## 1. Introduction



In 1964 John S. Bell, a native of Northern Ireland and a staff member
of CERN (European Organisation for Nuclear Research) whose primary
research concerned theoretical high energy physics, published a paper
(Bell 1964) in the short-lived journal *Physics*, which
eventually transformed the study of the foundation of quantum
mechanics.



The paper showed, under conditions that were relaxed in later work by
Bell (1971, 1976) himself and by his followers (Clauser, Horne,
Shimony, and Holt 1969, Clauser and Horne 1974, Aspect 1983, Mermin
1986), that, on the assumption of certain auxiliary conditions, no
physical theory that satisfies a certain locality condition, which may
be called *Bell locality*, can fully reproduce the quantum
probabilities for outcomes of experiments. Since that time, variants
on the theorem, with family resemblances, have been formulated.
"Bell's Theorem" is the collective name for the
entire family.



The theorem has roots in Bell's investigations into the status
of the hidden-variables program, and in earlier work concerning
quantum entanglement.



Bell presented several formulations of the theorem over the years
(Bell 1964, 1971, 1976, 1990), and variants of it have been presented
by others. The original derivation (1964) relied on a set-up involving
perfect anticorrelation of the results of spin experiments on pairs of
spin-1/2 particles prepared in the singlet state. Under this
condition, the Bell locality condition entails that outcomes of
experiments are predetermined by the complete specification of state,
a condition which we will call *outcome determinism* (OD).
Clauser, Horne, Shimony and Holt (1969) derived an inequality, the
CHSH inequality, that does not require this assumption. Though in
their proof they employed the condition OD, this condition is
unnecessary for the derivation of the inequality, as shown by Bell
(1971), who provided a proof of the CHSH inequality that relies on
neither the assumption of perfect anticorrelation nor an assumption of
outcome determinism.



One line of investigation in the prehistory of Bell's Theorem is
Bell's examination of the hidden-variables program. This program
involves supplementation of the quantum mechanical state of a system
by further "elements of reality", or "hidden
variables", the incompleteness of the quantum state being the
explanation for the statistical character of quantum mechanical
predictions concerning the system. A pioneering version of a hidden
variables theory was proposed by Louis de Broglie in 1926-7 (de
Broglie 1927, 1928), and revived by David Bohm in 1952 (Bohm 1952; see
also the entry on
 Bohmian mechanics).



In a paper (Bell 1966) that was written before the one in which
Bell's theorem first appeared, but, due to an editorial mishap,
was published later, Bell raises the question of the viability of a
hidden-variables theory that reproduces the statistical predictions of
quantum mechanics via averaging over better defined states that
uniquely determine the result of any experiment that could be
performed. In this paper he examines several theorems that had been
presented as no-go theorems for theories of this sort, and supplements
them with one of his own, a theorem that was independently formulated
by Specker (1960), and published by Kochen and Specker (1967), and has
come to be known as the *Kochen-Specker Theorem* or
*Bell-Kochen Specker Theorem* (see
 entry on the Kochen-Specker Theorem
 for more details). In each case Bell argues that the proof contains
premises that are physically unwarranted.



The Bell-Kochen-Specker theorem is a corollary of Gleason's
theorem (Gleason 1957), though Bell and Kochen-Specker obtain it
directly, and not via Gleason's theorem, whose proof is
considerably more intricate. The question addressed by Gleason has to
do with assignments of probabilities to closed subspaces of a Hilbert
space (or, equivalently, to projection operators onto such subspaces),
such that the probabilities assigned to orthogonal projections are
additive. Gleason proved that, in a Hilbert space of dimension 3 or
greater, any such assignment of probabilities can be represented by a
density operator. The BKS theorem deals with the special case in which
the assignments are confined to the values 1 or 0.



The assumption that a definite value (1 or 0) is to be assigned to
each projector on the system's Hilbert space, with the condition
that values assigned to commuting projectors be additive, is a
weakening of the assumption of the von Neumann no-go theorem (von
Neumann 1932), which assumes that the quantum mechanical additivity of
expectation values of all observables, whether represented by
commuting operators or not, extends to the hypothetical
dispersion-free states (see section 2 of entry on
 the Kochen-Specker Theorem).
 Despite the *prima facie* plausibility of this assumption,
Bell regards it, too, as physically unmotivated, and therefore, unlike
Kochen and Specker, does not regard the Bell-Kochen-Specker theorem as
a no-go theorem for hidden-variables theories. The reason for this is
that the assumption embodies a condition that was later to be known as
 *noncontextuality*.[1]
 In a Hilbert space of dimension greater than two, any projection
operator will be a member of more than one complete set of commuting
projections. For each of these complete sets, there will be an
experiment whose outcomes correspond to the projections in the set.
The assumption of noncontextuality amounts to the assumption that a
value can be assigned to a projection operator that is independent of
which of these experiments is to be performed, or as Bell puts it,
that "measurement of an observable must yield the same value
independently of what other measurements may be made
simultaneously." The assumption need not hold; "[t]he
result of an observation may reasonably depend not only on the state
of the system (including hidden variables) but also on the complete
disposition of the apparatus" (Bell 1966, 451; 1987b and 2004, 9
).



Noncontextual hidden-variables theories that reproduce the predictions
of quantum mechanics are ruled out by the Bell-Kochen-Specker theorem.
A natural question arises as to the possibility of a contextual
hidden-variables theory on which the unavoidable contextuality is
restricted to *local* dependencies. Is it possible to have a
theory on which the outcome of an experiment performed in some spatial
region \(A\) is determined by the complete state of a system in a way
that does not depend on the disposition of experimental apparatus at a
distance from \(A\)?



Bell's article ends with a brief exposition of the de
Broglie-Bohm theory, noting in particular the feature that "in
this theory an explicit causal mechanism exists whereby the
disposition of one piece of apparatus affects the results obtained
with a distant piece" (Bell 1966, 452; 1987b and 2004, 11). The
article ends with the remark:



> 
> Bohm of course was well aware of these features of his scheme, and has
> given them much attention. However, it must be stressed that, to the
> present writer's knowledge, there is no *proof* that
> *any* hidden variable account of quantum mechanics
> *must* have this extraordinary character. It would therefore be
> interesting, perhaps, to pursue some further 'impossibility
> proofs,' replacing the arbitrary axioms objected to above by
> some 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 been
found." The potential drama of the announcement was spoiled by
the fact that the follow-up paper containing the proof (Bell 1964) had
already been published (see Jammer 1974, 303, for an account of the
circumstances that led to the publication delay).



The fact that Bell's theorem has roots into investigations on
hidden-variables theories has led to a misconception that the theorem
is a no-go theorem for hidden-variables theories *tout court*.
There could be no such theorem, since, as Bell himself repeatedly
emphasized, there is a functioning hidden-variables theory, the de
Broglie-Bohm theory.



Another line of investigation leading to Bell's Theorem was the
investigation of quantum mechanical entangled states, that is, quantum
states of a composite system that cannot be expressed either as
products of quantum states of the individual components, or as
mixtures of product states. That quantum mechanics admits of such
entangled states was discovered by Erwin Schrodinger (1926) in
one of his pioneering papers, but the significance of this discovery
was not emphasized until the paper of Einstein, Podolsky, and Rosen
(1935). They examined correlations between the positions and the
linear momenta of two well separated spinless particles and concluded
that in order to avoid an appeal to nonlocality these correlations
could only be explained by "elements of physical reality"
in each particle -- specifically, both definite position and
definite momentum -- and since this description is richer than
permitted by the uncertainty principle of quantum mechanics their
conclusion is effectively an argument for a hidden variables
interpretation.
 [2]
 See also the entry on the
 Einstein-Podolsky-Rosen paradox.


## 2. Proof of a Theorem of Bell's Type



In the present section the pattern of Bell's 1964 paper will be
followed: formulation of a framework, derivation of an inequality,
demonstration of a discrepancy between certain quantum mechanical
expectation values and this inequality. As already mentioned,
Bell's 1964 derivation assumed an experiment involving perfect
anticorrelation of the results of aligned Stern-Gerlach experiments on
a pair of entangled spin-\(\frac{1}{2}\) particles. Experimental
tests, in which perfect anticorrelation (or correlation) may be
approximated but cannot be assumed to hold exactly, require this
assumption to be relaxed. Papers which took the steps from
Bell'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 in
 Section 6.



This conceptual framework first of all postulates an ensemble of pairs
of systems, the individual systems in each pair being labeled as 1 and
2. Each pair of systems is characterized by a "complete
state" \(\lambda\) which contains the entirety of the properties
of the pair at the moment of generation. No assumption whatsoever is
made 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 a
set whose elements are quantum states supplemented by additional
variables, or something more exotic, perhaps some state space as yet
unthought of.



We make the assumption, which remains tacit in most expositions, that
we have an appropriate choice of subsets of \(\Lambda\) to be regarded
as the measurable subsets, forming a measurable space to which
probabilistic considerations may be applied. It is assumed that the
mode of generation of the pairs establishes a probability distribution
\(\rho\) that is independent of the adventures of each of the two
systems after they separate. This does not preclude temporal evolution
of the properties of the two systems after separation. What is assumed
is that the state \(\lambda\) prescribes probabilities for subsequent
events (including any temporal evolution), and thereby probabilities
for 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 not
assumed that these parameters capture the complete state of the
experimental apparatus, which might have a range of microstates
corresponding to each experimental setting. It is assumed that the
preparation probability distribution \(\rho\) is independent of the
processes by which a choice of experiments to be performed is made.
This assumption we call the *Measurement Independence
Assumption*.



The result of an experiment with setting \(a\) on system 1 is labeled
by a real parameter \(s\), which can take on values from a discrete
set \(S\_a\) of real numbers in the interval [\(-1, 1\)]. Likewise, the
result of an experiment on 2 is labeled by a parameter \(t\), which
can take on any of a discrete set of real numbers \(T\_b\) in \([-1,
1]\). As suggested by the subscripts, the sets of potential outcomes
may depend on the experimental settings. The restriction of the values
of the outcome labels to lie in the interval \([-1, 1]\) is of no
physical significance, and is a choice made only for convenience.
Indeed, the use of numbers to label outcomes is merely a matter of
convenience. The inequality to be obtained places a condition on the
probabilities involved in it; if desired, the use of numbers as labels
on outcomes may be dispensed with, and the inequalities can expressed
solely in terms of the relevant probabilities, as in the CH
inequality, inequality (25), below. Bell's own version of his
theorem assumed experiments with two possible outcomes, labelled \(\pm
1\). Other variants of the theorem involve larger sets of potential
outcomes.



