Bell’s Theorem is the collective name for a family ofresults, all of which involve the derivation, from a condition onprobability distributions inspired by considerations of localcausality, together with auxiliary assumptions usually thought of asmild side-assumptions, of probabilistic predictions about the resultsof spatially separated experiments that conflict, for appropriatechoices of quantum states and experiments, with quantum mechanicalpredictions. These probabilistic predictions take the form ofinequalities that must be satisfied by correlations derived from anytheory satisfying the conditions of the proof, but which are violated,under certain circumstances, by correlations calculated from quantummechanics. Inequalities of this type are known asBellinequalities, or sometimes,Bell-type inequalities.Bell’s theorem shows that no theory that satisfies theconditions imposed can reproduce the probabilistic predictions ofquantum mechanics under all circumstances.

The principal condition used to derive Bell inequalities is acondition that may be calledBell locality, orfactorizability. It is, roughly, the condition that anycorrelations between distant events be explicable in local terms, asdue to states of affairs at the common source of the particles uponwhich the experiments are performed. See section 3.1 for a morecareful statement.

The incompatibility of theories satisfying the conditions that entailBell inequalities with the predictions of quantum mechanics permits anexperimental adjudication between the class of theories satisfyingthose conditions and the class, which includes quantum mechanics, oftheories that violate those conditions. At the time that Bellformulated his theorem, it was an open question whether, under thecircumstances considered, the Bell inequality-violating correlationspredicted by quantum mechanics were realized in nature. Beginning inthe 1970s, there has been a series of experiments of increasingsophistication to test whether the Bell inequalities are satisfied.With few exceptions, the results of these experiments have confirmedthe quantum mechanical predictions, violating the relevant BellInequalities. Prior to 2015, however, each of these experiments wasvulnerable to at least one of two loopholes, referred to as thecommunication, orlocality loophole, and thedetection loophole (see section 5). In 2015, experiments wereperformed that demonstrated violation of Bell inequalities with theseloopholes blocked. Experimental demonstration of violation of the Bellinequalities has consequences for our physical worldview, as theconditions that entail Bell inequalities are, arguably, an integralpart of the physical worldview that was accepted prior to the adventof quantum mechanics. If one accepts the lessons of the experimentalresults, then some one or other of these conditions must berejected.

For much of the interval between the original publication ofBell’s theorem and the experiments of Aspect and hiscollaborators, interest in Bell’s theorem was confined to ahandful of physicists and philosophers. During that period, much ofthe discussions on the foundations of physics occurred in amimeographed publication entitledEpistemological Letters. Inthe wake of the Aspect experiments (Aspect, Grangier, and Roger, 1982;Aspect, Dalibard, and Roger 1982), there was considerablephilosophical discussion of the implications of Bell’s theorem;see Cushing and McMullin, eds. (1989), for a snapshot of thephilosophical discussions of the time. Interest was also stimulated bythe publication of a collection of Bell’s papers on thefoundations of quantum mechanics (Bell 1987b). The rise of quantuminformation theory, which, among other things, explores the ways inwhich quantum entanglement can be used to perform tasks that would notbe feasible classically, also contributed to raising awareness of thesignificance of Bell’s theorem, which throws into sharp reliefthe difference between quantum entanglement-based correlations andclassical correlations. The year 2014 was the 50th anniversary of theoriginal publication of Bell’s theorem, and was marked by aspecial issue ofJournal of Physics A (47,number 42, 24 October 2014), a collection of essays (Bell and Gao,eds., 2016), and a large conference comprising over 400 attendees (seeBertlmann and Zeilinger, eds., 2017). The interested reader is urgedto consult these collections for an overview of current discussions ontopics surrounding Bell’s theorem.

The shift in attitude of the physics community towards the importanceof Bell’s theorem was dramatically illustrated by the awardingof the Nobel Prize in Physics for 2022 to Alain Aspect, John Clauser,and Anton Zeilinger “for experiments with entangled photons,establishing the violation of Bell inequalities and pioneering quantuminformation science.”

• 1. Introduction
• 2. Proof of a Theorem of Bell’s Type
• 3. The Assumptions of the Proof
  • 3.1 Locality and causality assumptions
  • 3.2 Supplementary assumptions
  • 3.3 On “local realism”
• 4. Early Experimental Tests of Bell’s Inequalities
• 5. The Communication and Detection Loopholes, and their Remedies
  • 5.1 The Communication Loophole, and its remedy
  • 5.2 The Detection Loophole and its remedy
  • 5.3 Loophole-free tests
• 6. Some Variants of Bell’s Theorem
• 7. Significance for Quantum Information Theory
• 8. Philosophical/Metaphysical Implications
  • 8.1 Options left open by experimental violation of Bell Inequalities
  • 8.2 Quantum mechanics and relativity
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

In 1964 John S. Bell, a native of Northern Ireland and a staff memberof CERN (European Organisation for Nuclear Research) whose primaryresearch concerned theoretical high energy physics, published a paper(Bell 1964) in the short-lived journalPhysics, whicheventually transformed the study of the foundation of quantummechanics.

The paper showed, under conditions that were relaxed in later work byBell (1971, 1976) himself and by his followers (Clauser, Horne,Shimony, and Holt 1969, Clauser and Horne 1974, Aspect 1983, Mermin1986), that, on the assumption of certain auxiliary conditions, nophysical theory that satisfies a certain locality condition, which maybe calledBell locality, can fully reproduce the quantumprobabilities for outcomes of experiments. Since that time, variantson the theorem, with family resemblances, have been formulated.“Bell’s Theorem” is the collective name for theentire family.

The theorem has roots in Bell’s investigations into the statusof the hidden-variables program, and in earlier work concerningquantum entanglement.

Bell presented several formulations of the theorem over the years(Bell 1964, 1971, 1976, 1990), and variants of it have been presentedby others. The original derivation (1964) relied on a set-up involvingperfect anticorrelation of the results of spin experiments on pairs ofspin-1/2 particles prepared in the singlet state. Under thiscondition, the Bell locality condition entails that outcomes ofexperiments are predetermined by the complete specification of state,a condition which we will calloutcome determinism (OD).Clauser, Horne, Shimony and Holt (1969) derived an inequality, theCHSH inequality, that does not require this assumption. Though intheir proof they employed the condition OD, this condition isunnecessary for the derivation of the inequality, as shown by Bell(1971), who provided a proof of the CHSH inequality that relies onneither the assumption of perfect anticorrelation nor an assumption ofoutcome determinism.

One line of investigation in the prehistory of Bell’s Theorem isBell’s examination of the hidden-variables program. This programinvolves supplementation of the quantum mechanical state of a systemby further “elements of reality”, or “hiddenvariables”, the incompleteness of the quantum state being theexplanation for the statistical character of quantum mechanicalpredictions concerning the system. A pioneering version of a hiddenvariables theory was proposed by Louis de Broglie in 1926–7 (deBroglie 1927, 1928), and revived by David Bohm in 1952 (Bohm 1952; seealso the entry onBohmian mechanics).

In a paper (Bell 1966) that was written before the one in whichBell’s theorem first appeared, but, due to an editorial mishap,was published later, Bell raises the question of the viability of ahidden-variables theory that reproduces the statistical predictions ofquantum mechanics via averaging over better defined states thatuniquely determine the result of any experiment that could beperformed. In this paper he examines several theorems that had beenpresented as no-go theorems for theories of this sort, and supplementsthem with one of his own, a theorem that was independently formulatedby Specker (1960), and published by Kochen and Specker (1967), and hascome to be known as theKochen-Specker Theorem orBell-Kochen Specker Theorem (seeentry on the Kochen-Specker Theorem for more details). In each case Bell argues that the proof containspremises that are physically unwarranted.

The Bell-Kochen-Specker theorem is a corollary of Gleason’stheorem (Gleason 1957), though Bell and Kochen-Specker obtain itdirectly, and not via Gleason’s theorem, whose proof isconsiderably more intricate. The question addressed by Gleason has todo with assignments of probabilities to closed subspaces of a Hilbertspace (or, equivalently, to projection operators onto such subspaces),such that the probabilities assigned to orthogonal projections areadditive. Gleason proved that, in a Hilbert space of dimension 3 orgreater, any such assignment of probabilities can be represented by adensity operator. The BKS theorem deals with the special case in whichthe assignments are confined to the values 1 or 0.

The assumption that a definite value (1 or 0) is to be assigned toeach projector on the system’s Hilbert space, with the conditionthat values assigned to commuting projectors be additive, is aweakening of the assumption of the von Neumann no-go theorem (vonNeumann 1932), which assumes that the quantum mechanical additivity ofexpectation values of all observables, whether represented bycommuting operators or not, extends to the hypotheticaldispersion-free states (see section 2 of entry onthe Kochen-Specker Theorem). Despite theprima facie plausibility of this assumption,Bell regards it, too, as physically unmotivated, and therefore, unlikeKochen and Specker, does not regard the Bell-Kochen-Specker theorem asa no-go theorem for hidden-variables theories. The reason for this isthat the assumption embodies a condition that was later to be known asnoncontextuality.^[1] In a Hilbert space of dimension greater than two, any projectionoperator will be a member of more than one complete set of commutingprojections. For each of these complete sets, there will be anexperiment whose outcomes correspond to the projections in the set.The assumption of noncontextuality amounts to the assumption that avalue can be assigned to a projection operator that is independent ofwhich of these experiments is to be performed, or as Bell puts it,that “measurement of an observable must yield the same valueindependently of what other measurements may be madesimultaneously.” The assumption need not hold; “[t]heresult of an observation may reasonably depend not only on the stateof the system (including hidden variables) but also on the completedisposition of the apparatus” (Bell 1966, 451; 1987b and 2004, 9).

Noncontextual hidden-variables theories that reproduce the predictionsof quantum mechanics are ruled out by the Bell-Kochen-Specker theorem.A natural question arises as to the possibility of a contextualhidden-variables theory on which the unavoidable contextuality isrestricted tolocal dependencies. Is it possible to have atheory on which the outcome of an experiment performed in some spatialregion \(A\) is determined by the complete state of a system in a waythat does not depend on the disposition of experimental apparatus at adistance from \(A\)?

Bell’s article ends with a brief exposition of the deBroglie-Bohm theory, noting in particular the feature that “inthis theory an explicit causal mechanism exists whereby thedisposition of one piece of apparatus affects the results obtainedwith a distant piece” (Bell 1966, 452; 1987b and 2004, 11). Thearticle ends with the remark:

Bohm of course was well aware of these features of his scheme, and hasgiven them much attention. However, it must be stressed that, to thepresent writer’s knowledge, there is noproof thatany hidden variable account of quantum mechanicsmust have this extraordinary character. It would therefore beinteresting, perhaps, to pursue some further ‘impossibilityproofs,’ replacing the arbitrary axioms objected to above bysome condition of locality, or of separability of distant systems.

To the second of the above-quoted sentences is attached a note:“Since the completion of this paper such a proof has beenfound.” The potential drama of the announcement was spoiled bythe fact that the follow-up paper containing the proof (Bell 1964) hadalready been published (see Jammer 1974, 303, for an account of thecircumstances that led to the publication delay).

The fact that Bell’s theorem has roots into investigations onhidden-variables theories has led to a misconception that the theoremis a no-go theorem for hidden-variables theoriestout court.There could be no such theorem, since, as Bell himself repeatedlyemphasized, there is a functioning hidden-variables theory, the deBroglie-Bohm theory.

Another line of investigation leading to Bell’s Theorem was theinvestigation of quantum mechanical entangled states, that is, quantumstates of a composite system that cannot be expressed either asproducts of quantum states of the individual components, or asmixtures of product states. That quantum mechanics admits of suchentangled states was discovered by Erwin Schrödinger (1926) inone of his pioneering papers, but the significance of this discoverywas not emphasized until the paper of Einstein, Podolsky, and Rosen(1935). They examined correlations between the positions and thelinear momenta of two well separated spinless particles and concludedthat in order to avoid an appeal to nonlocality these correlationscould only be explained by “elements of physical reality”in each particle — specifically, both definite position anddefinite momentum — and since this description is richer thanpermitted by the uncertainty principle of quantum mechanics theirconclusion is effectively an argument for a hidden variablesinterpretation.^[2] See also the entry on theEinstein-Podolsky-Rosen paradox.

2. Proof of a Theorem of Bell’s Type

In the present section the pattern of Bell’s 1964 paper will befollowed: formulation of a framework, derivation of an inequality,demonstration of a discrepancy between certain quantum mechanicalexpectation values and this inequality. As already mentioned,Bell’s 1964 derivation assumed an experiment involving perfectanticorrelation of the results of aligned Stern-Gerlach experiments ona pair of entangled spin-\(\frac{1}{2}\) particles. Experimentaltests, in which perfect anticorrelation (or correlation) may beapproximated but cannot be assumed to hold exactly, require thisassumption to be relaxed. Papers which took the steps fromBell’s 1964 demonstration to the one given here are Clauser,Horne, Shimony and Holt (1969), Bell (1971), Clauser and Horne (1974),Aspect (1983) and Mermin (1986).^[3] Other strategies for deriving Bell-type theorems will be mentioned inSection 6.

