along with freedom of settings as in (7), implies no-signalling.But the converse is not true in general. A number of authors have noted that inde Broglie-Bohm pilot wave mechanics, for instance, there is no-signalling in EPRBexperiments, even though Jarrett locality is clearly violated [9, 4, 44, 24, 8]. Theconfusion comes from shifting the definition of λ . When talking about no-signalling,Jarrett slips from treating past events λ as some hypothetical variables describingthe past which may not be operationally accessible (as in Bell’s analysis), to treatingthem as a known initial state s of an operational character, dropping considerationof any “hidden” past events. The value of such an s could be known to the agents,so that they can condition their probabilities on it, and then it is clear from (17) and(18) that changing the value of a setting could change the measured probabilities fora spacelike outcome if Jarrett locality was abandoned. But that is not necessarily thecase when we take Bell’s meaning for λ , as the pilot wave example makes clear.15e will add here that, were it not for the error in identifying Jarrett locality withno-signalling, Jarrett’s argument could be taken as a review of Bell’s theorem andno-signalling in the EPRB experiment. He merely notes that no-signalling relieves usof the most obvious potential conflict with relativity. So, were it not for this error,there would be nothing new in claiming that Jarrett locality is sufficient to ensure thelocality principle (as Howard does). This claim would merely be the assertion thatlocality is the same thing as no-signalling, against the warnings of Bell, as alreadyconsidered in section 2.4. In his writings on Bell’s theorem [17] ([3], Teller pp. 208-223), Teller is concernedwith reconciling relativity with quantum mechanics. He says he will “call any theorywhich embodies these local actions and no superluminal propagation requirementsRelativistic Causal theories”, that is, Relativistic Causality is for him (at least at thispoint in the paper) a combination of what are here called local action and locality.He then claims that “our unhesitating acceptance of relativistic causal theories...involves an assumption so basic to the thinking of most of us that we are not evenaware that we are making it”. This is the assumption of separability, which he calls anapplication of “particularism”: “[i]n application to relativistic theories, particularismtakes the form of supposing the theory to apply exclusively to spacetime points andtheir non-relational properties” (p.213). This definition agrees with the separabilityprinciple above, and is adequately represented by the separability condition given inthe framework of section 2. He goes on to add “we should question the unspokenassumption, particularism, which sets the precondition for getting any relativisticcausal theory off the ground ” (my italics). The following argument is offered later inthe article as a justification of this claim:But when (or insofar as) particularism is denied, the idea of causal lo-cality has no application. Causal locality concerns the lawlike connectionbetween nonrelational properties applying (in this discussion) to space-time points. To say that causal locality has been violated most plausiblyshould be taken to mean that there are nonrelational properties of space-time points which are related in some other way (lawlike dependencies)at a distance or through superluminal causal chains. On the other hand,when we are concerned with nonsupervening relations, this circle of ideashas no grip.([3], Teller p. 215).This is little more than a reiteration of the claim. It is supposedly implausible toattribute a violation of local causality to properties that do not apply at spacetimepoints, but no explanation is given for this. Instead, Teller again begs the questionby assuming that “causal locality” is only plausible as a principle when applied to aseparable theory. We have seen that this is not the case in the previous section.Later in the same article Teller makes the argument more specific for the prob-abilistic version of Bell’s theorem reviewed above. He gives a version of Jarrett’s16onditions and argues, with Jarrett, that outcome dependence is the part of Bell lo-cality that we must reject. The problem still remains, however, of explaining outcomedependence as an instance of failure of particularism rather than anything suggestiveof a conflict with relativity. To this end, Teller states a version of the principle ofcommon cause due to Lincoln Moses, claims that “Moses was unwittingly presuppos-ing particularism,” and adds the supposed assumption explicitly to the PCC. Addingrelativistic causal structure, Teller gives his weakened version of what is here calledlocality: “when particularism holds, relativistic causal theories exclude superluminalcausal propagation.” He then argues that “if one grants the assumptions about a par-ticularist world, failure of outcome independence must involve some kind of failureof particularism”.Let us set aside the fact that, in the same article, relativistic causality was pre-viously defined to mean local action and no superluminal propagation, but the argu-ment is now that relativistic causality only implies these things under the conditionof particularism, giving the impression that this attractive-sounding principle couldbe saved if one denies particularism. There are larger problems. Teller’s argumentdepends on the assumption that locality, and indeed the PCC itself, only make senseunder the assumption of separability (“particularism”), and otherwise do not. Thispoint is returned to often in the article, and is depended on at the conclusion. Butthere is no substantial argument to justify this assumption anywhere the article. Thepassages quoted above are the most explicit to be found. To make matters worse thisclaim is false, as has been demonstrated in section 2. Without such a justification,what remains is the form of argument referred to in the introduction: adding an un-necessary assumption to Bell’s theorem and then claiming to have saved locality byremoving it. One can only suppose that the incorrect and unsubstantiated assertionis implicitly taken to have been established in the literature already, perhaps from aninterpretation of Howard’s work.