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 implicit
averaging over possible states of the experimental
 apparatus.[4]
 Bell's own version of his theorem assumed experiments with two
possible outcomes, labelled \(\pm 1\). Other variants of the theorem
involve larger sets of potential outcomes. We can use these
probability functions -- which we will call *response
probabilities* -- to define marginal probabilities:



\[\begin{align} \tag{1a} p^1\_{a,b}(s \mid \lambda) &\equiv
\Sigma\_t p\_{a,b}(s,t \mid \lambda), \\ \tag{1b}
p^2\_{a,b}(t\mid\lambda) &\equiv \Sigma\_s p\_{a,b}(s,t\mid\lambda).
\end{align}\]


Here, and in what follows, it is to be understood that the sums be
taken over all \(s \in S\_a\) and \(t \in T\_b\). Define
\(A\_{\lambda}(a, b)\), \(B\_{\lambda}(a, b)\) as the expectation
values, for complete state \(\lambda\), of the outcomes of experiments
on system 1 and system 2, respectively, when the settings are \(a,
b\).

 
\[\begin{align}
\tag{2a} A\_{\lambda}(a,b) &\equiv \Sigma\_s s \: p^1\_{a,b}(s\mid\lambda), \\
 \tag{2b} B\_{\lambda}(a,b) &\equiv \Sigma\_t t \: p^2\_{a,b}(t\mid\lambda). 
\end{align}\]


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

 
\[\tag{3} E\_{\lambda}(a, b) \equiv \Sigma\_{s,t} s \: t \: p\_{a,b}(s,t\mid\lambda). \]


Bell-type inequalities follow from a condition, inspired by
considerations of locality and causality, which has been called
*Factorizability*, or *Bell 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 later
expositions of Bell's theorem (Bell 1976, 1990). Bell refers to
the factorizability condition (F) as the condition that the
correlations be *locally explicable* (Bell 1981, C2-55;
1987b and 2004, 152; see also 1990, 109; 2004, 243). This has two
components: that the correlations be explained, and not taken as
primitive, and that the explanation be local. This will be discussed
further in section 3.1. It should be noted that Bell regarded this
condition "not as the *formulation* of 'local
causality', 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 complete
state uniquely determines the outcome of any experiment, and quantum
uncertainty relations reflect incompleteness of the usual
specification of state. For such a theory, the response probabilities
\(p\_{ab}(s,t|\lambda)\) take on the extremal values 0 or 1. Let us
call this condition OD, for *outcome determinism*. It is also
sometimes referred to, misleadingly, as *realism* (see
discussion in
 section 3.3,
 below).



(OD)
For all \(\lambda\), \(a, b\), and all \(s \in S\_a\) and \(t \in
T\_b , p\_{ab}(s,t\mid\lambda) \in \{0, 1\}\).


Suppes and Zanotti (1976) showed that, for the special case of perfect
correlations between outcomes of the two experiments, in which an
outcome of an experiment on one system makes possible prediction with
probability one of the outcome of an experiment on the other, OD must
be satisfied if the factorizability condition (F) is. This applies to
the case considered by Bell 1964. In Bell 1971 and subsequent
expositions Bell provided a generalization that does not presume
perfect correlation and does not require OD, either as a supposition
or as a consequence of other
 suppositions.[5]



If the factorizability condition (F) is satisfied, then



\[\tag{4} E\_{\lambda} (a, b) = A\_{\lambda}(a) B\_{\lambda}(b). \]


The above definitions are valid for experiments with any number of
discrete outcomes. An important special case is that in which each
experiment has only two distinct outcomes, which we may label by \(\pm
1\). For the case of bivalent experiments, (4) is equivalent to
condition (F).



Now consider the quantities

 

\[\tag{5} S\_{\lambda}(a, a', b, b') = \lvert E\_{\lambda} (a, b) +
E\_{\lambda} (a, b') \rvert + \lvert E\_{\lambda}( a', b) - E\_{\lambda}
(a',b')\rvert. \]


Let *S\(\_{\varrho}\)* denote the corresponding relation between
the expectation values of the \(E\_{\lambda}s\), with respect to the
preparation distribution \(\rho\).

 

\[
\begin{align}
\tag{6}
S\_{\varrho}(a, a', b, b') 
 =\ &\lvert\langle E\_{\lambda}(a, b) \rangle\_{\varrho} + \langle E\_{\lambda}(a, b') \rangle\_{\varrho}\rvert \\
 &+ \lvert\langle E\_{\lambda}(a', b)
\rangle\_{\varrho} - \langle E\_{\lambda}( a', b')\rangle\_{\varrho}\rvert. 
\end{align}
\]


Note that, in taking the expectation values of the quantities
appearing on the right-hand side of this definition with respect to
the same distribution \(\rho\), we are
invoking the Measurement Independence Assumption.



Since the absolute value of the average of any random variable cannot
be greater than the average of its absolute value, is clear that



\[\tag{7} S\_{\varrho}(a, a', b, b') \leq \langle S\_{\lambda}(a, a', b,
b') \rangle\_{\varrho}. \]


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

 
\[\tag{8} S\_{\varrho}(a, a', b, b') \leq 2. \]


The second step consists of showing that there are quantum states and
experimental set-ups that are such that the quantum-mechanical
expectation values violate the inequality (8). This shows that no
theory satisfying the factorizability condition can reproduce the
statistical predictions of quantum mechanics in all situations.
Moreover, the inequality furnishes a bound on how close the
predictions of such a theory can come to reproducing quantum
mechanical predictions. The inequality (8), due to Clauser, Horne,
Shimony, and Holt (1969), is known as the *CHSH
inequality*.



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

 

\[\begin{align}\tag{9} 
&S\_{\lambda}(a, a', b, b') \\
&\ = \lvert A\_{\lambda}(a)(B\_{\lambda}(b) + B\_{\lambda}(b'))\rvert + 
 \lvert A\_{\lambda}(a')(B\_{\lambda}(b) - B\_{\lambda}(b'))\rvert \\ 
&\ \leq \lvert B\_{\lambda}(b) + B\_{\lambda}(b') \rvert + \lvert B\_{\lambda}(b)
- B\_{\lambda}(b')\rvert . \end{align}\]


It is easy to check that

 

\[\tag{10} \lvert B\_{\lambda}(b) + B\_{\lambda}(b') \rvert + \lvert
B\_{\lambda}(b) - B\_{\lambda}(b') \rvert\, = 2 \max( \lvert
B\_{\lambda}(b) \rvert , \lvert B\_{\lambda}(b')\rvert). \]


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

 
\[\tag{11} S\_{\lambda}(a, a', b, b') \leq 2. \]


Since this bound holds for every value of \(\lambda\), it must also
hold for the expectation value of \(S\_{\lambda}\).

 
\[\tag{12} \langle S\_{\lambda}(a, a', b, b')\rangle\_{\varrho} \leq 2. \]


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



The final step of the proof of our Bell-type theorem is to exhibit a
system, a quantum mechanical state, and a set of quantities for which
the statistical predictions violate inequality (8). The example used
by 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,

 

\[\tag{13} \ket{\Psi^-} = \frac{1}{\sqrt{2}} \left(\ket{\mathbf{n}+}\_1
\ket{\mathbf{n}-}\_2 - \ket{\mathbf{n}-}\_1 \ket{\mathbf{n}+}\_2\right),
\]


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

 

\[\tag{14} \begin{align} E\_{\Psi^-}(\mathbf{a}, \mathbf{b}) &=
\langle \Psi^- \mid \sigma^1\_{\mathbf{a}} \otimes
\sigma^2\_{\mathbf{b}} \mid \Psi^- \rangle \\ &= -
\cos(\theta\_{\mathbf{a}\mathbf{b}}), \end{align}\]


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 singlet
state is ubiquitous in the literature as an illustrative example,
polarization-entangled photons have been more significant for
experimental tests of Bell inequalities. Consider a pair of photons 1
and 2 propagating in the \(z\)-direction. Let \(|x\rangle\_j\) and
\(|y\rangle\_j\) represent states in which photon \(j \; (j =1, 2)\) is
linearly polarized in the \(x\)- and \(y\)- directions, respectively.
Consider the following state vector,

 

\[\tag{15} \ket{\Phi} = \frac{1}{\sqrt{2}}\left(\ket{x}\_1 \ket{x}\_2 +
\ket{y}\_1 \ket{y}\_2 \right), \]


which is invariant under rotation of the \(x\) and \(y\) axes in the
plane perpendicular to \(z\). The total quantum state of the pair of
photons 1 and 2 is invariant under the exchange of the two photons, as
required by the fact that photons are integral spin particles. Suppose
now that photons 1 and 2 impinge respectively on the faces of
birefringent crystal polarization analyzers I and II, with the
entrance face of each analyzer perpendicular to \(z\). Each analyzer
has the property of separating light incident upon its face into two
outgoing non-parallel rays, the *ordinary ray* and the
*extraordinary ray*. The transmission axis of the analyzer is a
direction with the property that a photon polarized along it will
emerge in the ordinary ray (with certainty if the crystals are assumed
to be ideal), while a photon polarized in a direction perpendicular to
\(z\) and to the transmission axis will emerge in the extraordinary
ray. See Figure 1:



![Schematic diagram of the experimental set-up, indicating particle pairs emitted at a source, and impinging on analyzers, from which they are deflected into one of two outgoing rays.](image1.jpg)

Figure 1
  

(*reprinted with permission*)




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



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

 

\[\tag{16} \begin{align} E\_{\Phi}(\mathbf{a}, \mathbf{b}) &=
\cos^2 (\theta\_{\mathbf{a} \mathbf{b}}) - \sin^2 (\theta\_{\mathbf{a}
\mathbf{b}}) \\ &= \cos(2\theta\_{\mathbf{a} \mathbf{b}}).
\end{align}\]


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



Often, in popular writings, the case of aligned devices is the only
one mentioned, and the perfect anticorrelation (for
spin-\(\frac{1}{2}\) particles in the singlet state) or correlation
(for photons in state \(\ket{\Phi}\)) of results in this case is
offered as evidence of "spooky action at a distance." In
fact, as Bell (1964, 1966) demonstrated by means of simple toy models,
this behaviour can be reproduced by entirely local means. An important
insight of Bell's was that it is important to consider the
less-than perfect correlations obtained when the device axes are not
aligned. In Bell's toy models, correlations fall off linearly
with the angle between the device axes, whereas the quantum
correlations (14, 16) fall off sinusoidally; the decrease in
correlations away from the case of perfect alignments is less steep
than in the toy models. Bell's theorem shows that this sort of
behaviour is not a peculiarity of his models; no model satisfying the
condition F can reproduce the quantum correlations for all angles.
This can be seen by considering the quantum predictions (14, 16) and
plugging them into the expression for \(S\). For a pair of
spin-\(\frac{1}{2}\) particles in the singlet state, we have,



\[\tag{17} S\_{\Psi^-} = \lvert \cos(\theta\_{\mathbf{a}\mathbf{b}}) +
\cos(\theta\_{\mathbf{a}\mathbf{b}'})\rvert + \lvert
\cos(\theta\_{\mathbf{a}'\boldsymbol{b}}) -
\cos(\theta\_{\mathbf{a}'\mathbf{b}'})\rvert . \]


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

 

\[\tag{18} S\_{\Psi^-}(\phi) = \lvert 2 \cos(\phi)\rvert + \lvert
\cos(\phi) - \cos(3\phi)\rvert . \]


This exceeds the CHSH bound (8) when \(0 \lt |\phi | \lt\)
\(\arccos\left((\sqrt{3} - 1)/2 \right) \approx 1.95\) radians, or
68deg, with maximum violation at \(\phi = \pm {\pi}/{4}\), or
45deg. For these angles, we have

 

\[\tag{19} S\_{\Psi^-}(\pi /4) = 2\sqrt{2} \approx 2.828. \]


