New Virtual Issue from the BJPS: Bell’s Theorem

The Editors of the BJPS are delighted to announce the publication of a new virtual issue. Following on from the success of previous issues on Kuhn and the philosophy of psychology and cognitive science, BJPS Associate Editor Peter Lewis has drawn together papers from the journal that have critically explored Bell’s theorem and its philosophical upshot. The full issue can be found here, but we’ve reproduced Peter’s introduction below. Enjoy!

Bell’s Theorem 55 Years On

Peter J. Lewis

1 Introduction

Bell’s theorem is a milestone in our understanding of the quantum world. Since it first appeared in print ([1964]), it has become the centrepiece of philosophical discussions of the metaphysical consequences of quantum mechanics. But this doesn’t mean that its import is transparently clear: there is considerable disagreement in the literature about precisely what Bell’s theorem tells us about the physical world. Much of that debate has occurred in the pages of BJPS. This virtual issue presents a collection of papers analysing Bell’s theorem, questioning its presuppositions, extending it, and evaluating its consequences.

Consider spin measurements performed on a pair of spin-1/2 particles in an entangled singlet state. Notoriously, quantum mechanics does not in general deliver definitive predictions concerning the outcomes of measurements like this, but instead gives us probability distributions over the various possibilities. Nevertheless, it is natural to assume that the particles must possess precise spin values prior to measurement, particularly in light of the argument to that effect by Einstein et al. ([1935]). Possessed values of this kind are usually called ‘hidden variables’, and accounts of quantum mechanics that supplement it with such values are called ‘hidden variable theories’. Bell’s theorem provides a mathematical constraint on hidden variable theories—but, notably, not only on hidden variable theories.

In fact, Bell proves two theorems concerning hidden variables: Shimony ([1984]) provides a helpful guide to the relation between them. The first theorem is a corollary of a fundamental result due to Gleason ([1957]), which shows that, except for very simple systems, precise values cannot be simultaneously assigned to all observable properties of a system without contradiction. Due to editorial difficulties, this theorem did not appear in print until 1966 (Bell [1966]). Despite this impossibility result, Bell knew that Bohm ([1952]) had explicitly constructed a hidden variable theory that is demonstrably empirically equivalent to standard quantum mechanics. The trick is that according to Bohm’s theory, most properties are contextual, in the sense that the value revealed by a measurement can depend on the physical context in which the measurement occurs, not just on the possessed property values of the system itself. Bell’s ([1966]) theorem assumes that properties are not contextual in this way.

Bell also knew that Bohm’s theory is non-local: a measurement on one subsystem can instantaneously affect the properties of another, distant subsystem. The question he asked himself was whether this non-locality is just an incidental feature of Bohm’s model. His ([1964]) theorem—the one that is now known as ‘Bell’s theorem’—constitutes an answer to this question. Bell assumes a locality condition, and derives an inequality that quantum mechanics violates. For a pair of particles in the singlet state and space-like separated spin measurements, quantum mechanics predicts (and experiment confirms) correlations between the outcomes stronger than can be produced by local hidden variables. Bell concludes that non-locality is inescapable for hidden variable theories.

One might take this to be a reason to reject hidden variable theories. However, Bell’s result can be generalized to cover probabilistic explanations of quantum measurement results (Clauser and Horne [1974]). Again, one assumes a locality condition, and derives an inequality that quantum mechanics violates: quantum mechanics predicts (and experiment confirms) stronger correlations between the outcomes than can be produced by any locally acting mechanism, deterministic or probabilistic. The conclusion seems to be that any explanation of quantum measurement results, deterministic or probabilistic, must be non-local—and hence that the non-locality of deterministic hidden variable theories such as Bohm’s should not be held against them.

2 Locality

Non-locality is an uncomfortable conclusion. Not only is action at a distance ‘spooky’, it also looks like it violates special relativity. Hence much of the initial philosophical work surrounding Bell’s theorem seeks to analyse the locality assumption that is a premise in Bell’s proof, and to clarify what its rejection entails. This focus is reflected in several of the papers in this issue.