Howard also wrote on separability, using a certain reading of Einstein as inspirationfor a treatment of Bell’s theorem [16] ([3], Howard pp. 224-253). His position isdistinct from Teller’s in several significant ways. In the earlier of the two articles,he writes that “separability says that spatially separated systems possess separatereal states”. In his later, more careful definition, Howard’s separability means thatdisjoint regions are such that “(1) each possesses its own, distinct physical state,and (2) the joint state of the two systems is wholly determined by these separatestates.” While (at least in the light of the discussion above) the first part mightsuggest something like the localised events principle, the second is the part that hasled to definitions of separability such as that used above. In the same place, Howarddistinguishes two kinds of separability:The more modest concerns the individuation of states; it is the claimthat spatio-temporally separated systems do not always possess separable states , that under certain circumstances either there are no separate states17r the joint state is not completely determined by the separate states. Icall this way of denying the separability principle the non-separability ofstates . The more radical denial may be called the non-separability ofsystems ; it is the claim that spatio-temporal separation is not a sufficientcondition for individuating systems themselves, that under certain circum-stances the contents of two spatio-temporally separated regions constitutea single system.([3], Howard p. 226, Howard’s italics).Howard’s inspiration comes from a number of quotes from Einstein, primarily thefollowing, which precedes an argument for the incompleteness of orthodox quantummechanics:If one asks what is characteristic of the realm of physical ideas inde-pendently of the quantum-theory, then above all the following attractsour attention: the concepts of physics refer to a real external world, i.e.,ideas are posited of things that claim a real existence independent of theperceiving subject (bodies, fields, etc.), and these ideas are, on the onehand, brought into as secure a relationship as possible with sense impres-sions. Moreover, it is characteristic of these physical things that theyare conceived of as being arranged in a space-time continuum. Further,it appears to be essential for this arrangement of the things introducedin physics that, at a specific time, these things claim an existence inde-pendent of one another, insofar as these things “lie in different parts ofspace.” Without such an assumption of the mutually independent exis-tence (the “being-thus”) of spatially distant things, an assumption whichoriginates in everyday thought, physical thought in the sense familiar tous would not be possible. Nor does one see how physical laws could beformulated and tested without such a clean separation. Field theory hascarried this principle to the extreme, in that it localizes within infinitelysmall (four-dimensional) space-elements the elementary things existing in-dependently of one another that it takes as basic, as well as the elementarylaws it postulates for them.For the relative independence of spatially distant things (A and B), thisidea is characteristic: an external influence on A has no immediate effecton B; this is known as the “principle of local action”, which is appliedconsistently only in field theory. The complete suspension of this basicprinciple would make impossible the idea of the existence of (quasi-) closedsystems and, thereby, the establishment of empiricaly testable laws in thesense familiar to us.([16] p. 187-188, translated from [45] pp. 321-322.)In the following section this quote will be discussed further, but for now the focuswill be on Howard’s reading of it. From this quote, Howard claims that “Einstein’s‘principle of local action’ and his ‘assumption of the mutually independent existence18f spatially distant things’ correspond, respectively, to the locality and separabilityprinciples.” ([3], Howard p.234). In other words, Howard’s claim is that Einstein hereis describing two different principles, separability in the first paragraph and localityin the second. He goes on to claim that separability and locality are independentassumptions of Bell’s theorem.Whatever Howard’s “non-separability of systems” is, “non-separability of states”is more clearly modeled on the “non-separability” apparent for entangled quantumstates. Howard explains that “at a minimum, the idea is that no information iscontained in the joint state that is not already contained in the separate states, or,alternatively, that no measurement result could be predicted on the basis of the jointstate that could not already be predicted on the basis of the separate states” ([3],Howard p. 226). This, along with the above definition, will be called the “separabilityof states principle” in the following. In the same place, Howard defines a “localityprinciple”, saying that “the relativistic version of the principle asserts that a system’sstate is unaffected by events in regions of spacetime separated from it by a spacelikeinterval” ([3], Howard p. 227). This is sufficiently similar to the locality principlestated above as to be identified with it for the purposes of this argument.For the EPRB experiment, Howard goes on, “I define a state as a conditionalprobability measure assigning probabilities to outcomes conditional upon the presenceof global measurement contexts”. He then formalises separability of states as a specificcondition in the case of the EPRB set-up by making the following identifications: µ ( a o | a s ∩ b s ∩ λ ) = µ α ( a o | a s ∩ b s ) , (19) µ ( b o | a s ∩ b s ∩ λ ) = µ β ( b o | a s ∩ b s ) , (20)“where α and β represent the separate states of the systems in the A and B wings,and λ represents the joint