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



\[\tag{20} S\_{\Phi}(\phi) = \lvert 2 \cos(2\phi)\rvert + \lvert
\cos(2\phi) - \cos(6\phi) \rvert. \]


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



\[\tag{21} S\_{\Phi}({\pi}/{8}) = 2\sqrt{2} \approx 2.828. \]


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


## 3. The Assumptions of the Proof



The distinctive condition giving rise to the Bell Inequality
assumption is the factorizability condition (F). This condition is
motivated by considerations concerning locality and causality.
Considerations of this sort have been the focus of the discussion of
the implications of Bell's theorem. However, in order to a
derive a conflict between the predictions for theories that satisfy
this assumption, other assumptions--some of which are the sort
usually accepted without question in scientific
experimentation--are needed. The analysis of Bell's theorem
has provoked careful scrutiny of the reasoning required to reach the
conclusion that F is to be rejected. As a result, some assumptions
that in another context would have been left implicit have been made
explicit, and each has been challenged by some authors. In this
section we outline a set of assumptions sufficient to ensure
satisfaction of Bell inequalities, which, therefore, constitute a set
of assumptions that cannot all be satisfied by any theory that yields,
in agreement with experiment, violations of Bell inequalities. There
are other paths to Bell inequalities; see, in particular, Wiseman and
Cavalcanti (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's
theorem first appeared is a follow-up to Bell (1966), which explores
the prospects for hidden-variables theories in which the outcome of an
experiment performed is predetermined by the complete state of the
system. The introduction to Bell (1964) begins with mention of
theories with additional variables that are to restore causality and
locality, and says that "In this note that idea will be
formulated mathematically and shown to be incompatible with the
statistical predictions of quantum mechanics." The locality
assumption is glossed as the requirement "that the result of a
measurement on one system be unaffected by operations on a distant
system with which it has interacted in the past." Applied to the
case at hand, of Stern-Gerlach experiments performed on an entangled
pair of spin-1/2 particles, this is "the hypothesis ... that if
the two measurements are made at places remote from one another the
orientation of one magnet does not influence the result obtained with
the other." Bell follows this with,



> 
> Since we can predict in advance the result of measuring any chosen
> component of \(\boldsymbol{\sigma}\_2\), by previously measuring the
> same component of \(\boldsymbol{\sigma}\_1\), it follows that the
> result of any such measurement must actually be predetermined (Bell
> 1964, 195; 1987b and 2004, 15).
> 



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



In Bell (1976) and (1990), Bell derives the factorizability condition
from a condition he calls the *Principle of Local Causality*
(see Norsen 2011 for discussion). In (1990) he begins his analysis
with a rehearsal of the reason that relativity should be taken to
prohibit superluminal causation. On the usual notion of causation, the
cause-effect relation is taken to be temporally asymmetric, with
causes temporally preceding their effects. In a relativistic
spacetime, events at spacelike separation are taken to have no
temporal order. Any system of coordinates will assign time coordinates
to each of any pair of events, but, if the events are spacelike
separated, the time coordinates assigned to a pair of events at
spacelike separation will differ in their ordering, depending on which
reference frame is being employed. If we take all of these reference
frames to be physically on a par, it must be concluded that there is
no temporal order between the events, as relations that are not
relativistically invariant have no physical significance.



On the basis of these considerations, that is, Lorentz invariance and
the assumption that causes temporally precede their effects, Bell
introduces what he calls the *principle of local
causality*.



(PLC-1)
The direct causes (and effects) of events are near by, and even
the indirect causes (and effects) are no further away than permitted
by the velocity of light.


This, says Bell, is "not yet sufficiently sharp and clean for
mathematics." For this reason, he introduces what he presents as
a sharpened version of the principle (refer to Figure 2).



(PLC-2)
A theory will be said to be locally causal if the probabilities
attached to values of local beables in a space-time region 1 are
unaltered by specification of values of local beables in a space-like
separated region 2, when what happens in the backward light cone of 1
is already sufficiently specified, for example by a full specification
of local beables in a spacetime region 3.


![Spacetime diagram depicting two regions, labelled 1 and 2, at spacelike separation from each other, and a third region, labelled 3, which is such that any timelike path that goes from the intersection of the past light cones of 1 and 2 to region 1 must go through region 3.](image2.jpg)

Figure 2




In Bell's terminology, a *beable* is any element of a
physical theory that is taken to correspond to something physically
real, and *local beables* pertaining to a space-time region are
those 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 for
mathematics that one is likely to throw the baby out with the
bathwater" (1990, 106; 2004, 239). The relation between them is
not further discussed in that article, but remarks in other papers
shed light on the transition between them.



In (1976), Bell motivates the formulation of the local causality
condition 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 mean
> that the two sets of events should be uncorrelated, for they could
> have common causes in the overlap of their backward light cones (Bell
> 1976, 14; 1985a, 88; 1987b and 2004, 54).
> 



He then precedes to formulate the condition of local causality, which
is that a full specification of beables in the overlap of the backward
light cones of the spacetime regions 1 and 2 should screen off
correlations between them. Implicit in this is that correlations
between two variables be susceptible to causal explanation, either via
a causal connection between the variables, or via a common cause. This
assumption was stated explicitly in a later article (Bell 1981), in
which he says that "the scientific attitude is that correlations
cry out for explanation" (Bell 1981, C2-55; 1987b and
2004, 152).



The assumption that correlations between two variables that are not in
a cause-effect relation to each other can be explained by some common
cause was named the *principle of the common cause* by
Reichenbach (1956, SS 19), and for this reason is often referred
to as *Reichenbach's Common Cause Principle*, though
Reichenbach made no pretense of originating the principle, and
regarded it as a codification of a mode of inference common in both
science and everyday life. See the entry on
 Reichenbach's common cause principle
 for more details. A variable \(C\) is a Reichenbachian common cause
of a correlation between two variables \(A\) and \(B\) if the
variables \(A\) and \(B\) are uncorrelated, conditional on
specification of the value of \(C\). Reichenbach's Common Cause
Principle says that, if two correlated variables are not in a
cause-effect relation with each other, there is a Reichenbachian
common cause of their correlations. The condition we have called PLC-1
does not, by itself, entail PLC-2, but it does follow from the
conjunction of PLC-1 and Reichenbach's Common Cause
Principle.



PLC-1 does not, by itself, entail that correlations be causally
explicable, and, indeed, does not commit to any sort of causal
relations in the world; it merely says that whatever causal relations
there are respect relativistic locality. For this reason, we will
refer to it as a *causal locality* principle. This is in
contradistinction to PLC-2, which requires there to be causal
relations of some sort wherever correlations are found. What Bell
calls the Principle of Local Causality, PLC-2, can be thought of as a
conjunction of (1) a causal locality condition along the lines of
PLC-1, restricting causes of an event to that event's past light
cone, and (2) Reichenbach's Common Cause Principle, which
requires that correlations be causally explicable. The former
condition can, itself, be regarded as following from relativistic
invariance and the principle that the cause of an event lie in its
temporal past. Bell's Principle of Local Causality thus follows
from the conjunction of three assumptions, all of which, in various
writings, were explicitly formulated by Bell:


* (Temporal Asymmetry of Causality) Any cause of an event lies in
its temporal past, and not in its temporal future.
* (Lorentz Invariance) The relation of temporal precedence is
invariant 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 the
particular set-up of Bell-type experiments, of Bell's Principle
of Local Causality. As we have seen, it can be thought of as the
conjunction of the condition of causal locality, and the common cause
principle. In this section we apply those conditions to the set-up of
Bell-type experiments.



On the assumption that the experimental settings can be treated as
free variables, whose values are determined exogenously, if the choice
of setting on one wing is made at spacelike separation from the
experiment on the other, a dependence of the probability of the
outcome of one experiment on the setting of the other would seem
straightforwardly 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 \(t \in T\_b\),
 
\[\begin{align}
\tag{22a} p^1\_{a,b}(s\mid \lambda) &= p^1\_{a,b'}(s\mid \lambda); \\
 \tag{22b} p^2\_{a,b}(t\mid \lambda) &= p^2\_{a',b}(t\mid \lambda). 
\end{align}\]


This is the condition that has come to be known as *parameter
independence*, following Shimony (1986, 1990).



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



(OI)
For each pair of parameter settings \(a, b\), and all \(\lambda\),
and all \(s\) in \(S\_a\) and \(t\) in \(T\_b\),
 
\[\tag{23} p\_{a,b}(s,t\mid \lambda) = p^1\_{a,b}(s\mid \lambda)\; p^2\_{a,b}(t\mid \lambda). \]


This is the condition that has come to be known as *outcome
independence*, following Shimony (1986, 1990). As demonstrated by
Jarrett (1983, 1984), the factorizability condition (F) is the
conjunction of parameter independence and outcome independence. The
two conditions bear different relations to the locality and causality
conditions discussed in the previous subsection. PI is a consequence
of the causal locality condition PLC-1 alone, whereas OI requires in
addition the assumption of the common cause principle.



The parsing of the Bell locality condition as a conjunction of PI and
OI is due to Clauser and Horne (1974), though they did not propose
distinct names for the conjuncts. Jarrett (1983, 1984) referred to the
conditions as *locality* and *completeness*. Jarrett
(1983, 1984, 1989) argued that a violation of PI would inevitably
permit superluminal signalling. The conclusion requires an additional
assumption, that the state of the system be
 controllable.[7]
 It would not hold for a theory on which there are principled
limitations on the ability of would-be signallers to control the
states of the systems they are dealing with; an example of a theory of
this sort is the de Broglie-Bohm theory, which violates PI but does
not permit superluminal signalling. Nonetheless, in some of the
literature PI has been treated as equivalent to no-signalling. For
some (see *e.g.* Ballentine and Jarrett 1987, fn. 6; Jarrett
1989, 70; Shimony 1993, 139), this stems from the thought that any
limitations on control that might prevent a violation of PI from being
exploited for signalling would involve only practical limitations
irrelevant to foundational concerns, which should have to do with what
is possible in principle.


### 3.2 Supplementary assumptions


#### 3.2.1 Unique experimental outcomes



Though it might seem that this goes without saying, the entire
analysis is predicated on the assumption that, of the potential
outcomes of a given experiment, one and only one occurs, and hence
that it makes sense to speak of *the* outcome of an experiment.
The reason that this assumption is worth mentioning is that there is a
family of approaches to the interpretation of quantum mechanics,
namely, Everettian, or "many-worlds" approaches, and some
variants of the relational approach, according to which all potential
outcomes actually occur, in what are effectively distinct worlds. See
entries on
 many-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 the
complete state \(\lambda\) is sampled from the same probability
distribution \(\rho\), no matter what choice of experiment is made,
and that, for this reason, the subset of experiments corresponding to
any given choice of settings is a fair sample of the distribution of
\(\lambda\). Clauser and Horne (1974, fn. 13) made this assumption
explicit, as did Bell, in his publications subsequent to an exchange
with Shimony, Clauser, and Horne (1976).



The assumption that experimental settings may treated as statistically
independent of the variable \(\lambda\) has variously been called the
*Free Will assumption*, the *Freedom of Choice
Assumption*, the *No-conspiracies Assumption*, and, in some
of the recent literature, the *Statistical Independence
Assumption*. In this article it is called the *Measurement
Independence Assumption*. It will be discussed in more detail in
section 8.1, below.