Shimony ([1984]) provides just such an analysis of Bell’s locality condition. Shimony calls this condition ‘factorizability’, both to avoid prejudging the question of its relationship with the locality of special relativity, and because of its mathematical form. For a two-particle system, what it says is that when the system has state λ, the probability of getting result A for a measurement of property a on particle 1 and result B for a measurement of property b on particle 2 can be factorized into the product of the probability of getting A for a on 1 and the probability of getting B for b on 2: Prλab(A&B) = Prλa(A).Prλb(B). This is essentially the combination of two independence conditions. The first, which Shimony ([1986]) later calls ‘outcome independence’, says that holding the measurements fixed, the probability of getting result A for particle 1 is independent of the outcome for particle 2 (and similarly for result B): Prλab(A&B) = Prλab(A).Prλab(B). The second, which he later calls ‘parameter independence’, says that the outcome for particle 1 is independent of the measurement performed on particle 2 (and vice versa): Prλab(A).Prλab(B) = Prλa(A).Prλb(B).

Shimony ([1984], p. 35) argues that the significance of Bell’s theorem is that it makes possible ‘some near decisive results in experimental metaphysics’. In particular, any hidden variable account of quantum mechanics must be contextual, and it must violate factorizability. He notes that this constitutes a threat to the relativistic conception of spacetime, since for space-like separated measurements, factorizability does indeed look like a relativistic locality condition. Nevertheless, he holds out hope for a ‘peaceful coexistence’ between quantum non-locality (that is, non-factorizability) and special relativity. Later (Shimony [1986]), he precisifies this hope as follows: Violations of parameter independence allow superluminal signalling, but violations of outcome independence do not, so we can make quantum mechanics (at least superficially) consistent with special relativity by rejecting outcome independence but not parameter independence. In particular, since Bohm’s hidden variable theory violates parameter independence, whereas standard quantum mechanics violates outcome dependence, Shimony takes Bohm’s theory to be nearly decisively ruled out.

Butterfield ([1992]) criticizes Shimony’s reasoning, noting that (i) the violation of parameter independence by Bohm’s theory cannot be controlled to send a signal, and (ii) in any event, the ability to send a signal is not a good criterion for the existence of causation. More generally, he argues that a prohibition on superluminal causation does not entail either parameter independence or outcome independence. In order to secure either entailment, we need another assumption, and the assumption in each case is a form of the common cause principle: if there is no direct causation between two correlated events, then there is an event in their common past that renders them probabilistically independent. Butterfield concludes that a prohibition on superluminal causation does not favour either parameter independence or outcome independence—and also that quantum mechanics gives us reason to be suspicious of common cause principles. This latter point is a recurring theme in the literature, as we will see.

Another recurring theme is the distinction between locality and separability. The intuitive distinction is clear enough: locality says that spatially separated systems cannot interact, whereas separability says that spatially separated systems each have their own distinct states. Several commentators take the lesson of Bell’s theorem to be that quantum mechanics violates separability rather than locality. For example, Howard ([1992]) associates locality with Shimony’s parameter independence and separability with outcome independence, and hence concludes that quantum mechanics is local but non-separable. Laudisa ([1995]) takes issue with such arguments, noting that separability is not among Bell’s premises: the quantum state is attributed to the system, not to each subsystem separately. Hence Laudisa concludes that non-locality, not non-separability, is the proper consequence of Bell’s theorem. Along the way, he considers several accounts of separability, including Shimony’s, and finds them wanting.

Together, these papers provide a careful and extensive analysis of Bell’s locality assumption, suggesting that Bell’s theorem tells us something deep and surprising about the causal structure of the world. The next set of papers in this virtual issue explores the nature of this implication.

3 Common Causes

Since its inception, quantum mechanics has been regarded as in some sense problematic, and Bell’s theorem clarifies the nature of the problem. Pitowsky ([1994]) underlines the significance of the theorem in this regard by tracing the inequality that Bell derives (and that quantum mechanics violates) back to Boole’s results concerning the relative frequencies of logically connected events. Pitowsky notes that this makes quantum mechanics more problematic than relativity: relativity threatens some physical intuitions, but quantum mechanics threatens logic itself. He then considers some solutions to the problem. He glosses Bohm’s theory as explaining the violation of Boole’s constraints by postulating that the measurements we perform bias the samples we obtain; essentially, this is to say that Bohm’s hidden variables are contextual. However, he rejects Bohm’s theory for its causal non-locality and instead proposes that the quantum phenomena in question simply have no causal explanation. Since the phenomena in question involve correlations between space-like separated measurement outcomes (so that there is no possibility of direct, local causation between them), what Pitowsky seems to be denying here is the existence of common cause explanations, in agreement with Butterfield ([1992]).