This conceptual framework first of all postulates an ensemble of pairsof systems, the individual systems in each pair being labeled as 1 and2. Each pair of systems is characterized by a “completestate” \(\lambda\) which contains the entirety of the propertiesof the pair at the moment of generation. No assumption whatsoever ismade about the nature of the state \(\lambda\). The state space\(\Lambda\), which is the totality of all possible complete states\(\lambda\), could be a set of quantum states and nothing more, or aset whose elements are quantum states supplemented by additionalvariables, or something more exotic, perhaps some state space as yetunthought of.

We make the assumption, which remains tacit in most expositions, thatwe have an appropriate choice of subsets of \(\Lambda\) to be regardedas the measurable subsets, forming a measurable space to whichprobabilistic considerations may be applied. It is assumed that themode of generation of the pairs establishes a probability distribution\(\rho\) that is independent of the adventures of each of the twosystems after they separate. This does not preclude temporal evolutionof the properties of the two systems after separation. What is assumedis that the state \(\lambda\) prescribes probabilities for subsequentevents (including any temporal evolution), and thereby probabilitiesfor outcomes of experiments to be performed on the systems.

Different experiments may be performed on each system. We will use\(a, a'\) as variables ranging over possible experiments on 1, and\(b, b'\) as variables that range over experiments on 2. It is notassumed that these parameters capture the complete state of theexperimental apparatus, which might have a range of microstatescorresponding to each experimental setting. It is assumed that thepreparation probability distribution \(\rho\) is independent of theprocesses by which a choice of experiments to be performed is made.This assumption we call theMeasurement IndependenceAssumption.

The result of an experiment with setting \(a\) on system 1 is labeledby a real parameter \(s\), which can take on values from a discreteset \(S_a\) of real numbers in the interval [\(-1, 1\)]. Likewise, theresult of an experiment on 2 is labeled by a parameter \(t\), whichcan take on any of a discrete set of real numbers \(T_b\) in \([-1,1]\). As suggested by the subscripts, the sets of potential outcomesmay depend on the experimental settings. The restriction of the valuesof the outcome labels to lie in the interval \([-1, 1]\) is of nophysical significance, and is a choice made only for convenience.Indeed, the use of numbers to label outcomes is merely a matter ofconvenience. The inequality to be obtained places a condition on theprobabilities involved in it; if desired, the use of numbers as labelson outcomes may be dispensed with, and the inequalities can expressedsolely in terms of the relevant probabilities, as in the CHinequality, inequality (25), below. Bell’s own version of histheorem assumed experiments with two possible outcomes, labelled \(\pm1\). Other variants of the theorem involve larger sets of potentialoutcomes.

We assume that, for each pair of settings \(a, b\), and every\(\lambda\) in \(\Lambda\), there is a probability function\(p_{a,b}(s,t \mid \lambda)\), which takes on values in the interval[0, 1] and sums to unity when summed over all \(s\) in \(S_a\) and\(t\) in \(T_b\). These response functions may include implicitaveraging over possible states of the experimental apparatus.^[4] Bell’s own version of his theorem assumed experiments with twopossible outcomes, labelled \(\pm 1\). Other variants of the theoreminvolve larger sets of potential outcomes. We can use theseprobability functions — which we will callresponseprobabilities — to define marginal probabilities:

Here, and in what follows, it is to be understood that the sums betaken over all \(s \in S_a\) and \(t \in T_b\). Define\(A_{\lambda}(a, b)\), \(B_{\lambda}(a, b)\) as the expectationvalues, for complete state \(\lambda\), of the outcomes of experimentson system 1 and system 2, respectively, when the settings are \(a,b\).

Now define the expectation value of the product \(st\) ofoutcomes:

Bell-type inequalities follow from a condition, inspired byconsiderations of locality and causality, which has been calledFactorizability, orBell locality.

(F)

For any \(a, b, \lambda\), there exist probability functions\(p^1_a(s\mid\lambda), \; p^2_b(t\mid\lambda)\), such that\(p_{a,b}(s,t\mid\lambda) = p^1_a (s\mid\lambda) \:p^2_b(t\mid\lambda).\)

This is the condition formulated explicitly by Bell in his laterexpositions of Bell’s theorem (Bell 1976, 1990). Bell refers tothe factorizability condition (F) as the condition that thecorrelations belocally explicable (Bell 1981, C2–55;1987b and 2004, 152; see also 1990, 109; 2004, 243). This has twocomponents: that the correlations be explained, and not taken asprimitive, and that the explanation be local. This will be discussedfurther in section 3.1. It should be noted that Bell regarded thiscondition “not as theformulation of ‘localcausality’, but as a consequence thereof” (Bell 1990, 109;2004, 243).

As we have seen, Bell’s investigations were stimulated, in part,by the question of the prospects for theories in which the completestate uniquely determines the outcome of any experiment, and quantumuncertainty relations reflect incompleteness of the usualspecification of state. For such a theory, the response probabilities\(p_{ab}(s,t|\lambda)\) take on the extremal values 0 or 1. Let uscall this condition OD, foroutcome determinism. It is alsosometimes referred to, misleadingly, asrealism (seediscussion insection 3.3, below).

(OD)

For all \(\lambda\), \(a, b\), and all \(s \in S_a\) and \(t \inT_b , p_{ab}(s,t\mid\lambda) \in \{0, 1\}\).

Suppes and Zanotti (1976) showed that, for the special case of perfectcorrelations between outcomes of the two experiments, in which anoutcome of an experiment on one system makes possible prediction withprobability one of the outcome of an experiment on the other, OD mustbe satisfied if the factorizability condition (F) is. This applies tothe case considered by Bell 1964. In Bell 1971 and subsequentexpositions Bell provided a generalization that does not presumeperfect correlation and does not require OD, either as a suppositionor as a consequence of other suppositions.^[5]

If the factorizability condition (F) is satisfied, then

The above definitions are valid for experiments with any number ofdiscrete outcomes. An important special case is that in which eachexperiment has only two distinct outcomes, which we may label by \(\pm1\). For the case of bivalent experiments, (4) is equivalent tocondition (F).

Now consider the quantities

LetS\(_{\varrho}\) denote the corresponding relation betweenthe expectation values of the \(E_{\lambda}s\), with respect to thepreparation distribution \(\rho\).

Note that, in taking the expectation values of the quantitiesappearing on the right-hand side of this definition with respect tothe same distribution\(\rho\), we areinvoking the Measurement Independence Assumption.

Since the absolute value of the average of any random variable cannotbe greater than the average of its absolute value, is clear that

We now have the materials in hand required to state and prove aBell-type theorem. The first step consists of showing that, if thefactorizability condition (F) is satisfied, then

The second step consists of showing that there are quantum states andexperimental set-ups that are such that the quantum-mechanicalexpectation values violate the inequality (8). This shows that notheory satisfying the factorizability condition can reproduce thestatistical predictions of quantum mechanics in all situations.Moreover, the inequality furnishes a bound on how close thepredictions of such a theory can come to reproducing quantummechanical predictions. The inequality (8), due to Clauser, Horne,Shimony, and Holt (1969), is known as theCHSHinequality.

We now prove the first part of the theorem, namely, that the CHSHinequality follows from the factorizability condition (F). If F issatisfied, then, using (4), and the fact that \(A_{\lambda}(a)\) and\(A_{\lambda}(a')\) lie in the interval \([-1,1]\),

It is easy to check that

Since \(B_{\lambda}(b)\) and \(B_{\lambda}(b')\) also lie in theinterval \([-1,1]\), from (9) and (10) we conclude that, for every\(\lambda\),

Since this bound holds for every value of \(\lambda\), it must alsohold for the expectation value of \(S_{\lambda}\).

This, together with (7), yields the CHSH inequality (8).

The final step of the proof of our Bell-type theorem is to exhibit asystem, a quantum mechanical state, and a set of quantities for whichthe statistical predictions violate inequality (8). The example usedby Bell stems from Bohm’s variant of the EPR thought-experiment(Bohm 1951, Bohm and Aharonov 1957). A pair of spin-\(\frac{1}{2}\)particles is produced in the singlet state,

where \(\mathbf{n}\) is an arbitrarily chosen direction, and\(|\mathbf{n}+\rangle , |\mathbf{n}-\rangle\) are spin-up andspin-down eigenstates of spin in the \(\mathbf{n}\) direction. Thestate is rotationally invariant, and hence the expression (13)represents the state for any direction \(\mathbf{n}\). IfStern-Gerlach experiments are performed on the two particles, then,regardless of the direction of the axes of the devices, for each sideof the experiment the two possible results have the same probability,one-half. If experiments are done with the axes of the two devicesaligned, the results are guaranteed to be opposite; spin-up on onewill be obtained in one experiment if and only if spin-down isobtained on the other. If the axes are at right angles, the resultsare probabilistically independent. In the general case, with deviceaxes in directions given by unit vectors \(\mathbf{a}, \mathbf{b}\),respectively, with results labelled \(\pm 1\), then the expectationvalue of the product of the outcomes is given by

where \(\theta_{\mathbf{a}\mathbf{b}} = \theta_{\mathbf{a}} -\theta_{\mathbf{b}}\) is the angle between the vectors\(\mathbf{a},\mathbf{b}\).

Though the example of spin-\(\frac{1}{2}\) particles in the singletstate is ubiquitous in the literature as an illustrative example,polarization-entangled photons have been more significant forexperimental tests of Bell inequalities. Consider a pair of photons 1and 2 propagating in the \(z\)-direction. Let \(|x\rangle_j\) and\(|y\rangle_j\) represent states in which photon \(j \; (j =1, 2)\) islinearly polarized in the \(x\)- and \(y\)- directions, respectively.Consider the following state vector,

which is invariant under rotation of the \(x\) and \(y\) axes in theplane perpendicular to \(z\). The total quantum state of the pair ofphotons 1 and 2 is invariant under the exchange of the two photons, asrequired by the fact that photons are integral spin particles. Supposenow that photons 1 and 2 impinge respectively on the faces ofbirefringent crystal polarization analyzers I and II, with theentrance face of each analyzer perpendicular to \(z\). Each analyzerhas the property of separating light incident upon its face into twooutgoing non-parallel rays, theordinary ray and theextraordinary ray. The transmission axis of the analyzer is adirection with the property that a photon polarized along it willemerge in the ordinary ray (with certainty if the crystals are assumedto be ideal), while a photon polarized in a direction perpendicular to\(z\) and to the transmission axis will emerge in the extraordinaryray. See Figure 1:

Figure 1
(reprinted with permission)

Photon pairs are emitted from the source, each pair quantummechanically described by \(\ket{\Phi}\) of Eq. (15). I and II arepolarization analyzers, with outcomes \(s=1\) and \(t=1\) designatingemergence in the ordinary ray, while \(s = -1\) and \(t = -1\)designate emergence in the extraordinary ray. The crystals are alsoidealized by assuming that no incident photon is absorbed, but eachemerges in either the ordinary or the extraordinary ray.

The expectation value, in state \(\ket{\Phi}\), of the product of\(s\) and \(t\), is

Note that this displays the same sort of sinusoidal dependence onangle exhibited by (14), with \(2\theta\) replacing \(\theta\).

Often, in popular writings, the case of aligned devices is the onlyone mentioned, and the perfect anticorrelation (forspin-\(\frac{1}{2}\) particles in the singlet state) or correlation(for photons in state \(\ket{\Phi}\)) of results in this case isoffered as evidence of “spooky action at a distance.” Infact, as Bell (1964, 1966) demonstrated by means of simple toy models,this behaviour can be reproduced by entirely local means. An importantinsight of Bell’s was that it is important to consider theless-than perfect correlations obtained when the device axes are notaligned. In Bell’s toy models, correlations fall off linearlywith the angle between the device axes, whereas the quantumcorrelations (14, 16) fall off sinusoidally; the decrease incorrelations away from the case of perfect alignments is less steepthan in the toy models. Bell’s theorem shows that this sort ofbehaviour is not a peculiarity of his models; no model satisfying thecondition F can reproduce the quantum correlations for all angles.This can be seen by considering the quantum predictions (14, 16) andplugging them into the expression for \(S\). For a pair ofspin-\(\frac{1}{2}\) particles in the singlet state, we have,

Choose coplanar unit vectors \(\mathbf{a},\) \(\mathbf{a}',\)\(\mathbf{b},\) \(\mathbf{b}'\) such that \(\theta_{\mathbf{b}'} -\theta_{\mathbf{a}}\, =\) \(\theta_{\mathbf{a}} -\theta_{\mathbf{b}}\, =\) \(\theta_{\mathbf{b}} - \theta_{\mathbf{a}'}= \phi\), and therefore, \(\theta_{\mathbf{b}'} - \theta_{\mathbf{a}'}= 3\phi\). This choice yields

This exceeds the CHSH bound (8) when \(0 \lt |\phi | \lt\)\(\arccos\left((\sqrt{3} - 1)/2 \right) \approx 1.95\) radians, or68°, with maximum violation at \(\phi = \pm {\pi}/{4}\), or45°. For these angles, we have

For the case of polarization-entangled photons, we have,

This takes on its maximum at \(\phi = {\pi}/{8}\), or 22.5°.