state” . For Howard, λ is taken to be the joint state ofthe two wings before the other relevant events (in Howard’s notation λ is also a labelon probability distributions). The separability of states condition is then µ ( a o ∩ b o | a s ∩ b s ∩ λ ) = µ α ( a o | a s ∩ b s ) µ β ( b o | a s ∩ b s ) . (21)With the above definitions, this “separability of states” turns out to be equivalent toJarrett’s “completeness” as in eqn. (15). Thus it is a consequence of the assumptions Laudisa also criticises Howard’s arguments on separability, and his main criticism turns on thispoint [23]. He argues that, if α and β are defined as representing these separate states, then theidentifications (19) and (20) are not justified. For instance, in Bohm’s theory we might be temptedon this basis to make α just the position of the particle headed to the A wing and similarly for β . But λ should also contain the quantum state in that case, as that is part of the ontology aswell, and so (19) and (20) would not necessarily hold in that case. However, the association of thewave-function to a region has been left ambiguous here: there is nothing inconsistent with Bohm’stheory in including a copy of the quantum state at, say, every point in space (an observation made byLucien Hardy, private communication), and so it could be in both α and β . The arguments againstHoward’s views given here are different from Laudisa’s: for instance, Laudisa does not challengeJarrett’s views on Bell’s theorem or Howard’s interpretation of Einstein. Related, though distinct,arguments are made in appendix B. For now, we treat the equations simply as identifications thatdefine the meaning of α and β , accepting Howard’s definitions, and assess the consequences of this.
19f Bell’s theorem. Now, by this point, the definition of separability of states has di-verged from the definition of separability given in section 2.1, and arguably Howard’sown original definition, but discussion of this point will be relegated to appendix Bin order not to distract from a more important problem with Howard’s argument.This is the basis on which Howard argues that separability and locality are in-dependent assumptions of Bell’s theorem. After repeating Jarrett’s erroneous claimthat Jarrett locality is equivalent to no-signalling, he explicitly takes Jarrett localityas in eqn. (17) to be an adequate representative of the (relativistic) locality principleset out above: “[o]n Jarrett’s analysis, a violation of the Bell inequality need not entail relativistic nonlocality, because it may result either from a violation of theJarrett locality condition, or from a violation of his completeness condition” ([3],Howard p.230, Howard’s italics).It has already been argued in section 3.1 that removing the reliance on Jarrettwould clarify such arguments. What remains after this is done is nothing but a claimthat no-signalling can be identified with locality, and stronger conditions of locality(such as Bell’s) are therefore unwarranted. This entails a disagreement with Bell’soriginal arguments for preferring Bell locality, as noted in section 2.4. Howard’sargument must boil down to some new reason to reject Bell’s arguments for localcausality, if it is to have any content.Simply rejecting the formal condition of “separability of states” as in eqn. (21)(which is implied by Bell locality) is not therefore the crux of the argument. Thecrux is, rather, the claim that “separability of states” is not an implication of thelocality principle, properly understood. Why not? Howard simply asserts that Jarrettlocality has more right to the name of locality, as it is equivalent to no-signalling. Hedoes not take on Bell’s justifications for Bell locality directly. Thus there is no newargument against Bell’s claim that no-signalling is not enough. Howard’s argumenthas no new content once Jarrett’s erroneous claim is removed.It remains to consider what Einstein and Bell wrote about separability and itsrelation to their arguments about locality. In this paper a number of different notions of “independence of events associated todisjoint/spacelike regions” have been distinguished and some of their consequenceshave been discussed. A number of quite different concepts have been revealed, whichare, however, often referred to with the same words. This raises questions overHoward’s interpretation of Einstein. In contrast to Howard’s reading of the quotedpassages, it is consistent to interpret Einstein’s claim that “things claim an existenceindependent of one another” with something along the lines of the localised eventscondition as given in section 2, or at least, something weaker than the separabilityprinciple. That is, to each region of spacetime is attributed certain events in anunambiguous manner. Dropping this more basic assumption, rather than separability,gives much more reason to state that “without such an assumption... physical thoughtin the sense familiar to us would not be possible.”20hen Einstein says that “[f]ield theory has carried this principle to the extreme”he gives a definition more in line with the separability principle used here: “it localizeswithin infinitely small (four-dimensional) space-elements the elementary things exist-ing independently of one another that it takes as basic.” If this is the “extreme” caseand not the general case that Einstein explains earlier in the paragraph, this surelyimplies that what he said earlier in the paragraph is not equivalent to separabilitybut something weaker.If we accept Howard’s argument, then we must attribute to Einstein an inabilityto imagine physical thought without separability. Howard provides a possible reasonfor this. For Howard, Einstein is arguing that “spatio-temporal separation is the onlyconceivable objective criterion of individuation” of systems, which may be necessaryfor any conceivable