#### 3.2.3 When experiments end



In experimental tests of Bell locality, care is taken that the
experiments on the two systems, from choice of experimental setting to
registration of results, take place at spacelike separation. It is
assumed that experiments have unique results. The question arises as
to *when* the unique result emerges. It is typically assumed
that the result is definite once a detector has been triggered or the
result is recorded in a computer memory. However, as Kent (2005) has
pointed out, proposals have been made according to which the quantum
state of the apparatus would remain in a superposition of terms
corresponding to distinct outcomes for a greater length of time. One
such proposal is the suggestion that state reduction takes place only
when the uncollapsed state involves a superposition of sufficiently
distinct gravitational fields (Diosi 1987, Penrose 1989, 1996).
Another is Wigner's suggestion that conscious awareness of the
result is required to induce collapse (Wigner 1961). This gives rise
to what Kent calls the *collapse locality loophole*. One can
consider theories--Kent calls the family of such theories
*causal quantum theories*--on which collapses are
localized events, and the probability of a collapse is independent of
events, including other collapses, at spacelike separation from it. A
theory of that sort would differ in its predictions from standard
quantum theory, but a test to discriminate between such a theory and
standard quantum mechanics would require a set-up in which the entire
experiment on one system, from arrival to satisfaction of the collapse
condition, takes place at spacelike separation from the experiment on
the other. If the experiments are taken to end, not when the detector
is triggered, but when the difference between outcomes amounts to
differences in mass configurations large enough to correspond to
significantly distinct gravitational fields, then, as Kent argued,
experiments extant at the time of writing (2005) were subject to this
loophole. The experiment of Salart *et al*. (2008) closed the
loophole for the particular proposals of Penrose and Diosi,
though, as Kent (2018) points out, altering the Penrose-Diosi
threshold by a few orders of magnitude would render them compatible
with the results of this experiment. No experiment to date has
addressed the collapse locality loophole if the collapse condition is
taken to be awareness of the result by a conscious observer. See Kent
(2018) for proposals of ways in which causal quantum theory could be
subjected to more stringent tests.


### 3.3 On "local realism"



It has become commonplace to say that (provided that the supplementary
assumptions are accepted), the class of theories ruled out by
experimental violations of Bell inequalities is the class of *local
realistic theories*, and that the worldview to be abandoned is
*local realism*. The ubiquity of the use of this terminology
tends to obscure the fact that not all who use it use it in the same
sense; further, it is not always clear what is meant when the phrase
is used.



The terminology of "local realistic theories" as the
targets of experimental tests of Bell inequalities was introduced by
Clauser and Shimony (1978), intended as a synonym for what Clauser and
Horne (1974) called "objective local theories." These are
theories that satisfy the factorizability assumption (F). The
terminology was adopted by d'Espagnat (1979) and Mermin (1980).
For Clauser and Shimony realism is "a philosophical view
according to which external reality is assumed to exist and have
definite properties, whether or not they are observed by
someone" (1978, 1883). In a similar vein, d'Espagnat
(1979) says that realism is "the doctrine that regularities in
observed phenomena are caused by some physical reality whose existence
is independent of human observers" (158). Mermin, on the other
hand, takes realism to involve the condition that we have called
*outcome determinism* (OD): "As I shall use the term
here, local realism holds that one can assign a definite value to the
result of an impending measurement of any component of the spin of
either of the two correlated particles, whether or not that
measurement is actually performed" (Mermin 1980, 356). This is
not a commitment of realism in the sense of Clauser and Shimony, who
explicitly consider stochastic local realistic theories.



It is Mermin's sense that seems to be most widely used in the
current literature. In this sense, *local realism*, applied to
the set-up of the Bell experiments, amounts to the conjunction of
Parameter Independence (PI) and outcome determinism (OD). Now, it is
true that, if PI and OD hold, so does factorizability (F), and hence
the Bell inequalities. But the condition OD is stronger than what is
required, as the conjunction of PI and the strictly weaker condition
OI also suffice. Thus, to say that violations of Bell inequalities
rule out *local realistic theories*, with "realism"
identified as outcome determinism, is true but misleading, as it may
suggest that one can retain locality by rejecting
"realism" in the sense of outcome determinism. However, if
one accepts the supplementary assumptions, one is obliged to reject
not merely the conjunction of OD and PI, but the weaker condition of
factorizability, which contains no assumption regarding predetermined
outcomes of experiments.



Further confusion arises if the two senses are conflated. This can
lead to the notion that the condition OD is equivalent to the
metaphysical thesis that physical reality exists and possesses
properties independent of their cognizance by human or other agents.
This would be an error, as stochastic theories, on which the outcome
of an experiment is not uniquely determined by the physical state of
the world prior to the experiment, but is a matter of chance, are
perfectly compatible with the metaphysical thesis. One occasionally
finds 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 to
an avowal of operationalism. If all one asks of a theory is that it
produce the correct probabilities for outcomes of experiments,
eschewing all questions about what sort of physical reality gives rise
to these outcomes, then this undercuts the motivation of the analysis
that leads to Bell's theorem. In this sense of
"realism", it is not an assumption of the theorem but a
motivation for formulating it.



Several authors (see, in particular, Norsen 2007; Maudlin 2014) have
argued that no clear sense of "realism" has been
identified such that realism, in that sense, is a particular
presupposition of the derivation of Bell inequalities (as
distinguished from a presupposition of all physics). These authors
urge rejection of the currently prevalent practice of saying that
"local realist" theories are the targets of experimental
tests of Bell inequalities. Nonetheless, other authors maintain that
there is, indeed, a sense of "realism" on which realism is
an assumption of the derivation of Bell inequalities, though they
differ on what it is that realism involves; see Zukowski and
Brukner (2014), Werner (2014), Zukowski (2017), and Clauser
(2017).


## 4. Early Experimental Tests of Bell's Inequalities



The path toward a conclusive experimental test of Bell inequalities
was 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 the
pairs 1 and 2 be photons produced in an atomic cascade from an initial
atomic state with total angular momentum \(J = 0\) to an intermediate
atomic state with \(J = 1\) to a final atomic state \(J = 0\), as in
an experiment performed with calcium vapor for other purposes by
Kocher and Commins (1967). The proposed test was first performed by
Freedman and Clauser (1972). The result obtained by Freedman and
Clauser was 6.5 standard deviations from the limit allowed by the CHSH
inequality and in good agreement with the quantum mechanical
prediction. This was a difficult experiment, requiring 200 hours of
running time, much longer than in most later tests of Bell's
Inequality, which were able to use lasers for exciting the sources of
photon pairs.



Since then, several dozen experiments have been performed to test
Bell's Inequalities. References will now be given to some of the
most noteworthy of these, along with references to survey articles
which provide information about others. A discussion of more recent
experiments addressed to close two serious loopholes in the early Bell
experiments, the "detection loophole" and the
"communication loophole", will be reserved for
 Section 5.



Holt and Pipkin completed in 1973 (Holt 1973) an experiment very much
like that of Freedman and Clauser, but examining photon pairs produced
in the \(9^1 P\_1 \rightarrow 7^3 S\_1\rightarrow 6^3 P\_0\) cascade in
the zero nuclear-spin isotope of mercury-198 after using electron
bombardment to pump the atoms to the first state in this cascade. The
result of Holt and Pipkin was in fairly good agreement with the CHSH
Inequality, and in disagreement with the quantum mechanical prediction
by nearly 4 standard deviations--contrary to the results of
Freedman and Clauser. Because of the discrepancy between these two
early experiments, Clauser (1976) repeated the Holt-Pipkin experiment,
using the same cascade and excitation method but a different spin-0
isotope of mercury, and his results agreed well with the quantum
mechanical predictions but violated Bell's Inequality. Clauser
also suggested a possible explanation for the anomalous result of
Holt-Pipkin: that the glass of the Pyrex bulb containing the mercury
vapor was under stress and hence was optically active, thereby giving
rise to erroneous determinations of the polarizations of the cascade
photons.



Fry and Thompson (1976) also performed a variant of the Holt-Pipkin
experiment, using a different isotope of mercury and a different
cascade and exciting the atoms by radiation from a narrow-bandwidth
tunable dye laser. Their results also agreed well with the quantum
mechanical predictions and disagreed sharply with Bell's
Inequality. They gathered data in only 80 minutes, as a result of the
high excitation rate achieved by the laser.



Four experiments in the 1970s -- by Kasday-Ullman-Wu,
Faraci-Gutkowski-Notarigo-Pennisi, Wilson-Lowe-Butt, and
Bruno-d'Agostino-Maroni -- used photon pairs produced in
positronium annihilation instead of cascade photons. Of these, all but
that of Faraci *et al*. gave results in good agreement with the
quantum mechanical predictions and in disagreement with Bell's
Inequalities. A discussion of these experiments is given in the review
article by Clauser and Shimony (1978), who regard them as less
convincing than those using cascade photons, because they rely upon
stronger auxiliary assumptions.



The first experiment using polarization analyzers with two exit
channels, thus realizing the theoretical scheme envisaged in
 Section 2,
 was performed in the early 1980s with cascade photons from
laser-excited calcium atoms by Aspect, Grangier, and Roger (1982). The
outcome confirmed the predictions of quantum mechanics over those
satisfying the Bell inequalities more dramatically than any of its
predecessors, with the experimental result deviating from the upper
limit in a Bell's Inequality by 40 standard deviations. An
experiment soon afterwards by Aspect, Dalibard, and Roger (1982),
which aimed at closing the communication loophole, will be discussed
in
 Section 5.
 The historical article by Aspect (1992) reviews these experiments and
also surveys experiments performed by Shih and Alley, by Ou and
Mandel, by Rarity and Tapster, and by others, using photon pairs with
correlated linear momenta produced by down-conversion in non-linear
crystals. Discussion of more recent Bell tests can be found in review
papers (Zeilinger 1999, Genovese 2005, 2016).



Pairs of photons have been the most common physical systems in Bell
tests because they are relatively easy to produce and analyze, but
there have been experiments using other systems. Lamehi-Rachti and
Mittig (1976) measured spin correlations in proton pairs prepared by
low-energy scattering. Their results agreed well with the quantum
mechanical prediction and violated Bell's Inequality, but, as in
the positronium experiments, strong auxiliary assumptions had to be
made.



The outcomes of the Bell tests provide dramatic confirmations of the
*prima facie* entanglement of many quantum states of systems
consisting of 2 or more constituents. Actually, the first confirmation
of entanglement antedated Bell's work, since Bohm and Aharonov
(1957) demonstrated that the results of Wu and Shaknov (1950), Compton
scattering 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 depend
upon independence conditions inspired by relativistic causality. In
the early tests of Bell's Inequalities it was plausible that
these conditions were satisfied just because the 1 and the 2 arms of
the experiment were spatially well separated in the laboratory frame
of reference. This satisfaction, however, is a mere contingency not
guaranteed by any law of physics, and hence it is physically possible
that the setting of the analyzer of 1 and its detection or
non-detection could influence the outcome of analysis and the
detection or non-detection of 2, and conversely. This is the
*communication loophole*, to which the early Bell tests were
susceptible. It is addressed by ensuring that the experiments on the
two systems take place at spacelike separation.