This value \(2\sqrt{2}\) appearing in equations (19) and (21) is themaximum violation of the CHSH inequality for any quantum state, asshown by Tsirelson (Cirel’son 1980). This bound on quantumviolations of the CHSH inequality is called theTsirelsonbound. As Gisin (1991) and Popescu and Rohrlich (1992)independently demonstrated, for any pure entangled quantum state of apair of systems, observables can be found yielding a violation of theCHSH inequality. Popescu and Rohrlich (1992) also show that themaximum amount of violation is achieved with a quantum state ofmaximum degree of entanglement. Incidentally, this is not true formixed states; there are entangled mixed states not violating any Bellinequality (Werner 1989).

3. The Assumptions of the Proof

The distinctive condition giving rise to the Bell Inequalityassumption is the factorizability condition (F). This condition ismotivated by considerations concerning locality and causality.Considerations of this sort have been the focus of the discussion ofthe implications of Bell’s theorem. However, in order to aderive a conflict between the predictions for theories that satisfythis assumption, other assumptions—some of which are the sortusually accepted without question in scientificexperimentation—are needed. The analysis of Bell’s theoremhas provoked careful scrutiny of the reasoning required to reach theconclusion that F is to be rejected. As a result, some assumptionsthat in another context would have been left implicit have been madeexplicit, and each has been challenged by some authors. In thissection we outline a set of assumptions sufficient to ensuresatisfaction of Bell inequalities, which, therefore, constitute a setof assumptions that cannot all be satisfied by any theory that yields,in agreement with experiment, violations of Bell inequalities. Thereare other paths to Bell inequalities; see, in particular, Wiseman andCavalcanti (2017), who offer an analysis similar to that found here,as well as other analyses.

3.1 Locality and causality assumptions

3.1.1 Bell’s Principle of Local Causality

As mentioned above, the article (Bell 1964) in which Bell’stheorem first appeared is a follow-up to Bell (1966), which exploresthe prospects for hidden-variables theories in which the outcome of anexperiment performed is predetermined by the complete state of thesystem. The introduction to Bell (1964) begins with mention oftheories with additional variables that are to restore causality andlocality, and says that “In this note that idea will beformulated mathematically and shown to be incompatible with thestatistical predictions of quantum mechanics.” The localityassumption is glossed as the requirement “that the result of ameasurement on one system be unaffected by operations on a distantsystem with which it has interacted in the past.” Applied to thecase at hand, of Stern-Gerlach experiments performed on an entangledpair of spin-1/2 particles, this is “the hypothesis ... that ifthe two measurements are made at places remote from one another theorientation of one magnet does not influence the result obtained withthe other.” Bell follows this with,

Since we can predict in advance the result of measuring any chosencomponent of \(\boldsymbol{\sigma}_2\), by previously measuring thesame component of \(\boldsymbol{\sigma}_1\), it follows that theresult of any such measurement must actually be predetermined (Bell1964, 195; 1987b and 2004, 15).

This suggests that OD is not being assumed, but, rather, derived, viaan EPR-type argument, from a locality assumption and the perfectanticorrelations predicted by the considered quantum state. This ishow Bell explained the reasoning in later publications (see Bell 1981,fn 10).^[6]

In Bell (1976) and (1990), Bell derives the factorizability conditionfrom a condition he calls thePrinciple of Local Causality(see Norsen 2011 for discussion). In (1990) he begins his analysiswith a rehearsal of the reason that relativity should be taken toprohibit superluminal causation. On the usual notion of causation, thecause-effect relation is taken to be temporally asymmetric, withcauses temporally preceding their effects. In a relativisticspacetime, events at spacelike separation are taken to have notemporal order. Any system of coordinates will assign time coordinatesto each of any pair of events, but, if the events are spacelikeseparated, the time coordinates assigned to a pair of events atspacelike separation will differ in their ordering, depending on whichreference frame is being employed. If we take all of these referenceframes to be physically on a par, it must be concluded that there isno temporal order between the events, as relations that are notrelativistically invariant have no physical significance.

On the basis of these considerations, that is, Lorentz invariance andthe assumption that causes temporally precede their effects, Bellintroduces what he calls theprinciple of localcausality.

(PLC-1)

The direct causes (and effects) of events are near by, and eventhe indirect causes (and effects) are no further away than permittedby the velocity of light.

This, says Bell, is “not yet sufficiently sharp and clean formathematics.” For this reason, he introduces what he presents asa sharpened version of the principle (refer to Figure 2).

(PLC-2)

A theory will be said to be locally causal if the probabilitiesattached to values of local beables in a space-time region 1 areunaltered by specification of values of local beables in a space-likeseparated region 2, when what happens in the backward light cone of 1is already sufficiently specified, for example by a full specificationof local beables in a spacetime region 3.

Figure 2

In Bell’s terminology, abeable is any element of aphysical theory that is taken to correspond to something physicallyreal, andlocal beables pertaining to a space-time region arethose contained within that region.

The transition from what we have called PLC-1 to PLC-2 should,according to Bell, “be viewed with the utmost suspicion”as “it is precisely in cleaning up intuitive ideas formathematics that one is likely to throw the baby out with thebathwater” (1990, 106; 2004, 239). The relation between them isnot further discussed in that article, but remarks in other papersshed light on the transition between them.

In (1976), Bell motivates the formulation of the local causalitycondition with the remark,

Now my intuitive notion of local causality is that events in[space-time region] 2 should not be “causes” of events in[spacelike separated region] 1, and vice versa. But this does not meanthat the two sets of events should be uncorrelated, for they couldhave common causes in the overlap of their backward light cones (Bell1976, 14; 1985a, 88; 1987b and 2004, 54).

He then precedes to formulate the condition of local causality, whichis that a full specification of beables in the overlap of the backwardlight cones of the spacetime regions 1 and 2 should screen offcorrelations between them. Implicit in this is that correlationsbetween two variables be susceptible to causal explanation, either viaa causal connection between the variables, or via a common cause. Thisassumption was stated explicitly in a later article (Bell 1981), inwhich he says that “the scientific attitude is that correlationscry out for explanation” (Bell 1981, C2–55; 1987b and2004, 152).

The assumption that correlations between two variables that are not ina cause-effect relation to each other can be explained by some commoncause was named theprinciple of the common cause byReichenbach (1956, § 19), and for this reason is often referredto asReichenbach’s Common Cause Principle, thoughReichenbach made no pretense of originating the principle, andregarded it as a codification of a mode of inference common in bothscience and everyday life. See the entry onReichenbach’s common cause principle for more details. A variable \(C\) is a Reichenbachian common causeof a correlation between two variables \(A\) and \(B\) if thevariables \(A\) and \(B\) are uncorrelated, conditional onspecification of the value of \(C\). Reichenbach’s Common CausePrinciple says that, if two correlated variables are not in acause-effect relation with each other, there is a Reichenbachiancommon cause of their correlations. The condition we have called PLC-1does not, by itself, entail PLC-2, but it does follow from theconjunction of PLC-1 and Reichenbach’s Common CausePrinciple.

PLC-1 does not, by itself, entail that correlations be causallyexplicable, and, indeed, does not commit to any sort of causalrelations in the world; it merely says that whatever causal relationsthere are respect relativistic locality. For this reason, we willrefer to it as acausal locality principle. This is incontradistinction to PLC-2, which requires there to be causalrelations of some sort wherever correlations are found. What Bellcalls the Principle of Local Causality, PLC-2, can be thought of as aconjunction of (1) a causal locality condition along the lines ofPLC-1, restricting causes of an event to that event’s past lightcone, and (2) Reichenbach’s Common Cause Principle, whichrequires that correlations be causally explicable. The formercondition can, itself, be regarded as following from relativisticinvariance and the principle that the cause of an event lie in itstemporal past. Bell’s Principle of Local Causality thus followsfrom the conjunction of three assumptions, all of which, in variouswritings, were explicitly formulated by Bell:

• (Temporal Asymmetry of Causality) Any cause of an event lies inits temporal past, and not in its temporal future.
• (Lorentz Invariance) The relation of temporal precedence isinvariant under Lorentz boosts.
• Reichenbach’s Common Cause Principle.

3.1.2 Parameter independence and outcome independence

The condition F of factorizability is the application, to theparticular set-up of Bell-type experiments, of Bell’s Principleof Local Causality. As we have seen, it can be thought of as theconjunction of the condition of causal locality, and the common causeprinciple. In this section we apply those conditions to the set-up ofBell-type experiments.

On the assumption that the experimental settings can be treated asfree variables, whose values are determined exogenously, if the choiceof setting on one wing is made at spacelike separation from theexperiment on the other, a dependence of the probability of theoutcome of one experiment on the setting of the other would seemstraightforwardly to be an instance of a nonlocal causal influence.The condition that this not occur can be formulated as follows.

(PI)

For all parameter settings \(a\), \(a'\), \(b\), \(b'\), all\(\lambda\), and all \(s \in S_a\) and \(t \in T_b\),

This is the condition that has come to be known asparameterindependence, following Shimony (1986, 1990).

For fixed values of the experimental settings, Bell’s Principleof Local Causality entails that the outcomes of the experiments on thetwo systems be independent, conditional on the specification\(\lambda\) of the complete state of the system at the source. This isthe condition

(OI)

For each pair of parameter settings \(a, b\), and all \(\lambda\),and all \(s\) in \(S_a\) and \(t\) in \(T_b\),

This is the condition that has come to be known asoutcomeindependence, following Shimony (1986, 1990). As demonstrated byJarrett (1983, 1984), the factorizability condition (F) is theconjunction of parameter independence and outcome independence. Thetwo conditions bear different relations to the locality and causalityconditions discussed in the previous subsection. PI is a consequenceof the causal locality condition PLC-1 alone, whereas OI requires inaddition the assumption of the common cause principle.

The parsing of the Bell locality condition as a conjunction of PI andOI is due to Clauser and Horne (1974), though they did not proposedistinct names for the conjuncts. Jarrett (1983, 1984) referred to theconditions aslocality andcompleteness. Jarrett(1983, 1984, 1989) argued that a violation of PI would inevitablypermit superluminal signalling. The conclusion requires an additionalassumption, that the state of the system be controllable.^[7] It would not hold for a theory on which there are principledlimitations on the ability of would-be signallers to control thestates of the systems they are dealing with; an example of a theory ofthis sort is the de Broglie-Bohm theory, which violates PI but doesnot permit superluminal signalling. Nonetheless, in some of theliterature PI has been treated as equivalent to no-signalling. Forsome (seee.g. Ballentine and Jarrett 1987, fn. 6; Jarrett1989, 70; Shimony 1993, 139), this stems from the thought that anylimitations on control that might prevent a violation of PI from beingexploited for signalling would involve only practical limitationsirrelevant to foundational concerns, which should have to do with whatis possible in principle.

3.2 Supplementary assumptions

3.2.1 Unique experimental outcomes

Though it might seem that this goes without saying, the entireanalysis is predicated on the assumption that, of the potentialoutcomes of a given experiment, one and only one occurs, and hencethat it makes sense to speak ofthe outcome of an experiment.The reason that this assumption is worth mentioning is that there is afamily of approaches to the interpretation of quantum mechanics,namely, Everettian, or “many-worlds” approaches, and somevariants of the relational approach, according to which all potentialoutcomes actually occur, in what are effectively distinct worlds. Seeentries onmany-worlds interpretation of quantum mechanics,Everett’s relative-state formulation of quantum mechanics.

3.2.2 Experimental settings as free parameters

Bell’s original analyses (1964, 1971) tacitly assumed that thecomplete state \(\lambda\) is sampled from the same probabilitydistribution \(\rho\), no matter what choice of experiment is made,and that, for this reason, the subset of experiments corresponding toany given choice of settings is a fair sample of the distribution of\(\lambda\). Clauser and Horne (1974, fn. 13) made this assumptionexplicit, as did Bell, in his publications subsequent to an exchangewith Shimony, Clauser, and Horne (1976).