physics ([3], Howard p.241). Whether this is correct or not, wecan now ask: is not the localised events assumption a good enough basis on which toindividuate systems? This principle supplies a definition of what is associated to aregion X and what is not (any non-separable events associated to regions overlappingbut not contained in X are not), which should be sufficient . Thus, most of Howard’sspeculations about Einstein’s motivations for assuming separability, arguably, actu-ally apply to the assumption of localised events only (Einstein’s enthusiasm for fieldtheories is also mentioned by Howard as a motivation, and might be taken to applyonly to separability, but this carries little weight when divorced from the larger ar-gument). If what Einstein is calling for is simply the existence of localised events,the problem of having to attribute arbitrary prejudices to Einstein is considerablyameliorated.This ambiguity shows through in other places, such as the following:...we might therefore all along have been testing not simply local hiddenvariable theories, but separable, local hidden variable theories.I suspect that most of our trouble in understanding so-called ‘quantumnonlocality’ is a result of this more basic confusion. We focus our at-tention on the apparent demonstration of non-local effects mysteriouslycommunicated between two systems separated by a spacelike interval,without pausing to ask the deeper question of whether there are really two systems, or just one . We can and should clear up this confusion.([16] p. 195-196, Howard’s italics).Here Howard identifies the proposition that the two wings of the EPRB experimentcan safely be treated as two systems, in some unspecified sense, with the separabilityprinciple. It is only through a series of unjustified identifications of this sort thatwe arrive, from Einstein’s quote, to Howard’s separability of states. Without theseproblematic identifications it is impossible to get from one to the other. Healey recognises this when he writes, referring to the same quote from Einstein, that “[e]achof A and B may be spatially localised and have its own state, even if the state of the nonlocalised AB does not supervene on the states of A and B ”([18], p.352). not grounding his discussionof the incompleteness of QM with the separability principle .If we suitably define “separability” then of course it may be identified with whatEinstein is calling for. In that case identifying the content of Einstein’s statementswith “separability of states” as in eqn. (21) is the step of Howard’s argument thatrequires justification. But if it is consistent to read Einstein as calling only forlocalised events, then any such justification will fail: the existence of localised eventsclearly is not equivalent to Howard’s separability of states.The main claim in this article has been that separability is not a prerequisite fordefining locality, and neither is it an assumption of Bell’s theorem. Here, the point isthat Einstein did not write that the separability principle was a prerequisite for hisargument that quantum mechanics is incomplete, which inspired Bell. He only claimsthat real things may be “brought into as secure a relationship as possible with senseimpressions” and that “these things claim an existence independent of one another,insofar as these things ‘lie in different parts of space.’ ” Though not as formal as theconditions set out above, these statements could be taken to rule out such moves asrejecting operational consistency of localisation, or the principle of localised events.The intention could be that any such move would lead to absurdities of the typediscussed in section 2.5, but Einstein is not specific about that here.In conclusion, there is a strong argument that the “localised events” interpretationof Einstein’s statement should be preferred to Howard’s “separability” interpretation.At the very least, when we keep in mind this possible alternative interpretation, thereis nothing in any quotes Howard uses that unambiguously goes further than this onseparability and its relation to the completeness of quantum mechanics. In this lightit seems like an unwarranted over-interpretation to say that Einstein was arguing forseparability in any strong sense, such as the separability principle used here. A remaining question is whether Bell intended separability to be thought of as agrounding assumption of his locality condition and of his theorem, as is claimedin the article by Harrigan and Spekkens [30]. Their definition of separability is asfollows.Suppose a region R can be divided into local regions R , R , ..., R n . Anontological model is said to be separable only if the ontic state space Λ R of region R is the Cartesian product of the ontic state spaces Λ R i .”([30] p.9). The ontic state space here plays a similar role to Ω above, with thedifference that this “state space” refers to states at one time rather than histories. He does say that local action is “applied consistently only in field theory,” which might seemto undermine the argument (originally from Healy [27]) that local action and separability are notlinked, but a more plausible reading is that field theory was the only type of theory available at thetime of writing in which local action is unambiguously enforced. .They add the following as justification of their claim that separability is “a necessarycomponent of any sensible notion of locality”:The assumption of separability is made, for instance, by Bell when herestricts his attention to theories of local beables. These are variablesparameterising the ontic state space ‘which (unlike for the total energy)can be assigned to some bounded space-time region’ ”.The quote within the quote is from Bell ([1], 1976, p.53). This claim has little effecton their illuminating discussion of Einstein’s arguments for