Aspect, Dalibard, and Roger (1982) published the results of an
experiment in which the choices of the orientations of the analyzers
of photons 1 and 2 were performed so rapidly that they were events
with space-like separation. No physical modification was made of the
analyzers themselves. Instead, switches consisting of vials of water
in which standing waves were excited ultrasonically were placed in the
paths of the photons 1 and 2. When the wave is switched off, the
photon propagates in the zeroth order of diffraction to polarization
analyzers respectively oriented at angles \(a\) and \(b\), and when it
is switched on the photons propagate in the first order of diffraction
to polarization analyzers respectively oriented at angles \(a'\) and
\(b'\). The complete choices of orientation require time intervals 6.7
ns and 13.37 ns respectively, much smaller than the 43 ns required for
a signal to travel between the switches in obedience to special
relativity theory. *Prima facie* it is reasonable that the
independence conditions are satisfied, and therefore that the
coincidence counting rates agreeing with the quantum mechanical
predictions constitute a refutation of the Bell inequality and hence
of the family of theories that entail it. There are, however, several
imperfections in the experiment. First of all, the choices of
orientations of the analyzers are not random, but are governed by
quasiperiodic establishment and removal of the standing acoustical
waves in each switch. A scenario can be invented according to which
clever hidden variables of each analyzer can inductively infer the
choice made by the switch controlling the other analyzer and adjust
accordingly its decision to transmit or to block an incident photon.
Also, coincident count technology is employed for detecting joint
transmission of 1 and 2 through their respective analyzers, and this
technology establishes an electronic link which could influence
detection rates. And because of the finite size of the apertures of
the switches there is a spread of the angles of incidence about the
Bragg angles, resulting in a loss of control of the directions of a
non-negligible percentage of the outgoing photons.



The experiment of Tittel, Brendel, Zbinden, and Gisin (1998) did not
directly address the communication loophole but threw some light
indirectly on this question and also provided dramatic evidence
concerning the maintenance of entanglement between particles of a pair
that are well separated. Pairs of photons were generated in Geneva and
transmitted via cables, with very small probability per unit length of
losing the photons, to two analyzing stations in suburbs of Geneva,
located 10.9 kilometers apart on a great circle. The counting rates
agreed well with the predictions of quantum mechanics and violated the
CHSH inequality. No precautions were taken to ensure that the choices
of orientations of the two analyzers were events with space-like
separation. The great distance between the two analyzing stations
makes it difficult to conceive a plausible scenario for a conspiracy
that would violate Bell's independence conditions. Furthermore
-- and this is the feature which seems most to have captured the
imagination of physicists -- this experiment achieved much
greater separation of the analyzers than ever before, thereby
providing a test of a conjecture by Schrodinger (1935) that
entanglement is a property that may dwindle with spatial separation.
More recently, Bell inequality violation was demonstrated even at 144
km distance (Scheidl *et al*., 2010) and, in 2017, from
satellite transmission with a 1200 km distance (Yin *et al*.,
2017).



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

 
\[\tag{24} \ket{\Psi} = \frac{1}{\sqrt{2}} \left(\ket{H}\_1 \ket{V}\_2 - \ket{V}\_1 \ket{H}\_2 \right), \]


where the ket \(\ket{H}\) represents horizontal polarization and
\(\ket{V}\) represents vertical polarization. Each photon pair is
produced from a photon of a laser beam by the down-conversion process
in a nonlinear crystal. The momenta, and therefore the directions, of
the daughter photons are strictly correlated, which ensures that a
non-negligible proportion of the pairs jointly enter the apertures
(very small) of two optical fibers, as was also achieved in the
experiment of Tittel *et al*.. The two stations to which the
photon pairs are delivered are 400 m apart, a distance which light in
vacuo traverses in \(1.3 \mu\)s. Each photon emerging from an optical
fiber enters a fixed two-channel polarizer (*i.e.*, its exit
channels are the ordinary ray and the extraordinary ray). Upstream
from each polarizer is an electro-optic modulator, which causes a
rotation of the polarization of a traversing photon by an angle
proportional to the voltage applied to the modulator. Each modulator
is controlled by amplification from a very rapid generator, which
randomly causes one of two rotations of the polarization of the
traversing photon. An essential feature of the experimental
arrangement is that the generators applied to photons 1 and 2 are
electronically independent. The rotations of the polarizations of 1
and 2 are effectively the same as randomly and rapidly rotating the
polarizer entered by 1 between two possible orientations \(a\) and
\(a'\) and the polarizer entered by 2 between two possible
orientations \(b\) and \(b'\). The output from each of the two exit
channels of each polarizer goes to a separate detector, and a
"time tag" is attached to each detected photon by means of
an atomic clock. Coincidence counting is done after all the detections
are collected by comparing the time tags and retaining for the
experimental statistics only those pairs whose tags are sufficiently
close to each other to indicate a common origin in a single
down-conversion process. Accidental coincidences will also enter, but
these are calculated to be relatively infrequent. This procedure of
coincidence counting eliminates the electronic connection between the
detector of 1 and the detector of 2 while detection is taking place,
which conceivably could cause an error-generating transfer of
information between the two stations. The total time for all the
electronic and optical processes in the path of each photon, including
the random generator, the electro-optic modulator, and the detector,
is conservatively calculated to be smaller than 100 ns, which is much
less than the \(1.3 \mu\)s required for a light signal between the two
stations.



The experimental result in the experiment of Weihs *et al*. is
\(2.73 \pm 0.02,\) in good agreement with the quantum mechanical
prediction, and it is 30 standard deviations away from the upper limit
of the CHSH inequality inequality (8). Aspect, who designed the first
experimental test of a Bell Inequality with rapidly switched analyzers
(Aspect, Dalibard, Roger 1982) appreciatively summarized the import of
this result:



> 
> I suggest we take the point of view of an external observer, who
> collects the data from the two distant stations at the end of the
> experiment, and compares the two series of results. This is what the
> Innsbruck team has done. Looking at the data a posteriori, they found
> that the correlation immediately changed as soon as one of the
> polarizers was switched, without any delay allowing for signal
> propagation: this reflects quantum non-separability. (Aspect 1999,
> 190)
> 



Even if some small imperfection prevented the experiment of Weihs
*et al*. from completely blocking the detection loophole, these
problems were overcome in subsequent experiments.


### 5.2 The Detection Loophole and its remedy



The CHSH inequality (8) is a relation between expectation values. An
experimental test, therefore, requires empirical estimation of the
probabilities of the outcomes of experiments. This estimation involves
computing a ratio of event-counts: the number of pair-production
events with a certain outcome to the total number of pair-production
events. Typically, in experiments involving photons, most of the pairs
produced fail to enter the analyzers. Furthermore, some photons that
enter the analyzers will fail to be detected; in addition, the
detector will occasionally register a detection even when no photon is
detected (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 the
normalization factor required to infer relative frequencies from
event-counts, as required by a test of the CHSH inequality. CHSH
(1969) proposed the assumption that, if a photon passes through an
analyser, its probability of detection is independent of the
analyser's orientation. Though physically plausible, this is not
a condition required by local causality.



The fact that an assumption of this sort is needed for the analysis of
experiments of this type was made clear by toy models constructed by
Pearle (1970) and Clauser and Horne (1974). In these models, the rates
at which the photon pairs pass through the polarization analyzers with
various orientations are consistent with an inequality of Bell's
type, but the hidden variables provide instructions to the photons and
the apparatus not only regarding passage through the analyzers but
also regarding detection, thereby violating the fair sampling
assumption. Detection or non-detection is selective in the model in
such a way that the detection rates violate the Bell-type inequality
and agree with the quantum mechanical predictions. Other models were
constructed 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 are
*ad hoc* and lack physical plausibility, they constitute
existence proofs that theories satisfying the local causality
condition can be consistent with the quantum mechanical predictions
provided that the detectors are properly selective.



A second strategy involves construction of an experimental set-up in
which the production of each particle-pair may be registered. Clauser
and Shimony (1978) referred to apparatus achieving this as
"event-ready" detectors; some recent literature has
referred to a process of this sort as "heralding."



A third strategy involves employment of an inequality that can be
shown to be violated without knowledge of the absolute value of the
probabilities involved. This eliminates the need for untestable
auxiliary assumptions. An inequality suitable for this purpose was
first derived by Clauser and Horne (1974) (henceforth CH). The set-up
is as before, with the exception that each analyzer will have only one
output channel, and the eventualities to be considered are detection
and non-detection. We want an inequality expressed in terms of
probabilities of detection alone. The same sort of reasoning that
leads to the CHSH inequality yields the *CH Inequality*:




\[\tag{25} \ - 1 \le p\_{a,b}(+, +) + p\_{a,b'}(+, +) + p\_{a',b}(+, +) -
p\_{a',b'}(+, +) - p^1\_a(+) - p^2\_b(+) \le 0. \]



The probabilities appearing in (25) can be estimated by dividing
event-counts registered in a run of an experiment by the total number
of pairs produced. If we assume that the production rate at the source
is independent of the analyzer settings, we can take the normalization
factor to be the same for each term, and hence the magnitude of this
factor need not be known in order to demonstrate a violation of the
upper bound of (25). Another useful observation was made by Eberhard
(1993), who demonstrated that the minimal detection efficiency for a
detection loophole free experiment can be reduced (from 82% to 67%)
for non-maximally entangled states (*i.e.* a bipartite
entangled state with different weight for the two components). This
involves starting with a specified efficiency level, and then choosing
a state and a set of observables that maximize violation of the CH
inequality at that efficiency level.



For the maximally entangled states we have been considering, in the
idealized case of perfect detection efficiency, inequality (25) is
maximally violated by the quantum predictions for the same settings
considered above for violation of the CHSH inequality. However, for
non-ideal experiments, the quantum predictions satisfy the inequality
unless detector efficiency is high, considerably higher than that of
any experiment that had been performed up until the time that CH were
writing. For that reason, CH introduced a new auxiliary assumption,
called the *no-enhancement assumption*: for any value of
\(\lambda\), and any setting of an analyzer, the probability of
detection with the analyzer present is no higher than the probability
of detection with the analyzer removed. Let \(p^1\_{\infty}\) and
\(p^2\_{\infty}\) be the probabilities of detection of particles 1 and
2 when their respective analyzers have been removed. This assumption
gives rise to what may be called the *second CH
inequality*:




\[\tag{26} - 1 \le p\_{a,b}(+, +) + p\_{a,b'}(+, +) + p\_{a',b}(+, +) - p\_{a',b'}(+, +) \le p^1\_{\infty} + p^2\_{\infty}. \]



As CH note, this is violated by the results of the Freedman and
Clauser experiment, and hence that experiment rules out theories
satisfying the factorizability condition (F) and the no-enhancement
assumption, though it does not rule out the toy model constructed by
CH.



Historically, the efforts toward a detection loophole-free experiment
followed two main paths, though a few other possibilities were also
explored. One of these other possibilities involved K or B mesons
(Selleri 1983, Go 2004), where the detection loophole reappears in
another form (Genovese, Novero, and Predazzi 2001). Another involved
solid state systems (Ansmann *et al*. 2009).