The assumption that experimental settings may treated as statisticallyindependent of the variable \(\lambda\) has variously been called theFree Will assumption, theFreedom of ChoiceAssumption, theNo-conspiracies Assumption, and, in someof the recent literature, theStatistical IndependenceAssumption. In this article it is called theMeasurementIndependence Assumption. It will be discussed in more detail insection 8.1, below.

3.2.3 When experiments end

In experimental tests of Bell locality, care is taken that theexperiments on the two systems, from choice of experimental setting toregistration of results, take place at spacelike separation. It isassumed that experiments have unique results. The question arises astowhen the unique result emerges. It is typically assumedthat the result is definite once a detector has been triggered or theresult is recorded in a computer memory. However, as Kent (2005) haspointed out, proposals have been made according to which the quantumstate of the apparatus would remain in a superposition of termscorresponding to distinct outcomes for a greater length of time. Onesuch proposal is the suggestion that state reduction takes place onlywhen the uncollapsed state involves a superposition of sufficientlydistinct gravitational fields (Diósi 1987, Penrose 1989, 1996).Another is Wigner’s suggestion that conscious awareness of theresult is required to induce collapse (Wigner 1961). This gives riseto what Kent calls thecollapse locality loophole. One canconsider theories—Kent calls the family of such theoriescausal quantum theories—on which collapses arelocalized events, and the probability of a collapse is independent ofevents, including other collapses, at spacelike separation from it. Atheory of that sort would differ in its predictions from standardquantum theory, but a test to discriminate between such a theory andstandard quantum mechanics would require a set-up in which the entireexperiment on one system, from arrival to satisfaction of the collapsecondition, takes place at spacelike separation from the experiment onthe other. If the experiments are taken to end, not when the detectoris triggered, but when the difference between outcomes amounts todifferences in mass configurations large enough to correspond tosignificantly distinct gravitational fields, then, as Kent argued,experiments extant at the time of writing (2005) were subject to thisloophole. The experiment of Salartet al. (2008) closed theloophole for the particular proposals of Penrose and Diósi,though, as Kent (2018) points out, altering the Penrose-Diósithreshold by a few orders of magnitude would render them compatiblewith the results of this experiment. No experiment to date hasaddressed the collapse locality loophole if the collapse condition istaken to be awareness of the result by a conscious observer. See Kent(2018) for proposals of ways in which causal quantum theory could besubjected to more stringent tests.

3.3 On “local realism”

It has become commonplace to say that (provided that the supplementaryassumptions are accepted), the class of theories ruled out byexperimental violations of Bell inequalities is the class oflocalrealistic theories, and that the worldview to be abandoned islocal realism. The ubiquity of the use of this terminologytends to obscure the fact that not all who use it use it in the samesense; further, it is not always clear what is meant when the phraseis used.

The terminology of “local realistic theories” as thetargets of experimental tests of Bell inequalities was introduced byClauser and Shimony (1978), intended as a synonym for what Clauser andHorne (1974) called “objective local theories.” These aretheories that satisfy the factorizability assumption (F). Theterminology was adopted by d’Espagnat (1979) and Mermin (1980).For Clauser and Shimony realism is “a philosophical viewaccording to which external reality is assumed to exist and havedefinite properties, whether or not they are observed bysomeone” (1978, 1883). In a similar vein, d’Espagnat(1979) says that realism is “the doctrine that regularities inobserved phenomena are caused by some physical reality whose existenceis independent of human observers” (158). Mermin, on the otherhand, takes realism to involve the condition that we have calledoutcome determinism (OD): “As I shall use the termhere, local realism holds that one can assign a definite value to theresult of an impending measurement of any component of the spin ofeither of the two correlated particles, whether or not thatmeasurement is actually performed” (Mermin 1980, 356). This isnot a commitment of realism in the sense of Clauser and Shimony, whoexplicitly consider stochastic local realistic theories.

It is Mermin’s sense that seems to be most widely used in thecurrent literature. In this sense,local realism, applied tothe set-up of the Bell experiments, amounts to the conjunction ofParameter Independence (PI) and outcome determinism (OD). Now, it istrue that, if PI and OD hold, so does factorizability (F), and hencethe Bell inequalities. But the condition OD is stronger than what isrequired, as the conjunction of PI and the strictly weaker conditionOI also suffice. Thus, to say that violations of Bell inequalitiesrule outlocal realistic theories, with “realism”identified as outcome determinism, is true but misleading, as it maysuggest that one can retain locality by rejecting“realism” in the sense of outcome determinism. However, ifone accepts the supplementary assumptions, one is obliged to rejectnot merely the conjunction of OD and PI, but the weaker condition offactorizability, which contains no assumption regarding predeterminedoutcomes of experiments.

Further confusion arises if the two senses are conflated. This canlead to the notion that the condition OD is equivalent to themetaphysical thesis that physical reality exists and possessesproperties independent of their cognizance by human or other agents.This would be an error, as stochastic theories, on which the outcomeof an experiment is not uniquely determined by the physical state ofthe world prior to the experiment, but is a matter of chance, areperfectly compatible with the metaphysical thesis. One occasionallyfinds traces of a conflation of this sort in the literature; see,e.g., d’Espagnat (1979) and Mermin (1981).

For other authors, rejection of realism seems to amount primarily toan avowal of operationalism. If all one asks of a theory is that itproduce the correct probabilities for outcomes of experiments,eschewing all questions about what sort of physical reality gives riseto these outcomes, then this undercuts the motivation of the analysisthat leads to Bell’s theorem. In this sense of“realism”, it is not an assumption of the theorem but amotivation for formulating it.

Several authors (see, in particular, Norsen 2007; Maudlin 2014) haveargued that no clear sense of “realism” has beenidentified such that realism, in that sense, is a particularpresupposition of the derivation of Bell inequalities (asdistinguished from a presupposition of all physics). These authorsurge rejection of the currently prevalent practice of saying that“local realist” theories are the targets of experimentaltests of Bell inequalities. Nonetheless, other authors maintain thatthere is, indeed, a sense of “realism” on which realism isan assumption of the derivation of Bell inequalities, though theydiffer on what it is that realism involves; see Żukowski andBrukner (2014), Werner (2014), Żukowski (2017), and Clauser(2017).

4. Early Experimental Tests of Bell’s Inequalities

The path toward a conclusive experimental test of Bell inequalitieswas long and with several intermediate steps.

A first proposal to test a Bell inequality was made by Clauser, Horne,Shimony, and Holt (1969), henceforth CHSH, who suggested that thepairs 1 and 2 be photons produced in an atomic cascade from an initialatomic state with total angular momentum \(J = 0\) to an intermediateatomic state with \(J = 1\) to a final atomic state \(J = 0\), as inan experiment performed with calcium vapor for other purposes byKocher and Commins (1967). The proposed test was first performed byFreedman and Clauser (1972). The result obtained by Freedman andClauser was 6.5 standard deviations from the limit allowed by the CHSHinequality and in good agreement with the quantum mechanicalprediction. This was a difficult experiment, requiring 200 hours ofrunning time, much longer than in most later tests of Bell’sInequality, which were able to use lasers for exciting the sources ofphoton pairs.

Since then, several dozen experiments have been performed to testBell’s Inequalities. References will now be given to some of themost noteworthy of these, along with references to survey articleswhich provide information about others. A discussion of more recentexperiments addressed to close two serious loopholes in the early Bellexperiments, the “detection loophole” and the“communication loophole”, will be reserved forSection 5.

Holt and Pipkin completed in 1973 (Holt 1973) an experiment very muchlike that of Freedman and Clauser, but examining photon pairs producedin the \(9^1 P_1 \rightarrow 7^3 S_1\rightarrow 6^3 P_0\) cascade inthe zero nuclear-spin isotope of mercury-198 after using electronbombardment to pump the atoms to the first state in this cascade. Theresult of Holt and Pipkin was in fairly good agreement with the CHSHInequality, and in disagreement with the quantum mechanical predictionby nearly 4 standard deviations—contrary to the results ofFreedman and Clauser. Because of the discrepancy between these twoearly experiments, Clauser (1976) repeated the Holt-Pipkin experiment,using the same cascade and excitation method but a different spin-0isotope of mercury, and his results agreed well with the quantummechanical predictions but violated Bell’s Inequality. Clauseralso suggested a possible explanation for the anomalous result ofHolt-Pipkin: that the glass of the Pyrex bulb containing the mercuryvapor was under stress and hence was optically active, thereby givingrise to erroneous determinations of the polarizations of the cascadephotons.

Fry and Thompson (1976) also performed a variant of the Holt-Pipkinexperiment, using a different isotope of mercury and a differentcascade and exciting the atoms by radiation from a narrow-bandwidthtunable dye laser. Their results also agreed well with the quantummechanical predictions and disagreed sharply with Bell’sInequality. They gathered data in only 80 minutes, as a result of thehigh excitation rate achieved by the laser.

Four experiments in the 1970s — by Kasday-Ullman-Wu,Faraci-Gutkowski-Notarigo-Pennisi, Wilson-Lowe-Butt, andBruno-d’Agostino-Maroni — used photon pairs produced inpositronium annihilation instead of cascade photons. Of these, all butthat of Faraciet al. gave results in good agreement with thequantum mechanical predictions and in disagreement with Bell’sInequalities. A discussion of these experiments is given in the reviewarticle by Clauser and Shimony (1978), who regard them as lessconvincing than those using cascade photons, because they rely uponstronger auxiliary assumptions.

The first experiment using polarization analyzers with two exitchannels, thus realizing the theoretical scheme envisaged inSection 2, was performed in the early 1980s with cascade photons fromlaser-excited calcium atoms by Aspect, Grangier, and Roger (1982). Theoutcome confirmed the predictions of quantum mechanics over thosesatisfying the Bell inequalities more dramatically than any of itspredecessors, with the experimental result deviating from the upperlimit in a Bell’s Inequality by 40 standard deviations. Anexperiment soon afterwards by Aspect, Dalibard, and Roger (1982),which aimed at closing the communication loophole, will be discussedinSection 5. The historical article by Aspect (1992) reviews these experiments andalso surveys experiments performed by Shih and Alley, by Ou andMandel, by Rarity and Tapster, and by others, using photon pairs withcorrelated linear momenta produced by down-conversion in non-linearcrystals. Discussion of more recent Bell tests can be found in reviewpapers (Zeilinger 1999, Genovese 2005, 2016).

Pairs of photons have been the most common physical systems in Belltests because they are relatively easy to produce and analyze, butthere have been experiments using other systems. Lamehi-Rachti andMittig (1976) measured spin correlations in proton pairs prepared bylow-energy scattering. Their results agreed well with the quantummechanical prediction and violated Bell’s Inequality, but, as inthe positronium experiments, strong auxiliary assumptions had to bemade.

The outcomes of the Bell tests provide dramatic confirmations of theprima facie entanglement of many quantum states of systemsconsisting of 2 or more constituents. Actually, the first confirmationof entanglement antedated Bell’s work, since Bohm and Aharonov(1957) demonstrated that the results of Wu and Shaknov (1950), Comptonscattering of the photon pairs produced in positronium annihilation,already showed the entanglement of the photon pairs.

5. The Communication and Detection Loopholes, and their Remedies

5.1 The Communication Loophole, and its remedy

The derivations of all the variants of Bell’s Inequality dependupon independence conditions inspired by relativistic causality. Inthe early tests of Bell’s Inequalities it was plausible thatthese conditions were satisfied just because the 1 and the 2 arms ofthe experiment were spatially well separated in the laboratory frameof reference. This satisfaction, however, is a mere contingency notguaranteed by any law of physics, and hence it is physically possiblethat the setting of the analyzer of 1 and its detection ornon-detection could influence the outcome of analysis and thedetection or non-detection of 2, and conversely. This is thecommunication loophole, to which the early Bell tests weresusceptible. It is addressed by ensuring that the experiments on thetwo systems take place at spacelike separation.

Aspect, Dalibard, and Roger (1982) published the results of anexperiment in which the choices of the orientations of the analyzersof photons 1 and 2 were performed so rapidly that they were eventswith space-like separation. No physical modification was made of theanalyzers themselves. Instead, switches consisting of vials of waterin which standing waves were excited ultrasonically were placed in thepaths of the photons 1 and 2. When the wave is switched off, thephoton propagates in the zeroth order of diffraction to polarizationanalyzers respectively oriented at angles \(a\) and \(b\), and when itis switched on the photons propagate in the first order of diffractionto polarization analyzers respectively oriented at angles \(a'\) and\(b'\). The complete choices of orientation require time intervals 6.7ns and 13.37 ns respectively, much smaller than the 43 ns required fora signal to travel between the switches in obedience to specialrelativity theory.Prima facie it is reasonable that theindependence conditions are satisfied, and therefore that thecoincidence counting rates agreeing with the quantum mechanicalpredictions constitute a refutation of the Bell inequality and henceof the family of theories that entail it. There are, however, severalimperfections in the experiment. First of all, the choices oforientations of the analyzers are not random, but are governed byquasiperiodic establishment and removal of the standing acousticalwaves in each switch. A scenario can be invented according to whichclever hidden variables of each analyzer can inductively infer thechoice made by the switch controlling the other analyzer and adjustaccordingly its decision to transmit or to block an incident photon.Also, coincident count technology is employed for detecting jointtransmission of 1 and 2 through their respective analyzers, and thistechnology establishes an electronic link which could influencedetection rates. And because of the finite size of the apertures ofthe switches there is a spread of the angles of incidence about theBragg angles, resulting in a loss of control of the directions of anon-negligible percentage of the outgoing photons.

