Non-separability Does Not Relieve the Problem of Bell’s Theorem

Abstract

This paper addresses arguments that “separability” is an assumption of Bell’s theorem, and that abandoning this assumption in our interpretation of quantum mechanics (a position sometimes referred to as “holism”) will allow us to restore a satisfying locality principle. Separability here means that all events associated to the union of some set of disjoint regions are combinations of events associated to each region taken separately.

In this article, it is shown that: (a) localised events can be consistently defined without implying separability; (b) the definition of Bell’s locality condition does not rely on separability in any way; (c) the proof of Bell’s theorem does not use separability as an assumption. If, inspired by considerations of non-separability, the assumptions of Bell’s theorem are weakened, what remains no longer embodies the locality principle. Teller’s argument for “relational holism” and Howard’s arguments concerning separability are criticised in the light of these results. Howard’s claim that Einstein grounded his arguments on the incompleteness of QM with a separability assumption is also challenged. Instead, Einstein is better interpreted as referring merely to the existence of localised events. Finally, it is argued that Bell rejected the idea that separability is an assumption of his theorem.

Appendices

Appendix A: A Complication: Local Action and Nouvelle Locality

In Sect. 2 it was argued that Bell locality can be formulated without first assuming separability. However, there is another definition of Bell’s “local causality” given in a later article ([1], 1990, pp. 232–248). It is reasonable to ask: it possible that, in order to use this definition of locality, we must assume separability? And would that be 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 the relevant regions. Instead we are asked to consider a “slice” of spacetime (a region between two spacelike hypersurfaces). Now, Eq. (5) must hold for the past region consisting of “at least those parts of [the slice] blocking the two backward lightcones,” that is, the intersection of the slice with the union of the causal pasts of $\mathcal {A}$ and $\mathcal {B}$. We will call this region $\mathcal {S}$ and this version of the principle “nouvelle locality”. Bell writes that “what happens in the backwards lightcone of [the regions in question]” should be “sufficiently specified, for example by a full specification of local beables in [a slice of the past light cone]” ([1], 1990, p. 240, my italics). This suggests that Bell intended the condition to hold for any slice as long as we have reason to think that it “sufficiently specifies” the past. But what does, and how do we know it does?

Although nouvelle locality might look like a stronger condition than Bell locality, it is possible for correlations between events in $\mathcal {A}$ and $\mathcal {B}$ to be introduced rather than removed by conditioning on more past events, leading to the so-called “Simpson’s paradox” [36, 37]; 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 $\mathcal {S}$? These include events associated to regions to the past of $\mathcal {S}$, as well as non-separable events lying partially to the past and partially to the future of it. It is reasonable to demand that our causal principle rules this out for the same reason as it should rule out correlations after conditioning on events in $\mathcal {S}$.

From this is seems that conditioning on Bell’s “example,” the region $\mathcal {S}$, only sufficiently specifies the past under certain conditions.

Strong Relativistic Local Action (SRLA)

for any two events X and Y belonging to regions $\mathcal {X}$ and $\mathcal {Y}$ such that $\mathcal {X}$ lies entirely to the past of $\mathcal {Y}$, the following condition holds:

$$ \mu(X|\lambda)\mu(Y|\lambda)= \mu(X \cap Y|\lambda) \quad\forall \lambda\in\varPhi\bigl( \mathcal {D}(\mathcal {A},\mathcal {B}) \bigr), $$

(22)

where $\varPhi ( \mathcal {D}(\mathcal {X},\mathcal {Y}) )$ is the set of full specifications of the region $\mathcal {D}(\mathcal {X},\mathcal {Y})$ which is the intersection of some slice with $J^{+}(\mathcal {X}) \cap J^{-}(\mathcal {Y}) \backslash(\mathcal {X}\cup \mathcal {Y})$.²⁰

This rule seems strong enough to prevent influences from propagating from some past event to $\mathcal {A}$ or $\mathcal {B}$ without also showing up in $\mathcal {S}$. I conjecture that, assuming SRLA for both cases, Bell locality and nouvelle locality will turn out to be equivalent.²¹ A proof could conceivably run along roughly the same lines as those given in [36] for other 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 specify the past, as Bell required.

It seems unlikely that Bell changed the shape of the past region with the intention of introducing an implicit local action assumption. Instead, the real intention of the change may have been to relieve worries expressed elsewhere. Bell refers to “the necessity of paying attention in such a study to the creation of the world” in a different context ([1], 1977, p. 102) and then, in a footnote, adds that “The invocation in [([1], 1975, pp. 52–62)] of a complete account of the overlap of the backward light cones is embarrassing in a related way, whether going back indefinitely or to a finite creation time… In more careful discussion the notion of completeness [full specification] should perhaps be replaced by that of sufficient completeness for a certain accuracy, with suitable epsilonics” ([1], 1977, p. 104). With nouvelle locality nothing arbitrarily far back into the past need be conditioned on, resolving this possible worry in a cleaner way. But there seems nothing wrong with Bell’s original pragmatic suggestion if we do not wish to make this move.