One of the main avenues of approach employed entangled ions. The use
of ions looked very promising, since for such experiments detection
efficiency is very high. The experiment of Rowe *et 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 only
were not space-like separated; there was a common measurement on the
two ions. More recently, the distance between ions was increased. For
instance, Matsukevich *et al*. (2008) entangled two ytterbium
ions via interference and joint detection of two emitted photons, with
the distance between the ions set to 1 meter. However, a conclusive
experiment of this sort that eliminated also the communication
loophole would require a separation of kilometers.



The other main avenue of approach, which paved the way to a conclusive
test of Bell inequalities, involved innovations in tests using
photons. First, efficient sources of photon entangled states were
realized by exploiting Parametric Down Conversion, a non-linear
optical phenomenon in which a photon of higher energy converts into
two lower frequency photons inside a non-linear medium in such a way
that energy and momentum are conserved. This allows a high collection
efficiency due to wave vector correlation of the emitted photons.
Next, high efficiency single photon Transition Edge Sensors were
produced. These advances led to detection loophole-free experiments
with photons (Giustina *et al*. 2013, Christensen *et
al*. 2013) and finally to the conclusive tests discussed in the
next section.


### 5.3 Loophole-free tests



In 2015 three papers appeared claiming a conclusive test of Bell
inequalities. The first (Hensen *et al*., 2015) achieved a
violation of the CHSH inequality via an event-ready scheme. This
experiment is based on using electronic spin associated with the
nitrogen-vacancy (NV) defect in two diamond chips located in distant
laboratories. In the experiment, each of these two spins is entangled
with 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. An
appropriate result of the measurement of the photons projects the
spins in the two diamond chips onto a maximally entangled state, on
which a Bell inequality test is realized. The high efficiency in spin
measurement and the distance between the laboratories allows closure
of the detection and communication loophole at the same time. However,
the experiment utilized only a small number, 245, of trials, and thus
the 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 of *Physical
Review Letters* (Giustina *et al*. 2015, Shalm *et
al.* 2015), are based on transmitting two polarization-entangled
photons, produced by Parametric Down Conversion, to two remote
laboratories, where they are measured by high detection efficiency
Transition Edge Sensors. These experiments use states that are not
maximally entangled, but are optimized, in accordance with the
analysis of Eberhard (1993), to produce a maximal violation of the CH
inequality, given the detection efficiency of the experiments. In both
of these experiments a violation of the CH inequality was obtained, at
a high degree of statistical significance. Shalm *et al*.
report a \(p\)-value of \(2.3 \times 10^{-7}\), whereas Giustina
*et al*. report a \(p\)-value of \(3.4 \times 10^{-31}\),
corresponding to an 11.5 standard deviation effect. A very careful
analysis of data (including spacelike separation of detection events),
of statistical significance, and of all possible loopholes leaves
really no space for doubts about their conclusiveness. Besides the
detection and communication loophole, these two experiments address
also the following issues:


* The number of emitted particles must be independent of measurement
settings (production rate loophole). This was excluded by an
absolutely random choice of settings.
* The presence of a coincidence window must not allow in a hidden
variable scheme a situation where the local setting may change the
time at which the local event happens (coincidence loophole). This was
excluded by using a pulsed source.
* An eventual memory of previous measurements must be considered in
the statistical analysis since the data might not be independently and
identically distributed (memory loophole). This is excluded by a
careful analysis of data.



Furthermore, independent random number generators based on laser phase
diffusion 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 the
conditions required for a conclusive test, unequivocally tested Bell
inequalities 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 that
depart in some respect from the conceptual framework presented in
 Section 2.
 Both are less general than the version in
 Section 2,
 because they rely on perfect correlations, which, together with the
factorizability condition (F), entail outcome determinism (OD). Then,
two other variants will be mentioned briefly but not summarized in
detail.



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

 
\[\tag{27} \ket{\Phi} = \frac{1}{\sqrt{3}} \left[ \ket{z,1}\_1 \ket{z,-1}\_2 - \ket{z,0}\_1 \ket{z,0}\_2 + \ket{z,-1}\_1 \ket{z,1}\_2 \right], \]


where \(\ket{z,i}\_1\), with \(i = -1\) or 0 or 1 is the spin state of
particle 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 pairwise
commute. However, the squares of these operators -- \(s\_x^2,
s\_y^2, s\_z^2\) -- do commute, and therefore, in view of the
considerations of
 Section 1,
 any two of them can constitute a context in the measurement of the
third. 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 the
measurement of \(s\_{z}^2\). As noted in
 Section 1
 Bell exhibited the possibility of a *contextual* hidden
variables theory for a quantum system whose Hilbert space has
dimension 3 or greater even though the Bell-Kochen-Specker theorem
showed the impossibility of a *non-contextual* hidden variables
theory for such a system. The strategy of the argument is to use the
entangled state of Eq. (27) to predict the outcome of measuring
\(s\_{z}^2\) for particle 2 (for any choice of \(z)\) by measuring its
counterpart on particle 1. A specific complete state \(\lambda\) would
determine whether \(s\_{z}^2\) of 1, measured together with a context
in 1, is 0 or 1. Agreement with the quantum mechanical prediction of
the entangled state of Eq. (27) implies that \(s\_z^2\) of 2 has the
same value 0 or 1. But if the factorizability condition (F) is
assumed, then the result of measuring \(s\_{z}^2\) on 2 must be
independent of the remote context, that is, independent of the choice
of \(s\_{x}^2\) and \(s\_y^2\) of 1, hence of 2 because of correlation,
for any pair of orthogonal directions \(x\) and \(y\) in the plane
perpendicular to \(z\). It follows that the hypothetical theory which
supplies 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 is
impossible in view of the Bell-Kochen-Specker theorem. The conclusion
is that no theory satisfying the factorizability condition (F) is
consistent with the quantum mechanical predictions of the entangled
state (29).



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

 
\[\begin{align}
\tag{28a} \ket{a,+}\_1 &= \cos\theta \ket{a',+}\_1 + \sin\theta \ket{a',-}\_1 \\
 \tag{28b} \ket{a,-}\_1 &= -\sin\theta \ket{a',+}\_1 + \cos\theta \ket{a',-}\_1; 
\end{align}\]


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

 

\[\begin{align}\tag{29} 
\ket{\Psi} &= {-\cos \theta \ket{a',+}\_1 \ket{b',-}\_2} -{\cos\theta \ket{a',-}\_1 \ket{b',+}\_2}\\
 &\qquad + {\sin\theta \ket{a',+}\_1 \ket{b',+}\_2} 
\end{align}\]


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

 
\[\begin{align}
\tag{30} \langle \Psi \mid a,+\rangle\_1 \ket{b',+}\_2 = 0; \\
 \tag{31} \langle \Psi \mid a',+\rangle\_1 \ket{b,+}\_2 = 0; \\
 \tag{32} \langle \Psi \mid a',-\rangle\_1 \ket{b',-}\_2 = 0; 
\end{align}\]


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



\[\tag{33} \langle \Psi \mid a,+\rangle\_1 \ket{b,+}\_2 \ne 0, \]


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 the
counterfactual proposition that if the spin of a 2 in \(E'\) had been
measured along \(b'\) then with certainty the result would have been
\(-\); and likewise Eq. (31) implies the counterfactual proposition
that if the spin of a 1 in \(E'\) had been measured along \(a'\) then
with certainty the result would have been -. It is in this step
that counterfactual reasoning is used in the argument. Since the
subensemble \(E'\) is non-empty, we have reached a contradiction with
Eq. (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 both
results are -. The incompatibility of a deterministic local
theory with quantum mechanics is thereby demonstrated.



An attempt was made by Stapp (1997) to demonstrate a strengthened
version of Bell's theorem which dispenses with the conceptual
framework outlined above, and to use instead the logic of
counterfactual conditionals. His intricate argument has been the
subject of a criticism by Shimony and Stein (2001, 2003), who are
critical of certain counterfactual conditionals that are asserted by
Stapp by means of a "possible worlds" analysis without a
grounding 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 this
section concern ensembles of pairs of particles. An entirely new
domain of variants is opened by studying ensembles of \(n\)-tuples of
particles with \(n \ge 3\). The prototype of this kind of theorem was
demonstrated 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 an
entangled quantum state was written for four spin-\(\frac{1}{2}\)
particles and the expectation value of the product of certain binary
measurements performed on the individual particles was calculated.
They then showed that the attempt to duplicate this expectation value
subject to the factorizability constraint (F) produces a
contradiction. A similar result was obtained by Mermin for a state of
3 spin-\(\frac{1}{2}\) particles and by GHSZ for a state of 3 photons
entangled with respect to their direction of propagation. Because of
the length limitations, the details will not be summarized here, it is
however worth mentioning that experimental tests were realized (Pan
*et al*. 2000, Genovese 2005) (also in this case with
additional hypotheses).



Finally, it is worth mentioning that, usually, in a Bell inequality
test the expectation values in (8) must be evaluated on different
pairs of entangled photons. This could eventually be overcome by
exploiting weak measurements (Aharonov 1988), i.e. measurements where
the measuring apparatus is weakly coupled to the measured quantum
state: this allows for avoiding the collapse of the wave function,
obtaining partial information on the observable. In the case of Bell
measurement an estimate of S, eq(8), can be obtained in a single
measurement. By exploiting sequential weak measurements (Piacentini
2016), after a few first experimental results involving either weak
measurements on a single arm only (Foletto 2020) or one weak plus one
strong measurement (White 2016, Calder 2020), then a full Bell
inequality test based on weak measurements only, which only partially
decoheres the entangled state, was recently achieved (Virzi et
al. 2023, Other Internet Resources).


## 7. Significance for Quantum Information Theory



The set-up envisaged in the proof of Bell's theorem highlights a
striking prediction of quantum theory, namely, long-distance
entanglement, and experimental tests of the Bell inequalities provide
convincing evidence that it is a feature of reality (recent satellite
experiments having reached a 1200 km distance, Yin 2017 and Yin 2020).
Moreover, Bell's theorem reveals that the entanglement-based
correlations predicted by quantum mechanics are strikingly different
from the sort of locally explicable correlations familiar in a
classical context.



Investigations into entanglement and the ways in which it can be
exploited to perform tasks that would not be feasible with only
classical resources forms a key part of the discipline of quantum
information theory (see Benatti, *et al*. eds. (2010), and the
entries on
 quantum computing
 and
 quantum entanglement and information).
 In drawing attention to the import of entanglement and the ways in
which it is different from anything in classical physics, Bell's
theorem and the experimental work derived from it provided, at least
indirectly, some of the impetus to the development of quantum
information theory.



Bell's theorem has played a direct role in the development of
device independent quantum cryptography. One can exploit quantum
correlations to devise a quantum key distribution protocol that is
provably secure on the assumption that, whatever the underlying
physics is, it does not permit superluminal signalling. The basic idea
is that, if one has Bell inequality-violating correlations at
spacelike separation, any predictability of the results beyond that
afforded by the quantum probabilities could be exploited for
superluminal signalling; the contrapositive of this is that
impossibility of signalling entails "absolute randomness"
-- absolute in the sense of being independent of the details of
the underlying physics beyond the prohibition on superluminal
 signalling.[9] 



A further interesting application is to exploit the phenomenon of
quantum non-locality with a loophole-free Bell test to build a
random-number generator that can produce output that is unpredictable
to 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 account
the possibility of a conspiracy aimed at deceiving the users of a
cryptographic key, and so, in this context, it is essential to
demonstrate security in the presence of such a conspiracy.