The experiment of Tittel, Brendel, Zbinden, and Gisin (1998) did notdirectly address the communication loophole but threw some lightindirectly on this question and also provided dramatic evidenceconcerning the maintenance of entanglement between particles of a pairthat are well separated. Pairs of photons were generated in Geneva andtransmitted via cables, with very small probability per unit length oflosing the photons, to two analyzing stations in suburbs of Geneva,located 10.9 kilometers apart on a great circle. The counting ratesagreed well with the predictions of quantum mechanics and violated theCHSH inequality. No precautions were taken to ensure that the choicesof orientations of the two analyzers were events with space-likeseparation. The great distance between the two analyzing stationsmakes it difficult to conceive a plausible scenario for a conspiracythat would violate Bell’s independence conditions. Furthermore— and this is the feature which seems most to have captured theimagination of physicists — this experiment achieved muchgreater separation of the analyzers than ever before, therebyproviding a test of a conjecture by Schrödinger (1935) thatentanglement is a property that may dwindle with spatial separation.More recently, Bell inequality violation was demonstrated even at 144km distance (Scheidlet al., 2010) and, in 2017, fromsatellite transmission with a 1200 km distance (Yinet al.,2017).

An experiment that came closer to closing the communication loopholeis that of Weihs, Jennewein, Simon, Weinfurter, and Zeilinger (1998).The pairs of systems used to test a Bell’s Inequality are photonpairs in the entangled polarization state

where the ket \(\ket{H}\) represents horizontal polarization and\(\ket{V}\) represents vertical polarization. Each photon pair isproduced from a photon of a laser beam by the down-conversion processin a nonlinear crystal. The momenta, and therefore the directions, ofthe daughter photons are strictly correlated, which ensures that anon-negligible proportion of the pairs jointly enter the apertures(very small) of two optical fibers, as was also achieved in theexperiment of Tittelet al.. The two stations to which thephoton pairs are delivered are 400 m apart, a distance which light invacuo traverses in \(1.3 \mu\)s. Each photon emerging from an opticalfiber enters a fixed two-channel polarizer (i.e., its exitchannels are the ordinary ray and the extraordinary ray). Upstreamfrom each polarizer is an electro-optic modulator, which causes arotation of the polarization of a traversing photon by an angleproportional to the voltage applied to the modulator. Each modulatoris controlled by amplification from a very rapid generator, whichrandomly causes one of two rotations of the polarization of thetraversing photon. An essential feature of the experimentalarrangement is that the generators applied to photons 1 and 2 areelectronically independent. The rotations of the polarizations of 1and 2 are effectively the same as randomly and rapidly rotating thepolarizer entered by 1 between two possible orientations \(a\) and\(a'\) and the polarizer entered by 2 between two possibleorientations \(b\) and \(b'\). The output from each of the two exitchannels of each polarizer goes to a separate detector, and a“time tag” is attached to each detected photon by means ofan atomic clock. Coincidence counting is done after all the detectionsare collected by comparing the time tags and retaining for theexperimental statistics only those pairs whose tags are sufficientlyclose to each other to indicate a common origin in a singledown-conversion process. Accidental coincidences will also enter, butthese are calculated to be relatively infrequent. This procedure ofcoincidence counting eliminates the electronic connection between thedetector of 1 and the detector of 2 while detection is taking place,which conceivably could cause an error-generating transfer ofinformation between the two stations. The total time for all theelectronic and optical processes in the path of each photon, includingthe random generator, the electro-optic modulator, and the detector,is conservatively calculated to be smaller than 100 ns, which is muchless than the \(1.3 \mu\)s required for a light signal between the twostations.

The experimental result in the experiment of Weihset al. is\(2.73 \pm 0.02,\) in good agreement with the quantum mechanicalprediction, and it is 30 standard deviations away from the upper limitof the CHSH inequality inequality (8). Aspect, who designed the firstexperimental test of a Bell Inequality with rapidly switched analyzers(Aspect, Dalibard, Roger 1982) appreciatively summarized the import ofthis result:

I suggest we take the point of view of an external observer, whocollects the data from the two distant stations at the end of theexperiment, and compares the two series of results. This is what theInnsbruck team has done. Looking at the data a posteriori, they foundthat the correlation immediately changed as soon as one of thepolarizers was switched, without any delay allowing for signalpropagation: this reflects quantum non-separability. (Aspect 1999,190)

Even if some small imperfection prevented the experiment of Weihset al. from completely blocking the detection loophole, theseproblems were overcome in subsequent experiments.

5.2 The Detection Loophole and its remedy

The CHSH inequality (8) is a relation between expectation values. Anexperimental test, therefore, requires empirical estimation of theprobabilities of the outcomes of experiments. This estimation involvescomputing a ratio of event-counts: the number of pair-productionevents with a certain outcome to the total number of pair-productionevents. Typically, in experiments involving photons, most of the pairsproduced fail to enter the analyzers. Furthermore, some photons thatenter the analyzers will fail to be detected; in addition, thedetector will occasionally register a detection even when no photon isdetected (the rate of occurrence of this is known as the“dark-count”).

Three strategies for addressing this issue have been pursued.

One is to employ an auxiliary assumption to yield an estimate of thenormalization factor required to infer relative frequencies fromevent-counts, as required by a test of the CHSH inequality. CHSH(1969) proposed the assumption that, if a photon passes through ananalyser, its probability of detection is independent of theanalyser’s orientation. Though physically plausible, this is nota condition required by local causality.

The fact that an assumption of this sort is needed for the analysis ofexperiments of this type was made clear by toy models constructed byPearle (1970) and Clauser and Horne (1974). In these models, the ratesat which the photon pairs pass through the polarization analyzers withvarious orientations are consistent with an inequality of Bell’stype, but the hidden variables provide instructions to the photons andthe apparatus not only regarding passage through the analyzers butalso regarding detection, thereby violating the fair samplingassumption. Detection or non-detection is selective in the model insuch a way that the detection rates violate the Bell-type inequalityand agree with the quantum mechanical predictions. Other models wereconstructed later by Fine (1982a) and corrected by Maudlin (1994) (the“Prism Model”) and by C.H. Thompson (1996) (the“Chaotic Ball model”). Although all these models aread hoc and lack physical plausibility, they constituteexistence proofs that theories satisfying the local causalitycondition can be consistent with the quantum mechanical predictionsprovided that the detectors are properly selective.

A second strategy involves construction of an experimental set-up inwhich the production of each particle-pair may be registered. Clauserand Shimony (1978) referred to apparatus achieving this as“event-ready” detectors; some recent literature hasreferred to a process of this sort as “heralding.”

A third strategy involves employment of an inequality that can beshown to be violated without knowledge of the absolute value of theprobabilities involved. This eliminates the need for untestableauxiliary assumptions. An inequality suitable for this purpose wasfirst derived by Clauser and Horne (1974) (henceforth CH). The set-upis as before, with the exception that each analyzer will have only oneoutput channel, and the eventualities to be considered are detectionand non-detection. We want an inequality expressed in terms ofprobabilities of detection alone. The same sort of reasoning thatleads to the CHSH inequality yields theCH Inequality:

The probabilities appearing in (25) can be estimated by dividingevent-counts registered in a run of an experiment by the total numberof pairs produced. If we assume that the production rate at the sourceis independent of the analyzer settings, we can take the normalizationfactor to be the same for each term, and hence the magnitude of thisfactor need not be known in order to demonstrate a violation of theupper bound of (25). Another useful observation was made by Eberhard(1993), who demonstrated that the minimal detection efficiency for adetection loophole free experiment can be reduced (from 82% to 67%)for non-maximally entangled states (i.e. a bipartiteentangled state with different weight for the two components). Thisinvolves starting with a specified efficiency level, and then choosinga state and a set of observables that maximize violation of the CHinequality at that efficiency level.

For the maximally entangled states we have been considering, in theidealized case of perfect detection efficiency, inequality (25) ismaximally violated by the quantum predictions for the same settingsconsidered above for violation of the CHSH inequality. However, fornon-ideal experiments, the quantum predictions satisfy the inequalityunless detector efficiency is high, considerably higher than that ofany experiment that had been performed up until the time that CH werewriting. For that reason, CH introduced a new auxiliary assumption,called theno-enhancement assumption: for any value of\(\lambda\), and any setting of an analyzer, the probability ofdetection with the analyzer present is no higher than the probabilityof detection with the analyzer removed. Let \(p^1_{\infty}\) and\(p^2_{\infty}\) be the probabilities of detection of particles 1 and2 when their respective analyzers have been removed. This assumptiongives rise to what may be called thesecond CHinequality:

As CH note, this is violated by the results of the Freedman andClauser experiment, and hence that experiment rules out theoriessatisfying the factorizability condition (F) and the no-enhancementassumption, though it does not rule out the toy model constructed byCH.

Historically, the efforts toward a detection loophole-free experimentfollowed two main paths, though a few other possibilities were alsoexplored. One of these other possibilities involved K or B mesons(Selleri 1983, Go 2004), where the detection loophole reappears inanother form (Genovese, Novero, and Predazzi 2001). Another involvedsolid state systems (Ansmannet al. 2009).

One of the main avenues of approach employed entangled ions. The useof ions looked very promising, since for such experiments detectionefficiency is very high. The experiment of Roweet al. (2001)employed beryllium ions, observing a CHSH inequality violation \(S =2.25 \pm 0.03\) with a total detection efficiency of about 98%.Nevertheless, in this set-up the measurements on two ions not onlywere not space-like separated; there was a common measurement on thetwo ions. More recently, the distance between ions was increased. Forinstance, Matsukevichet al. (2008) entangled two ytterbiumions via interference and joint detection of two emitted photons, withthe distance between the ions set to 1 meter. However, a conclusiveexperiment of this sort that eliminated also the communicationloophole would require a separation of kilometers.

The other main avenue of approach, which paved the way to a conclusivetest of Bell inequalities, involved innovations in tests usingphotons. First, efficient sources of photon entangled states wererealized by exploiting Parametric Down Conversion, a non-linearoptical phenomenon in which a photon of higher energy converts intotwo lower frequency photons inside a non-linear medium in such a waythat energy and momentum are conserved. This allows a high collectionefficiency due to wave vector correlation of the emitted photons.Next, high efficiency single photon Transition Edge Sensors wereproduced. These advances led to detection loophole-free experimentswith photons (Giustinaet al. 2013, Christensenetal. 2013) and finally to the conclusive tests discussed in thenext section.

5.3 Loophole-free tests

In 2015 three papers appeared claiming a conclusive test of Bellinequalities. The first (Hensenet al., 2015) achieved aviolation of the CHSH inequality via an event-ready scheme. Thisexperiment is based on using electronic spin associated with thenitrogen-vacancy (NV) defect in two diamond chips located in distantlaboratories. In the experiment, each of these two spins is entangledwith the emission time of a single photon. Then the two,indistinguishable, photons are transmitted to a remote beam splitter.A measurement is made on the photons after the beam splitter. Anappropriate result of the measurement of the photons projects thespins in the two diamond chips onto a maximally entangled state, onwhich a Bell inequality test is realized. The high efficiency in spinmeasurement and the distance between the laboratories allows closureof the detection and communication loophole at the same time. However,the experiment utilized only a small number, 245, of trials, and thusthe statistical significance (2 standard deviations) of the result \(S= 2.42 \pm 0.20\) is limited.