the incompleteness ofquantum mechanics and its relation to the epistemic view of quantum states. Itdoes however have some relevance to future directions. They comment at the endof their paper that “ontological models that are fundamentally relational might alsofail to be captured by the framework described here.” This seems to leave the dooropen to the possibility that non-separable or “holistic” models could escape fromthe conceptual problems of Bell’s theorem by coming up with a “sensible notionof locality” that does not rely on separability. It should be noted, however, thatSpekkens has recently expressed a different view, arguing that Bell did not in fact needto assume separability for his theorem, whilst still maintaining that Bell’s localitycondition, as he stated it, presupposes separability [34].Let us analyse the claim in [30] in the light of the definitions given in section 2.The quote from Bell only talks of being able to assign spacetime regions to beables, i.e. it only justifies the assumption of the existence of localised events. It says nothingabout whether events belonging to regions can be resolved into combinations of eventsin smaller regions. Neither is there anything else to suggest that he is making suchan assumption in the article referred to. It is more likely that this idea stems fromHoward’s interpretation of Einstein and Bell’s theorem, which is referenced as well.All this does not absolutely rule out the possibility that Bell intended separabilityto be thought of as an assumption of his theorem. However, as Spekkens notes in hislater treatment [34], the following quote suggests that he did not.It is notable that in this argument nothing is said of the locality, or evenlocalizability, of the variable λ [which has the same meaning as above].These variables could well include, for example, quantum mechanical statevectors, which have no particular localization in ordinary space-time. Itis assumed only that the outputs A and B [ a o and b o here], and theparticular inputs a and b [ a s and b s here], are well localized.(Bell 1981, p. 153) . In the first sentence quoted here, Bell seems to be deliberatelyrepudiating the idea that separability is an assumption of his theorem. How else to This definition is at the level of the state/history space whereas, in section 2.1, the slightlyhigher level of the algebra of events was used. In this article, the set of all events Σ( M ) can be any σ -algebra on the history space, and so a condition like Harrigan and Spekkens’ would not beappropriate. But if Ω is countable and Σ( M ) is fixed to be the Boolean algebra of subsets of Ω, thedefinitions coincide. Thanks to Michel Buck for pointing out this quote. .There seems to be nothing anywhere else in Bell’s published writings on the sub-ject to suggest that he believed separability to be important for his theorem. In someof the earliest writings, Bell uses the word “separability” to refer to quantum states([1], 1966, p.9), and he states that “It is the requirement of locality... that createsthe essential difficulty. There have been attempts to show that even without sucha separability or locality requirement no ‘hidden variable’ interpretation of quantummechanics is possible”([1], 1964, p.14). Here he seems to be using the word inter-changeably with locality, referring to only one previously mentioned principle. Afterthis he seems to drop the use of the term altogether. At several other points heunambiguously speaks of the existence of localised “beables”, as in the quote usedby Spekkens and Harrigan, but not about the need to assume separability, which isdistinct as has already been argued.This illustrates Bell’s genius for focusing on essentials: if he had included separa-bility as a condition, this would have been unnecessary, as we have seen. Hopefully the arguments in this article have bought some clarity to claims aboutnon-separability in the context of Bell’s theorem. The main points that have beenargued for are summarised here. There is a weaker principle, localised events, whichmerely states that all physical events can be associated to spacetime regions in aconsistent manner. This principle does not imply separability. The definition ofBell’s locality condition does not rely on separability in any way. The proof of Bell’stheorem does not use separability as an assumption. If, inspired by considerationsof non-separability, the assumptions of Bell’s theorem are weakened, what remainsno longer embodies the locality principle. Teller’s argument for relational holism asa solution to the problem of Bell’s theorem relies on the unjustified assumption thatseparability underpins Bell locality. Howard’s argument that separability of statesshould be dropped is, once reliance on Jarrett is identified and excised, only a callto give up Bell’s notion of locality in favour of a ban on superluminal signalling,something that has already been criticised in the literature. His claim that Einsteingrounded his arguments on non-locality and the incompleteness of QM with theassumption of separability is highly questionable; the quotes from Einstein are betterinterpreted as referring to the assumption of localised events, or something similar. Note that Bell does need his past variables to be associated to some past region, even if aninfinite one. Therefore he is not arguing against the existence of localised events in the weakestsense here, although the spatial localisation of the wave-function might most naturally be an infiniteregion, which seems almost equivalent to saying it is not localised anywhere. more than merely a banon superluminal signalling, plausibly satisfying Bell’s motivations in defining localcausality? Can it give the right answers for other uncontroversial examples of localityand non-locality ( e.g. an agent or robot rolling some dice and sending