However, Hofer-Szabó et al. ([1999]) defend the possibility of common cause explanations of quantum phenomena. They point out that the derivation of Bell’s inequality assumes that there is a single common cause for any measurements that might be performed on the two particles—that is, that there is a common common cause behind the various correlations that might be observed by measuring different combinations of spin properties. They show that without this assumption, any correlation can be given a common cause explanation, including those exhibited by quantum systems that violate Bell’s inequality. Graßhoff et al. ([2005]) show that plausible assumptions block the construction of common cause explanations of this kind, reinforcing the conclusion that quantum correlations cannot be explained via common causes.

Shrapnel ([2019]) argues that our inability to construct common cause explanations in Bell-type scenarios is the result of applying classical notions of causal structure to the quantum world. She suggests instead that we construct an explicitly quantum account of causal structure by adapting Pearl’s ([2000]) interventionist account of causation to the quantum case. The result is that Bell-type correlations can be given a common cause explanation of sorts, in the sense that an intervention on the source of the two particles can effect changes to the measurement results obtained on each particle.

4 Other Ways Out

The extent to which common causes can figure in explanations of Bell correlations remains contested, if only because it is not entirely clear what it takes to be a common cause. But if Bell correlations cannot be given common cause explanations, can they be explained at all? Several strategies have been identified in the pages of BJPS for doing just that.

To begin with, there is an uncontested sense in which a common cause explanation of Bell-type correlations is technically possible, namely, if the same mechanism acts as a common cause for both the particle properties and the measurements to be performed on the particles. Lewis ([2006]) considers such ‘superdeterministic’ accounts of Bell correlations, and compares them with ‘retrocausal’ accounts, in which the (later) choice of measurements causally influences the (earlier) particle properties, finding the latter more plausible. Cavalcanti ([2010]) exploits the fact that Bell’s set-up is analogous to the Newcomb problem to argue that a causal decision theorist should bet against the predictions of quantum mechanics, resulting in a loss. He takes this as an argument against causal decision theory; however, he also notes that a premise in his argument is violated by superdeterministic and retrocausal accounts, which might be taken as a point in their favour. Evans et al. ([2013]) provide an explicit argument in favour of a retrocausal account by constructing an analogy between Bell’s set-up, in which two measurements occur at space-like separation, and a ‘rotated’ set-up, in which two measurements occur at time-like separation. Causal explanation in the latter case is an entirely local affair, and the suggestion is that the same kind of local explanation is at work in the former case—although this requires that causes can occur later than their effects.

Allori et al. ([2011]) point out that it is a tacit assumption of Bell’s proof that the measurements have unique, determinate outcomes. But in many-worlds accounts of quantum mechanics, this assumption fails, as every outcome of a measurement occurs in some ‘world’. What consequences does this have for Bell’s conclusion that any explanation of quantum measurement results must be non-local? To answer this question, Allori et al. construct a version of the many-worlds theory in which the wave function describes the evolution of a matter density distribution over three-dimensional space. They show that such a theory is causally non-local, in the sense that a measurement on one particle can instantaneously affect the way that the matter density is distributed over ‘worlds’ for the other. Hence the many-worlds approach does not necessarily avoid Bell’s non-locality conclusion.

Fine ([1973], [1982]) suggests that we can avoid the conclusion of Bell’s theorem by using a different account of probability. As the discussion of Pitowsky ([1994]) makes clear, Bell’s derivation takes place in the context of a classical probability space—a space in which probabilities are assigned to conjunctions of events according to the standard addition rule. But Fine argues that this is problematic, since according to quantum mechanics, the conjunction of measurements of incompatible observables is physically meaningless, so we should not be forced to assign probabilities to such conjunctions. Generalized probability spaces allow such conjunctions to lack probability values. However, Feintzeig ([2015]) proves a theorem designed to show that generalized probability spaces cannot provide a way around Bell’s theorem.

5 Future Work

Thanks to the work of philosophers of physics over the past fifty-five years, much of it in the pages of BJPS, we now have a good idea of the nature of the assumptions underlying Bell’s theorem, the strategies available to respond to it, and the strategies that turn out to be dead-ends. But it remains an open question what the correct response is. Perhaps our classically tutored intuitions about what counts as a good causal explanation are to blame. Perhaps we should be looking for causal explanations in which an effect can precede its cause. Perhaps we should avoid appealing to causes at all in our explanations in the quantum realm. If non-locality really is inevitable in explanations of quantum correlations, then perhaps we must find a way to reconcile the non-locality exhibited by Bohm-type hidden variable theories (or by GRW-type spontaneous collapse theories) with relativity. Or perhaps we can adopt a version of the many-worlds theory and thereby escape such non-locality. In any case, Bell’s theorem and its consequences remain at the heart of investigations into the nature of the quantum world.