The other two experiments, published in the same issue ofPhysicalReview Letters (Giustinaet al. 2015, Shalmetal. 2015), are based on transmitting two polarization-entangledphotons, produced by Parametric Down Conversion, to two remotelaboratories, where they are measured by high detection efficiencyTransition Edge Sensors. These experiments use states that are notmaximally entangled, but are optimized, in accordance with theanalysis of Eberhard (1993), to produce a maximal violation of the CHinequality, given the detection efficiency of the experiments. In bothof these experiments a violation of the CH inequality was obtained, ata high degree of statistical significance. Shalmet al.report a \(p\)-value of \(2.3 \times 10^{-7}\), whereas Giustinaet al. report a \(p\)-value of \(3.4 \times 10^{-31}\),corresponding to an 11.5 standard deviation effect. A very carefulanalysis of data (including spacelike separation of detection events),of statistical significance, and of all possible loopholes leavesreally no space for doubts about their conclusiveness. Besides thedetection and communication loophole, these two experiments addressalso the following issues:

• The number of emitted particles must be independent of measurementsettings (production rate loophole). This was excluded by anabsolutely random choice of settings.
• The presence of a coincidence window must not allow in a hiddenvariable scheme a situation where the local setting may change thetime at which the local event happens (coincidence loophole). This wasexcluded by using a pulsed source.
• An eventual memory of previous measurements must be considered inthe statistical analysis since the data might not be independently andidentically distributed (memory loophole). This is excluded by acareful analysis of data.

Furthermore, independent random number generators based on laser phasediffusion guarantee the elimination of the freedom-of-choice loophole(except in the presence of superdeterminism or other hypotheses that,by definition, do not allow a test through Bell inequalities).

In summary, these experiments, having satisfied carefully all theconditions required for a conclusive test, unequivocally tested Bellinequalities without any additional hypothesis.

More recently, loophole-free experiments were realized with atoms(Rosenfeld et al., 2017) and superconductors (Storz et al. 2023)too.

6. Variants of Bell’s Theorem and of Bell inequalities experiments

We start by discussing two variants of Bell’s Theorem thatdepart in some respect from the conceptual framework presented inSection 2. Both are less general than the version inSection 2, because they rely on perfect correlations, which, together with thefactorizability condition (F), entail outcome determinism (OD). Then,two other variants will be mentioned briefly but not summarized indetail.

The first variant is due independently to Kochen,^[8] Stairs (1978, 1983), and Heywood and Redhead (1983). Its ensemble ofinterest consists of pairs of spin-1 particles in the entangledstate

where \(\ket{z,i}_1\), with \(i = -1\) or 0 or 1 is the spin state ofparticle 1 with a component of spin \(i\) along the axis \(z\), and\(\ket{z,i}_2\) has an analogous meaning for particle 2. If \(x,y,z\)is a triple of orthogonal axes in 3-space then the components \(s_x,s_y, s_z\) of the spin operator along these axes do not pairwisecommute. However, the squares of these operators — \(s_x^2,s_y^2, s_z^2\) — do commute, and therefore, in view of theconsiderations ofSection 1, any two of them can constitute a context in the measurement of thethird. If the operator of interest is \(s_z^2\), the axes \(x\) and\(y\) can be any pair of orthogonal axes in the plane perpendicular to\(z\), thus offering an infinite family of contexts for themeasurement of \(s_{z}^2\). As noted inSection 1 Bell exhibited the possibility of acontextual hiddenvariables theory for a quantum system whose Hilbert space hasdimension 3 or greater even though the Bell-Kochen-Specker theoremshowed the impossibility of anon-contextual hidden variablestheory for such a system. The strategy of the argument is to use theentangled state of Eq. (27) to predict the outcome of measuring\(s_{z}^2\) for particle 2 (for any choice of \(z)\) by measuring itscounterpart on particle 1. A specific complete state \(\lambda\) woulddetermine whether \(s_{z}^2\) of 1, measured together with a contextin 1, is 0 or 1. Agreement with the quantum mechanical prediction ofthe entangled state of Eq. (27) implies that \(s_z^2\) of 2 has thesame value 0 or 1. But if the factorizability condition (F) isassumed, then the result of measuring \(s_{z}^2\) on 2 must beindependent of the remote context, that is, independent of the choiceof \(s_{x}^2\) and \(s_y^2\) of 1, hence of 2 because of correlation,for any pair of orthogonal directions \(x\) and \(y\) in the planeperpendicular to \(z\). It follows that the hypothetical theory whichsupplies the complete state \(\lambda\) is not contextual after all,but maps the set of operators \(s_z^2\) of 2, for any direction \(z\),noncontextually into the pair of values (0, 1). But that isimpossible in view of the Bell-Kochen-Specker theorem. The conclusionis that no theory satisfying the factorizability condition (F) isconsistent with the quantum mechanical predictions of the entangledstate (29).

A simpler proof of Bell’s Theorem, also relying uponcounterfactual reasoning and based upon a deterministic local theory,is that of Hardy (1993), here presented in Laloë’s (2001)formulation. Consider an ensemble of pairs 1 and 2 ofspin\(-\frac{1}{2}\) particles, the spin of 1 measured alongdirections in the \(xz\)-plane making angles \(a=\theta /2\) and\(a'=0\) with the \(z\)-axis, and angles \(b\) and \(b'\) havinganalogous significance for 2. The quantum states for particle 1 withspins \(+\frac{1}{2}\) and \(-\frac{1}{2}\) relative to direction\(a'\) are respectively \(\ket{a',+}_1\) and \(\ket{a',-}_1\), andrelative to direction \(a\) are respectively

the spin states for 2 are analogous. The ensemble of interest isprepared in the entangled quantum state

(unnormalized, because normalization is not needed for the followingargument). Then for the specified \(a, a', b\), and \(b'\) thefollowing quantum mechanical predictions hold:

and for almost all values of the \(\theta\) of Eq. (31)

with the maximum occurring around \(\theta = 9\degr\). Inequality (33)asserts that for the specified angles there is a non-empty subensemble\(E'\) of pairs for which the results for a spin measurement along\(a\) for 1 and along \(b\) for 2 are both +. Eq. (30) implies thecounterfactual proposition that if the spin of a 2 in \(E'\) had beenmeasured along \(b'\) then with certainty the result would have been\(-\); and likewise Eq. (31) implies the counterfactual propositionthat if the spin of a 1 in \(E'\) had been measured along \(a'\) thenwith certainty the result would have been –. It is in this stepthat counterfactual reasoning is used in the argument. Since thesubensemble \(E'\) is non-empty, we have reached a contradiction withEq. (32), which asserts that if the spin of 1 is measured along \(a'\)and that of 2 is measured along \(b'\) then it is impossible that bothresults are –. The incompatibility of a deterministic localtheory with quantum mechanics is thereby demonstrated.

An attempt was made by Stapp (1997) to demonstrate a strengthenedversion of Bell’s theorem which dispenses with the conceptualframework outlined above, and to use instead the logic ofcounterfactual conditionals. His intricate argument has been thesubject of a criticism by Shimony and Stein (2001, 2003), who arecritical of certain counterfactual conditionals that are asserted byStapp by means of a “possible worlds” analysis without agrounding on a local deterministic theory, and a response by Stapp(2001) himself, who defends his argument with some modifications.

The three variants of Bell’s Theorem considered so far in thissection concern ensembles of pairs of particles. An entirely newdomain of variants is opened by studying ensembles of \(n\)-tuples ofparticles with \(n \ge 3\). The prototype of this kind of theorem wasdemonstrated by Greenberger, Horne, and Zeilinger (1989) (GHZ) for\(n=4\) and modified to \(n=3\) by Mermin (1990) and by Greenberger,Horne, Shimony, and Zeilinger (1990) (GHSZ). In the theorem of GHZ anentangled quantum state was written for four spin-\(\frac{1}{2}\)particles and the expectation value of the product of certain binarymeasurements performed on the individual particles was calculated.They then showed that the attempt to duplicate this expectation valuesubject to the factorizability constraint (F) produces acontradiction. A similar result was obtained by Mermin for a state of3 spin-\(\frac{1}{2}\) particles and by GHSZ for a state of 3 photonsentangled with respect to their direction of propagation. Because ofthe length limitations, the details will not be summarized here, it ishowever worth mentioning that experimental tests were realized (Panet al. 2000, Genovese 2005) (also in this case withadditional hypotheses).

Finally, it is worth mentioning that, usually, in a Bell inequalitytest the expectation values in (8) must be evaluated on differentpairs of entangled photons. This could eventually be overcome byexploiting weak measurements (Aharonov 1988), i.e. measurements wherethe measuring apparatus is weakly coupled to the measured quantumstate: this allows for avoiding the collapse of the wave function,obtaining partial information on the observable. In the case of Bellmeasurement an estimate of S, eq(8), can be obtained in a singlemeasurement. By exploiting sequential weak measurements (Piacentini2016), after a few first experimental results involving either weakmeasurements on a single arm only (Foletto 2020) or one weak plus onestrong measurement (White 2016, Calder 2020), then a full Bellinequality test based on weak measurements only, which only partiallydecoheres the entangled state, was recently achieved (Virzì etal. 2023, Other Internet Resources).

7. Significance for Quantum Information Theory

The set-up envisaged in the proof of Bell’s theorem highlights astriking prediction of quantum theory, namely, long-distanceentanglement, and experimental tests of the Bell inequalities provideconvincing evidence that it is a feature of reality (recent satelliteexperiments having reached a 1200 km distance, Yin 2017 and Yin 2020).Moreover, Bell’s theorem reveals that the entanglement-basedcorrelations predicted by quantum mechanics are strikingly differentfrom the sort of locally explicable correlations familiar in aclassical context.

Investigations into entanglement and the ways in which it can beexploited to perform tasks that would not be feasible with onlyclassical resources forms a key part of the discipline of quantuminformation theory (see Benatti,et al. eds. (2010), and theentries onquantum computing andquantum entanglement and information). In drawing attention to the import of entanglement and the ways inwhich it is different from anything in classical physics, Bell’stheorem and the experimental work derived from it provided, at leastindirectly, some of the impetus to the development of quantuminformation theory.

Bell’s theorem has played a direct role in the development ofdevice independent quantum cryptography. One can exploit quantumcorrelations to devise a quantum key distribution protocol that isprovably secure on the assumption that, whatever the underlyingphysics is, it does not permit superluminal signalling. The basic ideais that, if one has Bell inequality-violating correlations atspacelike separation, any predictability of the results beyond thatafforded by the quantum probabilities could be exploited forsuperluminal signalling; the contrapositive of this is thatimpossibility of signalling entails “absolute randomness”— absolute in the sense of being independent of the details ofthe underlying physics beyond the prohibition on superluminal signalling.^[9]

A further interesting application is to exploit the phenomenon ofquantum non-locality with a loophole-free Bell test to build arandom-number generator that can produce output that is unpredictableto any adversary that is limited only by general physical principles,such as special relativity (Bierhorst 2018).

Furthermore, cryptography, by its very nature must take into accountthe possibility of a conspiracy aimed at deceiving the users of acryptographic key, and so, in this context, it is essential todemonstrate security in the presence of such a conspiracy.

A result due to Colbeck and Renner (2011, 2016), building on work ofBranciardet al. (2008), shows that this cannot be done if PIis satisfied. This result has significance both at the operational andthe fundamental level. It can be applied at the fundamental level toconclude that any theory with sharper probabilities than the quantumpredictions must violate PI. In addition, even if some deterministictheory (such as the de Broglie-Bohm theory) applies at the fundamentallevel, the Colbeck-Renner theorem can be applied at the operationallevel, where the probabilities involved may indicate limitations onaccessible information about the physical state. A violation of PI atthe operational level would permit signalling. Thus, the theorem showsthat, as long as the no-signalling condition is satisfied, a would-beeavesdropper attempting to subvert the privacy of a key distributionscheme by intercepting the particle pairs and substituting ones thatwill yield results that she has some information about, cannot do sowithout disrupting the correlations between the particle pairs. SeeLeegwater (2016) for a clear exposition of the theorem.

8. Philosophical/metaphysical implications

Bell inequalities follow from a number of assumptions that haveintuitive plausibility and which, arguably, are rooted in the sort ofworld-view that results from reflection on classical physics andrelativity theory. If one accepts that the experimental evidence givesus strong reason to believe that Bell inequality-violatingcorrelations are features of physical reality, then one or more ofthese assumptions must be given up. Some of these assumptions are ofthe sort that have traditionally been regarded as metaphysicalassumptions. The fact that a conjunction of theses of the sort usuallyregarded as metaphysical has consequences that can be subjected toexperimental test has led Shimony to speak of the enterprise ofexperimental testing for violations of Bell inequalities as“experimental metaphysics” (Shimony 1984a, 35; 1993, 115).As may be expected, the conclusions of experimental metaphysics arenot unambiguous. Someprima facie plausible options areexcluded, leaving a number of options open. In this section these arebriefly outlined, with no attempt made to adjudicate between them.