A result due to Colbeck and Renner (2011, 2016), building on work of
Branciard *et al*. (2008), shows that this cannot be done if PI
is satisfied. This result has significance both at the operational and
the fundamental level. It can be applied at the fundamental level to
conclude that any theory with sharper probabilities than the quantum
predictions must violate PI. In addition, even if some deterministic
theory (such as the de Broglie-Bohm theory) applies at the fundamental
level, the Colbeck-Renner theorem can be applied at the operational
level, where the probabilities involved may indicate limitations on
accessible information about the physical state. A violation of PI at
the operational level would permit signalling. Thus, the theorem shows
that, as long as the no-signalling condition is satisfied, a would-be
eavesdropper attempting to subvert the privacy of a key distribution
scheme by intercepting the particle pairs and substituting ones that
will yield results that she has some information about, cannot do so
without disrupting the correlations between the particle pairs. See
Leegwater (2016) for a clear exposition of the theorem.


## 8. Philosophical/metaphysical implications



Bell inequalities follow from a number of assumptions that have
intuitive plausibility and which, arguably, are rooted in the sort of
world-view that results from reflection on classical physics and
relativity theory. If one accepts that the experimental evidence gives
us strong reason to believe that Bell inequality-violating
correlations are features of physical reality, then one or more of
these assumptions must be given up. Some of these assumptions are of
the sort that have traditionally been regarded as metaphysical
assumptions. The fact that a conjunction of theses of the sort usually
regarded as metaphysical has consequences that can be subjected to
experimental test has led Shimony to speak of the enterprise of
experimental testing for violations of Bell inequalities as
"experimental metaphysics" (Shimony 1984a, 35; 1993, 115).
As may be expected, the conclusions of experimental metaphysics are
not unambiguous. Some *prima facie* plausible options are
excluded, leaving a number of options open. In this section these are
briefly outlined, with no attempt made to adjudicate between them.



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


### 8.1 Rejecting the supplementary assumptions



As we are considering implications of accepting Bell inequality
violations as a feature of reality, even when the experiments are
performed at spacelike separation, we set aside the collapse locality
loophole. This leaves us with two options.


* *Reject unique outcomes*. This is the option taken by
Everettian, or Many-Worlds theories, and related interpretations of
quantum mechanics. The question arises whether locality can be
maintained on this option. Posing the question of whether Bell's
theorem has implications for locality on Everettian or relative-state
interpretations requires that one first consider how to formulate
locality conditions in such a context, as the conditions formulated in
section 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 Smerlak
and Rovelli (2007) for a discussion in the context of relational
interpretations.
* *Reject the measurement independence assumption.* There are
essentially two ways to do this. One is to suppose a common cause in
the past that determines both experimental settings and experimental
outcomes. In an interview, Bell refer to an account of this sort as
"super-deterministic," (Davies and Brown, 47), and the
approach has come to be called *superdeterminism*. Another is
to reject the usual assumption that causes must temporally precede
their effects. If causal influence from future to past is admitted,
then, even if the settings are regarded as free variables, they might
influence the state of the system at the moment of preparation,
contravening the assumption that the preparation probability
distribution be independent of experimental settings. This approach is
often referred to as *retrocausality*. Both of these approaches
have gained advocates in recent years.


#### 8.1.1 Retrocausality



On a retrocausal approach, experimental settings on each wing of the
experiment retroactively effect the state of the pair of particles at
the source, which in turns has an effect on the outcome on the other
wing. We thus have violations of both Measurement Independence and
Parameter Independence. This avenue was suggested by Costa de
Beauregard (1977), in a comment on the interchange between Bell and
Shimony, Holt, and Clauser. Costa de Beauregard objected that the
discussants were disregarding the possibility of retrocausality, a
possibility that he had advanced earlier (1976). In recent years it
has been advocated by Price and Wharton (Price 1994, 1996; Price and
Wharton 2015). It forms part of the Transactional Interpretation of
quantum mechanics (Cramer 1986, 1988, 2016; Kastner 2013). For more on
retrocausal 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 a
Bell-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 sense
> free variables -- say at the whim of the experimenters -- or
> in any case not determined in the overlap of the backward lightcones.
> Indeed without such freedom I would not know how to formulate
> *any* idea of local causality, even the modest human one (Bell
> 1976, 8; 1985a, 95; 1987b and 2004, 61).
> 



This assumption was not, however, *explicitly* invoked in the
derivation, and the role this assumption is meant to play was not made
sufficiently 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 fanciful
story of a conspiracy involving manufacturers of experimental
apparatus and physicists' assistants. The director of the
conspiracy concocts a set of correlation experiment data, consisting
of a sequence of pairs of experimental settings and results obtained.
The director instructs the manufacturer to preprogram the apparatus to
produce the desired outcomes, and the assistants of the physicists
performing the experiment to orchestrate the apparatus settings to
match those specified by the predetermined list. Clearly, the
conspirators may utilize any set of correlation data for their
nefarious schemes; hence, *any* set of correlation data can be
obtained as the outcomes of this sort of process, without any
violation of any sort of locality condition. Correlations obtained by
this means could include, not only correlations of the sort predicted
by quantum mechanics, but also correlations forbidden by quantum
mechanics, such as correlations violating the Tsirelson bound, or
correlations on which the result at one wing of the experiment is
informative about the setting on the other, giving the illusion of
superluminal signalling.



Shimony, Horne, and Clauser's aim was not to reject Bell's
conclusion, but rather to motivate an explicit invocation of an
assumption that had been implicit in Bell's reasoning. For the
sorts of experiments envisaged in tests of Bell inequalities, Shimony,
Horne, and Clauser consider the assumption of independence of settings
and the state of the particle pairs to be justified, even though
relativistic causality does not mandate this independence.



> 
> After all, the backward light cones of those two acts do eventually
> overlap, and one can imagine one region which controls the decision of
> the two experimenters who chose \(a\) and \(b\). We cannot deny such a
> possibility. But we feel that it is wrong on methodological grounds to
> worry seriously about it if no specific causal linkage is proposed. In
> any scientific experiment in which two or more variables are supposed
> to be randomly selected, one can always conjecture that some factor in
> the overlap of the backwards light cones has controlled the presumably
> random choices. But, we maintain, skepticism of this sort will
> essentially dismiss all results of scientific experimentation. Unless
> we proceed under the assumption that hidden conspiracies of this sort
> do not occur, we have abandoned in advance the whole enterprise of
> discovering the laws of nature by experimentation (Shimony, Horne and
> Clauser 1976, 6; 1985, 101; Shimony 1993b, 168).
> 



Appended to the last quoted sentence is an endnote referencing
Shimony's "Scientific Inference," in which an
approach to scientific methodology is advocated that is guided by the
idea that, although there can be no guarantee, whatever scientific
methodology we adopt, that it will lead us towards the truth, we can
and 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 pervasive
in experimental science. A methodology that countenanced rejection of
a posit of this sort whenever its employment would lead to an
undesired conclusion would amount to an abandonment of the enterprise
of testing hypotheses by experimentation, in contravention of
Peirce's maxim.



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



> 
> A respectable class of theories, including contemporary quantum theory
> as it is practised, have 'free' 'external'
> variables in addition to those internal to and conditioned by the
> theory. These variables are typically external fields or sources. They
> are invoked to represent experimental conditions. They also provide a
> point of leverage for 'free willed experimenters', if
> reference to such hypothetical metaphysical entities is permitted. I
> am inclined to pay particular attention to theories of this kind,
> which seem to me most simply related to our everyday way of looking at
> the world. (1977, 80; 1985b, 104; 1987b and 2004, 101)
> 



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



> 
> For me this means that the values of such variables have implications
> only in their future light cones. They are in no sense a record of,
> and do not give information about, what has gone before. In particular
> they have no implications for the hidden variables n in the overlap
> of the backward light cones ... .(1977, 79; 1985b, 103; 1987b and
> 2004, 100)
> 



As stated, this is too strong; if the dynamics are invertible, it is
not possible that the values of these variables have implications
*only* in the future light-cones of the setting events. What is
needed is something considerably weaker than the condition that the
variables not be determined in the overlap of the backward light cones
of the experiments; what is needed is that they be "*at least
effectively free for the purpose at hand*." Bell argues that
a deterministic randomizer that is extraordinarily sensitive to
initial conditions would suffice to provide the requisite
independence, and that variables of this type may be treated as if
they have implications only for events in their future light
cones.



> 
> Suppose that the instruments are set at the whim, not of experimental
> physicists, but of mechanical random number generators. Indeed it
> seems less impractical to envisage experiments of this kind, with
> space-like separation between the outputs of two such devices, than to
> hope to realize such a situation with human operators. Could the
> outputs of such mechanical devices reasonably be regarded as
> sufficiently free for the purpose at hand? I think so.
> 
> 
> Consider the extreme case of a 'random' generator which is
> in fact perfectly deterministic in nature -- and, for simplicity,
> perfectly isolated. In such a device the complete final state
> perfectly determines the complete initial state -- nothing is
> forgotten. And yet for many purposes, such a device is precisely a
> "forgetting machine". A particular output is the result of
> combining so many factors, of such a lengthy and complicated dynamical
> chain, that it is quite extraordinarily sensitive to minute variations
> of many initial conditions. It is the familiar paradox of classical
> statistical mechanics that such exquisite sensitivity to initial
> conditions is practically equivalent to complete forgetfulness of
> them. To illustrate the point, suppose that the choice between two
> possible outputs, corresponding to *a* and *a'*,
> depended on the oddness or evenness of the digit in the millionth
> decimal place of some input variable. Then fixing *a* or
> *a'* indeed fixes something about the input -- i.e.,
> whether the millionth digit is odd or even. But this peculiar piece of
> information is unlikely to be the vital piece for any distinctively
> different purpose, i.e., it is otherwise rather useless. With a
> physical shuffling machine, we are unable to perform the analysis to
> the point of saying just what peculiar input is remembered in the
> output. But we can quite reasonably assume it is not relevant for
> other purposes. In this sense the output of such a device is indeed a
> sufficiently 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 physical
> randomizers are just wrong -- for the purpose at hand. A theory
> may appear in which such conspiracies inevitably occur, and these
> conspiracies may then seem more digestible than the non-localities of
> other theories. When that theory is announced I will not refuse to
> listen, either on methodological or other grounds. But I will not
> myself try to make such a theory. (1977, 83; 1985b, 106; 1987b and
> 2004, 103)
> 



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



In actual experiments, a variety of randomizing strategies have been
employed that, on the basis of plausible physical assumptions, would
be expected to ensure the requisite independence of experimental
setting and conditions at the source of the pair of particles. The
experiment of Scheidl *et al*. (2010) was the first to employ
random-number generators operating at spacelike separation from the
generation of the particle pairs at the source. Following a suggestion
of Clauser, other tests have employed measurements on photons from
Milky Way stars (Handsteiner *et al*. 2017) and from distant
quasars (Rauch *et al*. 2018). In addition, the "Big Bell
Test" ran experiments utilizing inputs provided by approximately
100,000 volunteers (Abellan *et al*. 2018). In the
experiment of Shalm *et al*. (2015), the measurement decisions
were determined by applying an XOR (exclusive or) operation to three
bits from three independent sources, one of which was a pseudorandom
sequence constructed from digits of \(\pi\) and binary strings derived
from various popular culture sources, including episodes of the movies
*Monty Python and the Holy Grail*, the *Back to the
Future* trilogy, and episodes of *Doctor Who*, *Saved by
the Bell*, and *Star Trek*.