Department of Philosophy
Dartmouth College


Bell, J. S. [1964]: ‘On the Einstein–Podolsky–Rosen Paradox’, Physics, 1, pp. 195–200.

Bell, J. S. [1966]: ‘On the Problem of Hidden Variables in Quantum Mechanics’, Reviews of Modern Physics, 38, pp. 447–52.

Bohm, D. [1952]: ‘A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables, I & II’, Physical Review, 85, pp. 166–93.

Butterfield, J. [1992]: ‘Bell’s Theorem: What It Takes’, British Journal for the Philosophy of Science, 43, pp. 41–83.

Cavalcanti, E. G. [2010]: ‘Causation, Decision Theory, and Bell’s Theorem: A Quantum Analogue of the Newcomb Problem’, British Journal for the Philosophy of Science, 61, pp. 569–97.

Clauser, J. F. and Horne, M. A. [1974]: ‘Experimental Consequences of Objective Local Theories’, Physical Review D, 10, pp. 526–35.

Einstein, A., Podolsky, B. and Rosen, N. [1935]: ‘Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?’, Physical Review, 47, pp. 777–80.

Evans, P. W., Price, H. and Wharton, K. B. [2013]: ‘New Slant on the EPR–Bell Experiment’, British Journal for the Philosophy of Science, 64, pp. 297–324.

Feintzeig, B. [2015]: ‘Hidden Variables and Incompatible Observables in Quantum Mechanics’, British Journal for the Philosophy of Science, 66, pp. 905–27.

Fine, A. [1973]: ‘Probability and the Interpretation of Quantum Mechanics’, British Journal for the Philosophy of Science, 24, pp. 1–37.

Fine, A. [1982]: ‘Joint Distributions, Quantum Correlations, and Commuting Observables’, Journal of Mathematical Physics, 23, pp. 1306–10.

Hofer-Szabó, G., Rédei, M. and Szabó, L. E. [1999]: ‘On Reichenbach’s Common Cause Principle and Reichenbach’s Notion of Common Cause’, British Journal for the Philosophy of Science, 50, pp. 377–99.

Howard, D. [1992]: ‘Locality, Separability, and the Physical Implications of the Bell Experiments’, in A. van der Merwe, F. Selleri and G. Tarozzi (eds), Bell’s Theorem and the Foundations of Modern Physics, Singapore: World Scientific, pp. 306–14.

Gleason, A. M. [1957]: ‘Measures on the Closed Subspaces of a Hilbert Space’, Journal of Mathematics and Mechanics, 6, pp. 885–93.

Graßhoff, G., Portmann, S. and Wüthrich, A. [2005]: ‘Minimal Assumption Derivation of a Bell-type Inequality’, British Journal for the Philosophy of Science, 56, pp. 663–80.

Laudisa, F. [1995]: ‘Einstein, Bell, and Nonseparable Realism’, British Journal for the Philosophy of Science, 46, pp. 309–29.

Lewis, P. J. [2006]: ‘Conspiracy Theories of Quantum Mechanics’, British Journal for the Philosophy of Science, 57, pp. 359–81.

Pearl, J. [2000), Causality: Models, Reasoning, and Inference, Cambridge: Cambridge University Press.

Pitowsky, I. [1994]: ‘George Boole’s “Conditions of Possible Experience” and the Quantum Puzzle’, British Journal for the Philosophy of Science, 45, pp. 95–125.

Shimony, A. [1984]: ‘Contextual Hidden Variables Theories and Bell’s Inequalities’, British Journal for the Philosophy of Science, 35, pp. 25–45.

Shimony, A. [1986]: ‘Events and Processes in the Quantum World’, in R. Penrose and C. Isham (eds), Quantum Concepts in Space and Time, Oxford: Oxford University Press.

Shrapnel, S. [2017]: ‘Discovering Quantum Causal Models’, British Journal for the Philosophy of Science.

Valia, A., Goldstein, S., Tumulka, R. and Zanghì, N. [2011]: ‘Many Worlds and Schrödinger’s First Quantum Theory’, British Journal for the Philosophy of Science, 62, pp. 1–27.