the results tomany distant locations, or a comet travelling at twice the speed of light relative to theEarth)? Also (something that was not relevant above but nevertheless may scupperother approaches), does it continue to hold good after past events are conditionedon, avoiding “Simpson’s paradox” [37, 36]?There are many other further directions for research suggested by the argumentsgiven in this paper. Now that relevant terms have been clarified and some resultsderived, advocates the idea of non-separability could move the debate forward byengaging with this position in its specifics. On another point, Howard’s interpretationof Einstein has been highly influential, but it has been challenged here. This briefchallenge could be expanded into a more comprehensive reinterpretation of Einstein’sviews on the subject. For instance, what was Einstein’s attitude towards the principleof local action, when it is distinguished from locality and separability as above?Another fairly obvious extension would be to apply the same style of discussion toother approaches to reconciling locality and QM. For example, there are a numberof approaches that, broadly speaking, rely on denying reality to the measurementoutcome in one wing of the EPRB experiment [10, 47, 48, 42] while others argue In [10] Brown and Timpson make a careful distinction between causal principles based on thePCC and what they define as locality, and so their approach is anything but na¨ıve to the sort ofquestions being raised here. Such reasoning could be applied to the other approaches listed here aswell. Nevertheless they all explicitly attempt to reconcile locality with quantum mechanics, usingdefinitions of locality similar to that employed here. indefiniteness , allowing us topreface standard propositions with “it is definite that...” and “it is indefinite that...”(this is similar to Reichenbach’s idea of introducing a third truth value [46], althoughwith different rules and motivations). This introduces a number of new choices on howto define causal influences, and thus locality. In forthcoming work, it will be claimedthat at least one of these choices allows quantum correlations while satisfying theminimal requirements listed above, offering an interesting way forward that arguablysatisfies some of both Einstein’s and Bohr’s intuitions [11].If this point is conceded, the outstanding question will be whether this argumentreally provides a satisfying alternative to Bell’s conditions that can do all (or any)of the work we expect a causal principle to do. Constructing simple models thatemploy this idea, and reproduce interesting features of quantum mechanics, might bea reasonable next step here. All this would be part of the ongoing task of restoringa meaningful, and useful, picture of the microworld to modern physics.
Acknowledgments
The author would like to thank Michel Buck for pointing out a useful quote from Bell,Fay Dowker and Rafael Sorkin for many conversations about causality and relativity,and especially to Harvey Brown for pointing to previous work on separability such asthat of Healey. Thanks are also due to Travis Norsen and Rob Spekkens for edifyingdiscussions of an earlier version on the manuscript. This work was made possiblethrough the support of a grant from the John Templeton Foundation.
A A complication: local action and nouvel le lo-cality
In section 2 it was argued that Bell locality can be formulated without first assumingseparability. However, there is another definition of Bell’s “local causality” given ina later article ([1], 1990, pp. 232-248). It is reasonable to ask: it possible that, inorder to use this definition of locality, we must assume separability? And would thatbe problematic for the main theses of this paper?In this case, the past region is not taken to be whole causal past of any of therelevant regions. Instead we are asked to consider a “slice” of spacetime (a regionbetween two spacelike hypersurfaces). Now, equation (5) must hold for the pastregion consisting of “at least those parts of [the slice] blocking the two backward26ightcones,” that is, the intersection of the slice with the union of the causal pastsof A and B . We will call this region S and this version of the principle “ nouvelle locality”. Bell writes that “what happens in the backwards lightcone of [the regionsin question]” should be “sufficiently specified, for example by a full specification oflocal beables in [a slice of the past light cone]’ ’([1], 1990, p.240, my italics). Thissuggests that Bell intended the condition to hold for any slice as long as we havereason to think that it “sufficiently specifies” the past. But what does, and how dowe know it does?Although nouvelle locality might look like a stronger condition than Bell locality,it is possible for correlations between events in A and B to be introduced rather thanremoved by conditioning on more past events, leading to the so-called “Simpson’sparadox” [37, 36]; this is essentially the reason why we must “sufficiently specify”the past. What if such a “Simpson’s paradox” arose due to past events outside of S ?These include events associated to regions to the past of S , as well as non-separableevents lying partially to the past and partially to the future of it. It is reasonable todemand that our causal principle rules this out for the same reason as it should ruleout correlations after conditioning on events in S .From this is seems that conditioning on Bell’s “example,” the region S , onlysufficiently specifies the past under certain conditions. Strong relativistic local action (SRLA): for any two events X and Y belongingto regions X and Y such that X lies entirely to the past of Y , the following conditionholds: µ ( X | λ ) µ ( Y | λ ) = µ ( X ∩ Y | λ ) ∀ λ ∈ Φ (cid:0) D ( A , B ) (cid:1) , (22)where Φ (cid:0) D ( X , Y ) (cid:1) is the