There is relevance to the discussion of separability. The condition rules out correlations due to a common cause amongst non-separable events that span $\mathcal {S}$, or a correlation mediated to $\mathcal {A}$ and $\mathcal {B}$ by non-separable events that span $\mathcal {S}$. This does not 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 trivialise their dynamics. But note that events associated to some subset of a spacelike hypersurface in $\mathcal {S}$ do not face this problem because they cannot span $\mathcal {S}$. So non-separable events of this kind cause no more problems when conditioning only on a slice. In one way the opposite is true: Bell does not rule out the extension of the past region to the whole of a slice in the above definition, in which case the specification could indeed include things like the quantum state which have no obvious “locality, or even localizability” in space at all.²²

In any case, the version of Bell locality used in Sect. 2.2 arguably is sensible despite Bell’s “embarrassment”. It is not obviously a problem to condition on events arbitrarily far into the past, because events do not have to be empirically accessible to be conditioned on in a model, as argued in [39]. The point of Bell’s theorem is that we cannot 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 and local action are different principles, and any condition that enforces locality need not also enforce local action.

In conclusion, whichever version of Bell’s condition we prefer, separability as defined in Sect. 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 alternative formulation of Bell locality is considered.

Appendix B: Separability vs. Howard’s Separability of States

In this appendix, we will discuss Howard’s formalisation of separability of states, given in Sect. 3.3, and find the connection to the definition of separability in Sect. 2.1. The relevant definitions are (19) and (20) and the condition itself, Eq. (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 associated to one point. In this case α and β clearly do not label initial states of two spatially disjoint systems corresponding to the two wings. However, it is possible to add an extra assumption that the relevant past region $\mathcal {P}$ is the disjoint union of two regions, $\mathcal {P}_{\mathcal {A}}$ and $\mathcal {P}_{\mathcal {B}}$, which lie to the past of the $\mathcal {A}$ and $\mathcal {B}$ wings respectively (this would accord with local action and nouvelle locality as in the preceeding appendix).

Even making this assumption, it is easy to see that separability does not imply Eq. (21). Take for example a theory with correlations between the outcomes, but a past with no non-trivial events, separable or otherwise (or any model that gives these probabilities after conditioning on some separable past events). Maudlin gives a similar argument, adding to the picture some tachyons to make it more intuitive and “mechanical” (maintaining local action for instance) ([4], pp. 97–98).²³

To put a finer point on this, consider for example a “back-yard” EPRB-like experiment, which can be modelled as above, but in which the wings are not required to be spacelike. Let us imagine that one outcome A _o (random, according to the theory we apply) triggers a ping-pong ball to be fired into the $\mathcal {B}$ wing. By some simple mechanism, this affects the outcome there. To complete the argument, note that Howard’s original definitions of separability say nothing about whether the “regions of 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 “fundamental ontological principle,” separability. It follows that the formal separability of states condition does not faithfully express a ban on non-separable events overlapping the two wings, the idea that the two wings are in fact two separate systems, the claim that basic 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 such scenarios. We already know that Howard’s separability of states condition follows from Bell’s locality condition, and it is best interpreted simply as a consequence of that well-founded condition.

Conversely Eq. (21) does not imply separability, or even the relevant consequence of separability: that the are no events associated to $\mathcal {P}_{\mathcal {A}}\cup \mathcal {P}_{\mathcal {B}}$ that are not logical combinations of events associated to $\mathcal {P}_{\mathcal {A}}$ and $\mathcal {P}_{\mathcal {B}}$. The equation means nothing more or less than it explicitly says: that μ(a _o ∩b _o |a _s ∩b _s ∩λ), where λ is the state of $\mathcal {P}$, which may 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 question to answer is why the conclusions reached in this framework differ from those already in the literature. There is an implicit assumption in Howard’s treatment that resolves this problem. Howard and those who have developed his argument [25, 26] must keep the meaning of λ that Bell gave, in order to keep their arguments about separability of states in contact with the derivation of the Bell inequalities. Now, λ, for Bell, ranges over all full specifications of $\mathcal {P}$ (see Sect. 2.2). However, Howard formally defines his “states” λ as p _λ (x|m), which translates to μ(a _o ∩b _o |a _s ∩b _s ∩λ) here ([3], Howard p. 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 is one reason for the discrepancy in conclusions. More importantly, the assumption is problematic. For example, if we add a third setting value, requiring the new condition for μ _α (a _o |a _s ∩b _s ) does not imply the same condition for its restriction back to two settings. In other words, the argument goes through when our theory only allows our apparatus to have two marks on its dial, but not if we imagine a third unused mark!²⁴

Even given this assumption, there is another problem. Winsberg and Fine also argue that separability does not imply (21), but on the grounds that, to satisfy Howard’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 some justification: 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 Winsberg and Fine conclude that both locality and separability can be preserved in the wake of 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 Sects. 3.3 and 3.4 that (a) even if we allow the definition of separability of states to go unquestioned, Howard’s argument that locality can be saved by dropping this condition is flawed, and (b) the attribution of the separability assumption to Einstein is questionable. We can now add that (c) the definition in Eq. (21) is not a good formalisation of the separability principle as Howard defines and discusses it.

This analysis is relevant to the so-called PBR theorem, which rules out the epistemic interpretation of quantum states under a number of conditions, including a so-called “preparation independence” condition [32] that bears some similarity to a separability condition. As the debate on this theorem and its conditions goes on, care must be taken to not to read too much into this condition. In this case, however, the line between epistemological and ontological concepts has been made clear by the original authors, giving hope that the discussion will remain clear-cut on this issue.