Suppose that one accepts that the experimental evidence indicates thatBell-inequality violating correlations are a real feature of theworld, even when the experiments are conducted at spacelikeseparation, on the most stringent of conditions that advocates of acollapse locality loophole might wish to impose (see section 3.2.3).Acceptance of the reality of such correlations requires rejection ofthe conjunction of any set of assumptions sufficient to entail a Bellinequality. The analysis of the assumptions of the proof outlined insection 3 affords a taxonomy of positions that one might adopt inlight of their experimental violation, as at least one of theassumptions must be rejected. The distinctive assumption is thePrinciple of Local Causality, which, when applied to the setup ofBell-type experiments, is embodied in the factorizability condition.This condition can be maintained only if one of the supplementaryassumptions is rejected. We begin with options rejecting supplementaryassumptions, then proceed to considering options for accepting theauxiliary assumptions and rejecting local causality

8.1 Rejecting the supplementary assumptions

As we are considering implications of accepting Bell inequalityviolations as a feature of reality, even when the experiments areperformed at spacelike separation, we set aside the collapse localityloophole. This leaves us with two options.

• Reject unique outcomes. This is the option taken byEverettian, or Many-Worlds theories, and related interpretations ofquantum mechanics. The question arises whether locality can bemaintained on this option. Posing the question of whether Bell’stheorem has implications for locality on Everettian or relative-stateinterpretations requires that one first consider how to formulatelocality conditions in such a context, as the conditions formulated insection 3.1, presuppose unique experimental outcomes. See Vaidman(1994), Bacciagaluppi (2002), Chapter 8 of Wallace (2012), Tipler(2014), Vaidman (2016), Brown and Timpson (2016), and Myrvold (2016)for discussions of locality in Everettian interpretations, and Smerlakand Rovelli (2007) for a discussion in the context of relationalinterpretations.

• Reject the measurement independence assumption. There areessentially two ways to do this. One is to suppose a common cause inthe past that determines both experimental settings and experimentaloutcomes. In an interview, Bell refer to an account of this sort as“super-deterministic,” (Davies and Brown, 47), and theapproach has come to be calledsuperdeterminism. Another isto reject the usual assumption that causes must temporally precedetheir effects. If causal influence from future to past is admitted,then, even if the settings are regarded as free variables, they mightinfluence the state of the system at the moment of preparation,contravening the assumption that the preparation probabilitydistribution be independent of experimental settings. This approach isoften referred to asretrocausality. Both of these approacheshave gained advocates in recent years.

8.1.1 Retrocausality

On a retrocausal approach, experimental settings on each wing of theexperiment retroactively effect the state of the pair of particles atthe source, which in turns has an effect on the outcome on the otherwing. We thus have violations of both Measurement Independence andParameter Independence. This avenue was suggested by Costa deBeauregard (1977), in a comment on the interchange between Bell andShimony, Holt, and Clauser. Costa de Beauregard objected that thediscussants were disregarding the possibility of retrocausality, apossibility that he had advanced earlier (1976). In recent years ithas been advocated by Price and Wharton (Price 1994, 1996; Price andWharton 2015). It forms part of the Transactional Interpretation ofquantum mechanics (Cramer 1986, 1988, 2016; Kastner 2013). For more onretrocausal approaches, see the entry,Retrocausality in Quantum Mechanics, and Wharton and Argaman (2020).

8.1.2 Superdeterminism

At the end of Bell (1976), in which Bell provides an exposition of aBell-type theorem, we find a remark regarding the independence of\(\lambda\) and the experimental settings.

It has been assumed that the settings of instruments are in some sensefree variables — say at the whim of the experimenters — orin any case not determined in the overlap of the backward lightcones.Indeed without such freedom I would not know how to formulateany idea of local causality, even the modest human one (Bell1976, 8; 1985a, 95; 1987b and 2004, 61).

This assumption was not, however,explicitly invoked in thederivation, and the role this assumption is meant to play was not madesufficiently clear in that article.

That an assumption of this sort is required was emphasized by Shimony,Horne and Clauser (1976), who illustrated the point via a fancifulstory of a conspiracy involving manufacturers of experimentalapparatus and physicists’ assistants. The director of theconspiracy concocts a set of correlation experiment data, consistingof a sequence of pairs of experimental settings and results obtained.The director instructs the manufacturer to preprogram the apparatus toproduce the desired outcomes, and the assistants of the physicistsperforming the experiment to orchestrate the apparatus settings tomatch those specified by the predetermined list. Clearly, theconspirators may utilize any set of correlation data for theirnefarious schemes; hence,any set of correlation data can beobtained as the outcomes of this sort of process, without anyviolation of any sort of locality condition. Correlations obtained bythis means could include, not only correlations of the sort predictedby quantum mechanics, but also correlations forbidden by quantummechanics, such as correlations violating the Tsirelson bound, orcorrelations on which the result at one wing of the experiment isinformative about the setting on the other, giving the illusion ofsuperluminal signalling.

Shimony, Horne, and Clauser’s aim was not to reject Bell’sconclusion, but rather to motivate an explicit invocation of anassumption that had been implicit in Bell’s reasoning. For thesorts of experiments envisaged in tests of Bell inequalities, Shimony,Horne, and Clauser consider the assumption of independence of settingsand the state of the particle pairs to be justified, even thoughrelativistic causality does not mandate this independence.

After all, the backward light cones of those two acts do eventuallyoverlap, and one can imagine one region which controls the decision ofthe two experimenters who chose \(a\) and \(b\). We cannot deny such apossibility. But we feel that it is wrong on methodological grounds toworry seriously about it if no specific causal linkage is proposed. Inany scientific experiment in which two or more variables are supposedto be randomly selected, one can always conjecture that some factor inthe overlap of the backwards light cones has controlled the presumablyrandom choices. But, we maintain, skepticism of this sort willessentially dismiss all results of scientific experimentation. Unlesswe proceed under the assumption that hidden conspiracies of this sortdo not occur, we have abandoned in advance the whole enterprise ofdiscovering the laws of nature by experimentation (Shimony, Horne andClauser 1976, 6; 1985, 101; Shimony 1993b, 168).

Appended to the last quoted sentence is an endnote referencingShimony’s “Scientific Inference,” in which anapproach to scientific methodology is advocated that is guided by theidea that, although there can be no guarantee, whatever scientificmethodology we adopt, that it will lead us towards the truth, we canand should guide our methodological choices so that they will,whenever possible, be sensitive to the truth, in accord with C. S.Peirce’s maxim, “Do not block the way of inquiry”(Peirce 1931). As the authors note, randomization posits are pervasivein experimental science. A methodology that countenanced rejection ofa posit of this sort whenever its employment would lead to anundesired conclusion would amount to an abandonment of the enterpriseof testing hypotheses by experimentation, in contravention ofPeirce’s maxim.

In response, Bell (1977) acknowledged that his formulation in (1976)had been inadequate, and explained his reasoning as “primarilyan analysis of certain kinds of physical theory.”

A respectable class of theories, including contemporary quantum theoryas it is practised, have ‘free’ ‘external’variables in addition to those internal to and conditioned by thetheory. These variables are typically external fields or sources. Theyare invoked to represent experimental conditions. They also provide apoint of leverage for ‘free willed experimenters’, ifreference to such hypothetical metaphysical entities is permitted. Iam inclined to pay particular attention to theories of this kind,which seem to me most simply related to our everyday way of looking atthe world. (1977, 80; 1985b, 104; 1987b and 2004, 101)

He makes it clear, however, that no metaphysical hypothesis ofexperimenters exempt from the laws of physics need be invoked.Bell’s preliminary statement of what it means for variables tobe ‘free’ is,

For me this means that the values of such variables have implicationsonly in their future light cones. They are in no sense a record of,and do not give information about, what has gone before. In particularthey have no implications for the hidden variables ν in the overlapof the backward light cones ... .(1977, 79; 1985b, 103; 1987b and2004, 100)

As stated, this is too strong; if the dynamics are invertible, it isnot possible that the values of these variables have implicationsonly in the future light-cones of the setting events. What isneeded is something considerably weaker than the condition that thevariables not be determined in the overlap of the backward light conesof the experiments; what is needed is that they be “at leasteffectively free for the purpose at hand.” Bell argues thata deterministic randomizer that is extraordinarily sensitive toinitial conditions would suffice to provide the requisiteindependence, and that variables of this type may be treated as ifthey have implications only for events in their future lightcones.

Suppose that the instruments are set at the whim, not of experimentalphysicists, but of mechanical random number generators. Indeed itseems less impractical to envisage experiments of this kind, withspace-like separation between the outputs of two such devices, than tohope to realize such a situation with human operators. Could theoutputs of such mechanical devices reasonably be regarded assufficiently free for the purpose at hand? I think so.

Consider the extreme case of a ‘random’ generator which isin fact perfectly deterministic in nature — and, for simplicity,perfectly isolated. In such a device the complete final stateperfectly determines the complete initial state — nothing isforgotten. And yet for many purposes, such a device is precisely a“forgetting machine”. A particular output is the result ofcombining so many factors, of such a lengthy and complicated dynamicalchain, that it is quite extraordinarily sensitive to minute variationsof many initial conditions. It is the familiar paradox of classicalstatistical mechanics that such exquisite sensitivity to initialconditions is practically equivalent to complete forgetfulness ofthem. To illustrate the point, suppose that the choice between twopossible outputs, corresponding toa anda′,depended on the oddness or evenness of the digit in the millionthdecimal place of some input variable. Then fixinga ora′ indeed fixes something about the input — i.e.,whether the millionth digit is odd or even. But this peculiar piece ofinformation is unlikely to be the vital piece for any distinctivelydifferent purpose, i.e., it is otherwise rather useless. With aphysical shuffling machine, we are unable to perform the analysis tothe point of saying just what peculiar input is remembered in theoutput. But we can quite reasonably assume it is not relevant forother purposes. In this sense the output of such a device is indeed asufficiently free variable for the purpose at hand. (1977,82–83; 1985b, 105–106; 1987b and 2004, 102–103)

Bell concluded with the remark,

Of course it might be that these reasonable ideas about physicalrandomizers are just wrong — for the purpose at hand. A theorymay appear in which such conspiracies inevitably occur, and theseconspiracies may then seem more digestible than the non-localities ofother theories. When that theory is announced I will not refuse tolisten, either on methodological or other grounds. But I will notmyself try to make such a theory. (1977, 83; 1985b, 106; 1987b and2004, 103)

The upshot of the exchange was substantial agreement between Bell andShimony, Horne, and Clauser.

In actual experiments, a variety of randomizing strategies have beenemployed that, on the basis of plausible physical assumptions, wouldbe expected to ensure the requisite independence of experimentalsetting and conditions at the source of the pair of particles. Theexperiment of Scheidlet al. (2010) was the first to employrandom-number generators operating at spacelike separation from thegeneration of the particle pairs at the source. Following a suggestionof Clauser, other tests have employed measurements on photons fromMilky Way stars (Handsteineret al. 2017) and from distantquasars (Rauchet al. 2018). In addition, the “Big BellTest” ran experiments utilizing inputs provided by approximately100,000 volunteers (Abellánet al. 2018). In theexperiment of Shalmet al. (2015), the measurement decisionswere determined by applying an XOR (exclusive or) operation to threebits from three independent sources, one of which was a pseudorandomsequence constructed from digits of \(\pi\) and binary strings derivedfrom various popular culture sources, including episodes of the moviesMonty Python and the Holy Grail, theBack to theFuture trilogy, and episodes ofDoctor Who,Saved bythe Bell, andStar Trek.

Nonetheless, the project of constructing a theory with purely localdynamics, on which the observed correlations are due to a failure ofthese and other purported randomization procedures to eliminatecorrelations of just the right sort needed to produce statisticsclosely matching the quantum-mechanical predictions, has gained somesupport in recent years. It is a feature of ’t Hooft’sCellular Automaton Interpretation of quantum mechanics (’t Hooft2016; see also Scardagliet al. 2019 ), and has beenadvocated by Hossenfelder and Palmer (2020) and Hance, Hossenfelderand Palmer (2022). These authors rely on the possibility that systemsthought to have dynamics that play the randomizing role of which Bellspeaks might, in fact, have dynamics of a different sort, and whatappears to be extraordinary sensitivity to minute variations ininitial conditions might not be.

Hossenfelder and Palmer (2020) take the charges thatsuperdeterministic theories are “conspiratorial” orrequire “fine-tuning” to mean: the set of initialconditions that lead to the requisite correlations between parametersettings and the initial states is small, according to some physicallysalient measure. The strategy of their response is to consider thepossibility of a theory with a state space whose possibilities aremore restricted than one might ordinarily think, and argue that, onsuch a space, a physically salient measure might be one on which theset of initial conditions that lead to the requisite correlations islarge. That is, a theory is imagined in which initial conditionslacking the requisite correlations are excluded as physicalpossibilities.

For further considerations regarding superdeterminism, see Chen(2022), Andreoletti and Vervoort (2022), Baas and Le Bihan (2023), andCiepielewski, Okon and Sudarsky (2023).