set of full specifications of the region D ( X , Y ) which is theintersection of some slice with J + ( X ) ∩ J − ( Y ) \ ( X ∪ Y ) .This rule seems strong enough to prevent influences from propagating from somepast event to A or B without also showing up in S . I conjecture that, assuming SRLAfor both cases, Bell locality and nouvelle locality will turn out to be equivalent . Aproof could conceivably run along roughly the same lines as those given in [36] forother variations on the shape of the past region. Without the SRLA assumption,conditioning only on events in a slice is problematic: they may not sufficiently specifythe past, as Bell required.It seems unlikely that Bell changed the shape of the past region with the intentionof introducing an implicit local action assumption. Instead, the real intention of the This condition is called “strong” as it is possible to imagine weaker version of the condition thatset D to be the whole of J + ( X ) ∩ J − ( X ) or a slightly smaller region. However such versions wouldnot save nouvelle locality, as they would still allow causal propagation via non-separable eventswhich might span S . Something similar is probably true of other examples in which the whole past is not conditionedon, like Percival and Penrose’s causal condition in which the past region is a “wedge” such thatthe remainder of J + ( A ) ∪ J − ( B ) when the wedge is removed is a disjoint union of two regions, onecontaining A and the other B [52]. complete account of the overlap of the backward light cones isembarrassing in a related way, whether going back indefinitely or to a finite creationtime... In more careful discussion the notion of completeness [full specification] shouldperhaps be replaced by that of sufficient completeness for a certain accuracy, withsuitable epsilonics” ([1], 1977, p.104). With nouvelle locality nothing arbitrarily farback into the past need be conditioned on, resolving this possible worry in a cleanerway. But there seems nothing wrong with Bell’s original pragmatic suggestion if wedo not wish to make this move.There is relevance to the discussion of separability. The condition rules out cor-relations due to a common cause amongst non-separable events that span S , or acorrelation mediated to A and B by non-separable events that span S . This doesnot imply separability, but it does mean that it is not necessary to specify these non-separable events in order to screen off spacelike events. This threatens to trivialisetheir dynamics. But note that events associated to some subset of a spacelike hyper-surface in S do not face this problem because they cannot span S . So non-separableevents of this kind cause no more problems when conditioning only on a slice. Inone way the opposite is true: Bell does not rule out the extension of the past regionto the whole of a slice in the above definition, in which case the specification couldindeed include things like the quantum state which have no obvious “locality, or evenlocalizability” in space at all .In any case, the version of Bell locality used in section 2.2 arguably is sensibledespite Bell’s “embarrassment”. It is not obviously a problem to condition on eventsarbitrarily far into the past, because events do not have to be empirically accessible tobe conditioned on in a model, as argued in [39]. The point of Bell’s theorem is that wecannot find a Bell local model that gives the same correlations as quantum mechanics no matter what we hypothesise about past events in our model. Also, locality andlocal action are different principles, and any condition that enforces locality need notalso enforce local action.In conclusion, whichever version of Bell’s condition we prefer, separability as de-fined in section 2.1 need not be assumed in order to make the condition sensible,and so the main arguments in the paper are not threatened when this alternativeformulation of Bell locality is considered. This is also of relevance to Healey’s thesis that locality and local action can both be preservedin gauge field theory only if separability is sacrificed [27]. The events there are defined on closedspacelike curves. It would be interesting to formulate local action and locality as specifically as isdone here, and to explicitly show that they are not violated for certain gauge theories in Healey’spicture of them. Separability vs. Howard’s separability of states
In this appendix, we will discuss Howard’s formalisation of separability of states, givenin section 3.3, and find the connection to the definition of separability in section 2.1.The relevant definitions are (19) and (20) and the condition itself, eqn. (21).It is useful to note at the outset that the definitions of µ α ( a o | a s ∩ b s ) and µ β ( b o | a s ∩ b s ) do not imply much about the meaning of the “states” α and β [23]. For example,consider a model of the EPRB experiment in which all the past events are associatedto one point. In this case α and β clearly do not label initial states of two spatiallydisjoint systems corresponding to the two wings. However, it is possible to add anextra assumption that the relevant past region P is the disjoint union of two regions, P A and P B , which lie to the past of the A and B wings respectively (this would accordwith local action and nouvelle locality as in the preceeding appendix).Even making this assumption, it is easy to see that separability does not implyeqn. (21). Take for example a theory with correlations between the outcomes, buta past with no non-trivial events, separable or otherwise (or any model that givesthese probabilities after conditioning on some separable past events). Maudlin givesa similar argument, adding to the picture some tachyons to make it more intuitiveand “mechanical” (maintaining local action for instance) ([4], pp. 97-98) .To put a finer point on this, consider for