Nonetheless, the project of constructing a theory with purely local
dynamics, on which the observed correlations are due to a failure of
these and other purported randomization procedures to eliminate
correlations of just the right sort needed to produce statistics
closely matching the quantum-mechanical predictions, has gained some
support in recent years. It is a feature of 't Hooft's
Cellular Automaton Interpretation of quantum mechanics ('t Hooft
2016; see also Scardagli *et al*. 2019 ), and has been
advocated by Hossenfelder and Palmer (2020) and Hance, Hossenfelder
and Palmer (2022). These authors rely on the possibility that systems
thought to have dynamics that play the randomizing role of which Bell
speaks might, in fact, have dynamics of a different sort, and what
appears to be extraordinary sensitivity to minute variations in
initial conditions might not be.



Hossenfelder and Palmer (2020) take the charges that
superdeterministic theories are "conspiratorial" or
require "fine-tuning" to mean: the set of initial
conditions that lead to the requisite correlations between parameter
settings and the initial states is small, according to some physically
salient measure. The strategy of their response is to consider the
possibility of a theory with a state space whose possibilities are
more restricted than one might ordinarily think, and argue that, on
such a space, a physically salient measure might be one on which the
set of initial conditions that lead to the requisite correlations is
large. That is, a theory is imagined in which initial conditions
lacking the requisite correlations are excluded as physical
possibilities.



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


#### 8.1.3 Rejecting Local Causality



If the auxiliary assumptions are accepted, then the lesson to be drawn
from the experimental violation of Bell inequalities is that the
condition that Bell called the *Principle of Local Causality*
must be rejected. As we have seen, this condition is a conjunction of
two conditions: a causal locality condition (PLC-1, above), and the
Principle of the Common Cause. The causal locality condition itself
can be regarded as stemming from the assumption that causes temporally
precede their effects and Lorentz invariance of the relation of
temporal precedence. If the relation of temporal precedence is to be
Lorentz invariant, then either it is the trivial relation that holds
between any two spacetime points, or else the past of a spacetime
point is its past light cone (Stein 1991, 2009).



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



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



These considerations yield a taxonomy of options for accepting the
supplementary assumptions while also accepting Bell-inequality
violating correlations.


* *Reject PLC-1*. One option is to reject the assumption PLC-1 of
causal locality, and accept that there are causal relations between
events that are outside of each others' light-cones. There are
two ways to do this.


	+ *Introduce non-relativistic spatiotemporal structure*. One
	option, and perhaps the most straightforward one, is to reject the
	assumption of Lorentz invariant spatiotemporal structure. This is the
	position that is involved in acceptance of the de Broglie-Bohm theory
	or any similar theory. On the de Broglie-Bohm theory, the velocity of
	a particle is a function not only of its own location but also of the
	location of any other particles it is entangled with. This means that
	the dynamics of the theory requires a privileged relation of distant
	simultaneity. The non-relativistic spatiotemporal structure could be
	introduced dynamically, as in the family of models outlined by
	Durr *et al*. (1999).
	+ *Retrocausality*. Another option is to maintain the
	assumption that the past of an event is its past light cone, and
	thereby 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 in
	either temporal direction; thus, every event is at least potentially
	in the causal domain of the other. For this reason, an approach of
	this sort involves a rejection of causal locality, even though it may
	respect the idea that a causal influence must be mediated by a
	continuous process along a timelike worldline.


* *Reject the Principle of Common Cause*. A stochastic theory,
such as a dynamical collapse theory, that reproduces quantum
probabilities for Bell experiments, will involve correlated events at
spacelike separation. It need not, however, involve any events in the
common past of these events that screen off the correlations; these
correlations will be built-in to the laws of the theory yielding
probabilities of events. Whether one is willing to extend talk of
cause-effect relations to refer to the relation between such events is
merely a matter of terminology; it should be noted, however, that
there is nothing of the usual asymmetry between cause and effect in
the relation between these events. To accept this relation as a new
sort of symmetric cause-effect relation removes any reason there is to
think that cause-effect relations between spacelike separated events
are incompatible with relativistic spacetime structure.



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



In all of this it is assumed that the spacetime topology matches that
of Minkowski spacetime. One might also reject this assumption; it is
possible that events thought to be spacelike separated are, in fact,
causally connectable, because of a spacetime topology different from
that usually assumed. On this approach, locality is restored either by
considering wormholes, the so-called ER = EPR (namely, Einstein-Rosen
wormhole is related to EPR paradox) conjecture (Maldacena and Susskind
2013) or more wrapped dimensions (Genovese and Gramegna 2019).


### 8.2 Quantum mechanics and relativity



Does Bell's theorem show that quantum theory is incompatible
with relativity?



The answer, of course, depends on what one takes relativity theory to
require. It can be shown (Eberhard 1978, Ghirardi, Rimini & Weber
1980, Page 1982) that, in the absence of nonlocal interaction terms in
the Hamiltonian, quantum correlations cannot be exploited to send
signals superluminally. There has been a tendency in some of the
literature to take this by itself to indicate compatibility with
relativity. That this is insufficient can be seen from the fact that
there can be theories, such as the de Broglie-Bohm theory, that
require nonrelativistic spacetime structure for the formulation of
their dynamical laws, while not permitting signalling (at least, as
long as the usual distribution postulate for particle positions is
satisfied).



Take a *relativistic spacetime structure* to be a structure
that includes a relation of temporal precedence on which the past of a
spacetime 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 nor
future. One can ask whether a given account of the goings-on in the
world is compatible with a relativistic spacetime structure. If a
theory requires events outside of its lightcone to be partitioned into
those that are past, future, and simultaneous with a point, as does
the de-Broglie Bohm theory, it is not compatible with a relativistic
spacetime structure. Compatibility of a theory with relativistic
spacetime structure, in this sense, is a distinct issue from Lorentz
covariance of the equations expressing the theory's dynamical
laws. One can construct theories on which nonrelativistic spacetime
structure is introduced dynamically, as is done by Durr,
Goldstein, Munch-Berndl and Zanghi (1999), who formulate a
Bohmian theory against a background of Minkowski spacetime by
introducing an auxiliary field that picks out a distinguished
foliation that is then used to formulate the dynamics of the theory
(see Durr *et al*. 2014 for discussion). Berndl,
Durr, Goldstein, and Zanghi (1996) consider the class of
trajectory theories--that is, theories that, like the de
Broglie-Bohm theory, assign definite trajectories to
particles--and prove that such theories could not satisfy the
Born-rule distribution postulate along arbitrary spacelike
hyperplanes. The argument generalizes to any theories, such as modal
interpretations, that assign definite values to variables other than
position (Dickson and Clifton 1998, Arntzenius 1998, Myrvold 2002).
Theories of this sort, therefore, must invoke a distinguished
foliation.



A deterministic theory must satisfy the condition of outcome
independence, and hence if one accepts \((a)\) that a violation of PI,
that is, a situation in which the choice of an experiment on one side
changes the probability of an outcome on the other, is an instance of
causation, and \((b)\) that a cause must temporally precede its
effect, and then it follows that a deterministic theory that satisfies
the supplementary conditions and reproduces the quantum correlations
is incompatible with relativistic spacetime structure. Such a theory
may, however, be Lorentz invariant at the phenomenal level, if (as in
the de Broglie-Bohm theory) the distinguished foliation is
unobservable.



A stochastic theory, such as a dynamical collapse theory, must involve
correlations between spacelike separated events. A relation like that,
however, is symmetric, and does not require that one of the events be
in the past of the other. There is no *prima facie* need for
nonrelativistic spacetime structure. Indeed, it is possible to
formulate a fully relativistic dynamical collapse theory. A
relativistic generalization of the Ghirardi-Rimini-Weber (GRW) theory
was 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 extract
from them a sensible account of a world that includes macroscopic
objects, see Pearle (1997), Bedingham *et al*. (2014), and
Myrvold (2019). For more on the bearing of relativity on various
approaches 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 "peaceful
coexistence" between special relativity and quantum theory. The
meaning of this varied. In its initial formulation (1978),
"peaceful coexistence" had to do with regarding
experimental outcomes as transitions from potentiality to actuality,
something that, Shimony says, requires further investigation to be
understood. In later writings, peaceful coexistence is
"suggested" by the fact that quantum correlations cannot
be exploited for superluminal signalling (Shimony 1983), though still
associated with the notion that quantum mechanics motivates a change
in our conception of an event (Shimony 1984a), and involves the
requirement of a coherent meshing of events as described with respect
to different reference frames (1986). In other works peaceful
coexistence seems to be simply identified with the impossibility of
exploiting quantum correlations for signalling (Shimony 1984b, 1990,
1991). Convinced by Bell (1990) that anthropocentric considerations
such as manipulability have no place in considerations of fundamental
physics, Shimony became dissatisfied with this avenue of approach to
reconciling relativity and quantum theory (Shimony 2009, 489).



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



> 
> In a theory in which parameters are added to quantum mechanics to
> determine the results of individual measurements, without changing the
> statistical predictions, there must be a mechanism whereby the setting
> of one measuring device can influence the reading of another
> instrument, however remote. Moreover, the signal involved must
> propagate instantaneously, so that such a theory could not be Lorentz
> invariant (Bell 1964, 199; 1987b and 2004, 20).
> 



At this point he is claiming incompatibility with relativity only for
deterministic 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, we
have an apparent incompatibility, at the deepest level, between the
two fundamental pillars of contemporary theory..." (Bell
1987b and 2004, 172; these are remarks from a meeting held in 1984).
Note that this is hedged, and he speaks of "apparently
essential" conflict only. In the same year, he wrote, "I
am unable to prove, or even formulate clearly, the proposition that a
sharp formulation of quantum field theory, such as that set out here,
must disrespect serious Lorentz invariance. But it seems to me that
this is probably so" (1984, 7; 1987b and 2004, 180).



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



> 
> I am particularly struck by the fact that the model is as Lorentz
> invariant as it could be in the nonrelativistic version. It takes away
> the ground of my fear that any exact formulation of quantum mechanics
> must conflict with fundamental Lorentz invariance (1987a, 14; 1987b
> and 2004, 209).
> 



In a lecture delivered in Trieste in the last year of his life, Bell
discussed the prospects for a genuinely relativistic version of a
dynamical collapse theory, and concluded that the difficulties
encountered by Ghirardi, Grassi, and Pearle in producing a genuinely
relativistic version of the Continuous Spontaneous Localization theory
(CSL), a theory that would be "Lorentz invariant, not just for
all practical purposes but deeply, in the sense of Einstein,
eliminating entirely any privileged reference system from the
theory" (2007, 2931), were "Second-Class
Difficulties," technical difficulties, and not deep conceptual
ones. This seems to have been borne out by the construction of the
fully relativistic collapse theories already mentioned.