8.1.3 Rejecting Local Causality

If the auxiliary assumptions are accepted, then the lesson to be drawnfrom the experimental violation of Bell inequalities is that thecondition that Bell called thePrinciple of Local Causalitymust be rejected. As we have seen, this condition is a conjunction oftwo conditions: a causal locality condition (PLC-1, above), and thePrinciple of the Common Cause. The causal locality condition itselfcan be regarded as stemming from the assumption that causes temporallyprecede their effects and Lorentz invariance of the relation oftemporal precedence. If the relation of temporal precedence is to beLorentz invariant, then either it is the trivial relation that holdsbetween any two spacetime points, or else the past of a spacetimepoint is its past light cone (Stein 1991, 2009).

In this context, it is useful to recall that the factorizabilitycondition can be thought of as a conjunction of outcome independence(OI) and parameter independence (PI). PI is a consequence of causallocality (PLC-1), applied to the set-up of the Bell experiments,whereas OI is a consequence of causal locality and the common causeprinciple. This leaves us with a dilemma. A rejection offactorizability involves a rejection of PI or OI. A rejection of PIinvolves a rejection of causal locality. If causal locality is to bemaintained, then OI, and hence the Common Cause Principle, must berejected.

Any deterministic theory must satisfy OI, and hence, a deterministictheory that rejects factorizability must reject causal locality.

These considerations yield a taxonomy of options for accepting thesupplementary assumptions while also accepting Bell-inequalityviolating correlations.

• Reject PLC-1. One option is to reject the assumption PLC-1 ofcausal locality, and accept that there are causal relations betweenevents that are outside of each others’ light-cones. There aretwo ways to do this.

  • Introduce non-relativistic spatiotemporal structure. Oneoption, and perhaps the most straightforward one, is to reject theassumption of Lorentz invariant spatiotemporal structure. This is theposition that is involved in acceptance of the de Broglie-Bohm theoryor any similar theory. On the de Broglie-Bohm theory, the velocity ofa particle is a function not only of its own location but also of thelocation of any other particles it is entangled with. This means thatthe dynamics of the theory requires a privileged relation of distantsimultaneity. The non-relativistic spatiotemporal structure could beintroduced dynamically, as in the family of models outlined byDürret al. (1999).
  • Retrocausality. Another option is to maintain theassumption that the past of an event is its past light cone, andthereby maintain Lorentz invariance of the spatiotemporal structure,but to deny that causes must temporally precede their effects.Typically, those who take this option accept retrocausality,maintaining that causal processes can propagated within light cones ineither temporal direction; thus, every event is at least potentiallyin the causal domain of the other. For this reason, an approach ofthis sort involves a rejection of causal locality, even though it mayrespect the idea that a causal influence must be mediated by acontinuous process along a timelike worldline.

• Reject the Principle of Common Cause. A stochastic theory,such as a dynamical collapse theory, that reproduces quantumprobabilities for Bell experiments, will involve correlated events atspacelike separation. It need not, however, involve any events in thecommon past of these events that screen off the correlations; thesecorrelations will be built-in to the laws of the theory yieldingprobabilities of events. Whether one is willing to extend talk ofcause-effect relations to refer to the relation between such events ismerely a matter of terminology; it should be noted, however, thatthere is nothing of the usual asymmetry between cause and effect inthe relation between these events. To accept this relation as a newsort of symmetric cause-effect relation removes any reason there is tothink that cause-effect relations between spacelike separated eventsare incompatible with relativistic spacetime structure.

  A more common view is that the lesson of Bell’s theorem is thatthere may be correlations that are not explicable in terms ofcause-effect relations. This involves rejection of the Common CausePrinciple (see,e.g., van Fraassen 1982, Fine 1989,Butterfield 1992, Elby 1992). Others have maintained that, though theCommon Cause Principle as formulated by Reichenbach is to be rejected,the principle has been formulated too narrowly, and needs to bereformulated in light of quantum phenomena. Some have suggested thatdespite not satisfying Reichenbach’s principle, correlationsviolating Bell inequalities are due to a common cause in their past,namely, the process that created entanglement (Unruh 2002, 136).Hofer-Szabo, Rédei, and Szabo (1999, 2002) have suggested amodification of the common cause principle, as have Leifer andSpekkens (Leifer 2006, Leifer and Spekkens 2011). See Cavalcanti andLal (2014) for discussion and a critique of these proposals.

In all of this it is assumed that the spacetime topology matches thatof Minkowski spacetime. One might also reject this assumption; it ispossible that events thought to be spacelike separated are, in fact,causally connectable, because of a spacetime topology different fromthat usually assumed. On this approach, locality is restored either byconsidering wormholes, the so-called ER = EPR (namely, Einstein-Rosenwormhole is related to EPR paradox) conjecture (Maldacena and Susskind2013) or more wrapped dimensions (Genovese and Gramegna 2019).

8.2 Quantum mechanics and relativity

Does Bell’s theorem show that quantum theory is incompatiblewith relativity?

The answer, of course, depends on what one takes relativity theory torequire. It can be shown (Eberhard 1978, Ghirardi, Rimini & Weber1980, Page 1982) that, in the absence of nonlocal interaction terms inthe Hamiltonian, quantum correlations cannot be exploited to sendsignals superluminally. There has been a tendency in some of theliterature to take this by itself to indicate compatibility withrelativity. That this is insufficient can be seen from the fact thatthere can be theories, such as the de Broglie-Bohm theory, thatrequire nonrelativistic spacetime structure for the formulation oftheir dynamical laws, while not permitting signalling (at least, aslong as the usual distribution postulate for particle positions issatisfied).

Take arelativistic spacetime structure to be a structurethat includes a relation of temporal precedence on which the past of aspacetime point is the set of points on or within its past lightcone,and its future, the set of points on or within its future lightcone,and other points are temporally unrelated to it, neither past norfuture. One can ask whether a given account of the goings-on in theworld is compatible with a relativistic spacetime structure. If atheory requires events outside of its lightcone to be partitioned intothose that are past, future, and simultaneous with a point, as doesthe de-Broglie Bohm theory, it is not compatible with a relativisticspacetime structure. Compatibility of a theory with relativisticspacetime structure, in this sense, is a distinct issue from Lorentzcovariance of the equations expressing the theory’s dynamicallaws. One can construct theories on which nonrelativistic spacetimestructure is introduced dynamically, as is done by Dürr,Goldstein, Münch-Berndl and Zanghì (1999), who formulate aBohmian theory against a background of Minkowski spacetime byintroducing an auxiliary field that picks out a distinguishedfoliation that is then used to formulate the dynamics of the theory(see Dürret al. 2014 for discussion). Berndl,Dürr, Goldstein, and Zanghì (1996) consider the class oftrajectory theories—that is, theories that, like the deBroglie-Bohm theory, assign definite trajectories toparticles—and prove that such theories could not satisfy theBorn-rule distribution postulate along arbitrary spacelikehyperplanes. The argument generalizes to any theories, such as modalinterpretations, that assign definite values to variables other thanposition (Dickson and Clifton 1998, Arntzenius 1998, Myrvold 2002).Theories of this sort, therefore, must invoke a distinguishedfoliation.

A deterministic theory must satisfy the condition of outcomeindependence, and hence if one accepts \((a)\) that a violation of PI,that is, a situation in which the choice of an experiment on one sidechanges the probability of an outcome on the other, is an instance ofcausation, and \((b)\) that a cause must temporally precede itseffect, and then it follows that a deterministic theory that satisfiesthe supplementary conditions and reproduces the quantum correlationsis incompatible with relativistic spacetime structure. Such a theorymay, however, be Lorentz invariant at the phenomenal level, if (as inthe de Broglie-Bohm theory) the distinguished foliation isunobservable.

A stochastic theory, such as a dynamical collapse theory, must involvecorrelations between spacelike separated events. A relation like that,however, is symmetric, and does not require that one of the events bein the past of the other. There is noprima facie need fornonrelativistic spacetime structure. Indeed, it is possible toformulate a fully relativistic dynamical collapse theory. Arelativistic generalization of the Ghirardi-Rimini-Weber (GRW) theorywas constructed by Dove (1996) and, independently, by Tumulka (2006).Relativistic versions of the Continuous Spontaneous Localization (CSL)theory have been constructed by Bedingham (2011) and Pearle (2015).For discussions of ontology for such theories, required to extractfrom them a sensible account of a world that includes macroscopicobjects, see Pearle (1997), Bedinghamet al. (2014), andMyrvold (2019). For more on the bearing of relativity on variousapproaches to the interpretation of quantum theory, see Myrvold(2022).

Shimony, in several of his writings (Shimony 1978, 1983, 1984a,b,1986, 1988, 1989, 1990, 1991) spoke of “peacefulcoexistence” between special relativity and quantum theory. Themeaning of this varied. In its initial formulation (1978),“peaceful coexistence” had to do with regardingexperimental outcomes as transitions from potentiality to actuality,something that, Shimony says, requires further investigation to beunderstood. In later writings, peaceful coexistence is“suggested” by the fact that quantum correlations cannotbe exploited for superluminal signalling (Shimony 1983), though stillassociated with the notion that quantum mechanics motivates a changein our conception of an event (Shimony 1984a), and involves therequirement of a coherent meshing of events as described with respectto different reference frames (1986). In other works peacefulcoexistence seems to be simply identified with the impossibility ofexploiting quantum correlations for signalling (Shimony 1984b, 1990,1991). Convinced by Bell (1990) that anthropocentric considerationssuch as manipulability have no place in considerations of fundamentalphysics, Shimony became dissatisfied with this avenue of approach toreconciling relativity and quantum theory (Shimony 2009, 489).

Bell’s own attitude towards the question of whether Bell’stheorem indicates a fundamental incompatibility between quantum theoryand relativity seems to have varied with time. At the end of the 1964article, he wrote

In a theory in which parameters are added to quantum mechanics todetermine the results of individual measurements, without changing thestatistical predictions, there must be a mechanism whereby the settingof one measuring device can influence the reading of anotherinstrument, however remote. Moreover, the signal involved mustpropagate instantaneously, so that such a theory could not be Lorentzinvariant (Bell 1964, 199; 1987b and 2004, 20).

At this point he is claiming incompatibility with relativity only fordeterministic hidden-variables theories. Later, however, he spoke of“the apparently essential conflict between any sharp formulation[of quantum theory] and fundamental relativity. That is to say, wehave an apparent incompatibility, at the deepest level, between thetwo fundamental pillars of contemporary theory…” (Bell1987b and 2004, 172; these are remarks from a meeting held in 1984).Note that this is hedged, and he speaks of “apparentlyessential” conflict only. In the same year, he wrote, “Iam unable to prove, or even formulate clearly, the proposition that asharp formulation of quantum field theory, such as that set out here,must disrespect serious Lorentz invariance. But it seems to me thatthis is probably so” (1984, 7; 1987b and 2004, 180).

A shift in attitude was occasioned by the publication of the GRWdynamical collapse theory (Ghirardi, Rimini, and Weber, 1986). In acommentary on this theory, Bell wrote,

I am particularly struck by the fact that the model is as Lorentzinvariant as it could be in the nonrelativistic version. It takes awaythe ground of my fear that any exact formulation of quantum mechanicsmust conflict with fundamental Lorentz invariance (1987a, 14; 1987band 2004, 209).

In a lecture delivered in Trieste in the last year of his life, Belldiscussed the prospects for a genuinely relativistic version of adynamical collapse theory, and concluded that the difficultiesencountered by Ghirardi, Grassi, and Pearle in producing a genuinelyrelativistic version of the Continuous Spontaneous Localization theory(CSL), a theory that would be “Lorentz invariant, not just forall practical purposes but deeply, in the sense of Einstein,eliminating entirely any privileged reference system from thetheory” (2007, 2931), were “Second-ClassDifficulties,” technical difficulties, and not deep conceptualones. This seems to have been borne out by the construction of thefully relativistic collapse theories already mentioned.

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• Goldstein, S., T. Norsen, D. V. Tausk, D. and N. Zanghì, “Bell’s Theorem,”Scholarpedia entry.
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Acknowledgments

A.S. acknowledges valuable conversations with John Clauser and EdwardFry. W.M. acknowledges valuable correspondence with Travis Norsen andHoward Wiseman, and is grateful to Sebastian Murgueitio Ramírezfor providing digitized copies of the cited issues ofEpistemological Letters. Springer Verlag and Alain Aspecthave kindly given permission to reproduce Figure 9.1 on p. 121 ofAspect’s article, “Bell’s Theorem: the NaïveView of an Experimentalist,” pp. 119–153 inQuantum[Un]speakables, R.A. Bertlmann and A. Zeilinger (eds.),Berlin-Heidelberg-New York: Springer Verlag, 2002; this Figure wasused as Figure 1 in the present article.