example a “back-yard” EPRB-like exper-iment, which can be modelled as above, but in which the wings are not required tobe spacelike. Let us imagine that one outcome A o (random, according to the theorywe apply) triggers a ping-pong ball to be fired into the B wing. By some simplemechanism, this affects the outcome there. To complete the argument, note thatHoward’s original definitions of separability say nothing about whether the “regionsof spacetime” concerned are spacelike or timelike to each other. Now, this back-yard experiment violates Howard’s formal condition (21) as applied to its “wings,”but obviously this simple scenario does not rely on violating Howard’s “fundamentalontological principle,” separability. It follows that the formal separability of statescondition does not faithfully express a ban on non-separable events overlapping thetwo wings, the idea that the two wings are in fact two separate systems, the claim thatbasic properties are all associated to points, lack of mysterious“passion-at-a-distance”or anything of the sort. There is nothing motivating the condition – unless that is,we put the wings spacelike to each other, and invoke locality in order to prevent suchscenarios. We already know that Howard’s separability of states condition followsfrom Bell’s locality condition, and it is best interpreted simply as a consequence ofthat well-founded condition.Conversely eqn. (21) does not imply separability, or even the relevant consequence Other models can be imagined: for example, a spherical wave propagating out from one outcomethat affects the other, or a string between a pair of entangled particles along which local actions couldpass. If the question is what chance there is of formulating a natural and empirically adequate theorythat manifests outcome dependence and separability, thoughts of this kind might be of interest (see[24] for a discussion); the question at hand, however, is whether separability implies separabilityof states over all theories, and so these considerations are not relevant. In this sense, Maudlin’sargument against Howard is not reliant on any ill-defined “intuitions about tachyons” [24].
29f separability: that the are no events associated to P A ∪ P B that are not logicalcombinations of events associated to P A and P B . The equation means nothing moreor less than it explicitly says: that µ ( a o ∩ b o | a s ∩ b s ∩ λ ), where λ is the state of P , whichmay well have associated to it non-separable events , equals µ α ( a o | a s ∩ b s ) µ β ( b o | a s ∩ b s ).So much for the comparison itself. As in the main text, the more difficult questionto answer is why the conclusions reached in this framework differ from those alreadyin the literature. There is an implicit assumption in Howard’s treatment that resolvesthis problem. Howard and those who have developed his argument [25, 26] must keepthe meaning of λ that Bell gave, in order to keep their arguments about separability ofstates in contact with the derivation of the Bell inequalities. Now, λ , for Bell, rangesover all full specifications of P (see section 2.2). However, Howard formally defineshis “states” λ as p λ ( x | m ), which translates to µ ( a o ∩ b o | a s ∩ b s ∩ λ ) here ([3], Howardp. 226). The two definitions are only consistent if the following assumption holds: µ ( a o ∩ b o | a s ∩ b s ∩ λ ) completely determines the full specification λ , and similarly for α and β . This assumption is not part of the framework used in this paper, which isone reason for the discrepancy in conclusions. More importantly, the assumption isproblematic. For example, if we add a third setting value, requiring the new conditionfor µ α ( a o | a s ∩ b s ) does not imply the same condition for its restriction back to twosettings. In other words, the argument goes through when our theory only allowsour apparatus to have two marks on its dial, but not if we imagine a third unusedmark !Even given this assumption, there is another problem. Winsberg and Fine alsoargue that separability does not imply (21), but on the grounds that, to satisfyHoward’s principle, the joint “state” could determine the “states” of the wings [25]in any way, not necessarily through a product (see also Fogel’s detailed account [26]).With Howard’s implicit assumption, Winsberg and Fine’s weaker condition has somejustification: if, as separability implies, the full specifications α and β determine λ ,then the assumption implies that µ α ( a o | a s ∩ b s ) and µ β ( b o | a s ∩ b s ) will determine µ ( a o ∩ b o | a s ∩ b s ∩ λ ) (Fogel’s “functionwise” composition). However, when Winsbergand Fine conclude that both locality and separability can be preserved in the wakeof Bell’s theorem, they are relying on the rest of Howard’s (and Jarrett’s) arguments,including this implicit, and flawed, assumption.In conclusion, we saw in sections 3.3 and 3.4 that ( a ) even if we allow the definitionof separability of states to go unquestioned, Howard’s argument that locality can besaved by dropping this condition is flawed, and ( b ) the attribution of the separabilityassumption to Einstein is questionable. We can now add that ( c ) the definition ineqn. (21) is not a good formalisation of the separability principle as Howard definesand discusses it.This analysis is relevant to the so-called PBR theorem, which rules out the epis-temic interpretation of quantum states under a number of conditions, including aso-called “preparation independence” condition [32] that bears some similarity to a This argument is closely related to Norsen’s criticism of Jarrett’s interpretation of Bell’s theorem,which also depends on the definition of λ , although Norsen’s account goes into more depth. It canalso be usefully compared to Laudisa’s criticism of Howard [23]. References [1] J. Bell,
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