Most physicists agree that the Einstein–Podolsky–Rosen–Bell paradox exemplifies much of the strange behavior of quantum mechanics, but argument persists about what assumptions underlie the paradox. To clarify what the debate is about, we employ a simple and well-known thought experiment involving two correlated photons to help us focus on the logical assumptions needed to construct the EPR and Bell arguments. The view presented in this paper is that the minimal assumptions behind Bell’s inequality are locality and counterfactual definiteness but not scientific realism, determinism, or hidden variables as are often suggested. We further examine the resulting constraints on physical theory with an illustration from the many-worlds interpretation of quantum mechanics—an interpretation that we argue is deterministic, local, and realist but that nonetheless violates the Bell inequality.

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6.
Much has been written on the limitations of real experiments that purport to test Bell’s theorem. I apologetically ignore this discussion to focus on more central issues. See
Philippe
Grangier
, “
Count them all
,”
Nature (London)
409
,
774
775
(
2001
), for a description of the two most popular complaints and how experimenters have approached them.
7.
Whenever a hidden variable specifies the outcome of an individual measurement, it is naturally taken to represent some form of underlying reality, following logic similar to that used by EPR to define their elements of reality. In this sense, hidden-variable theories are usually considered realist.
8.
Hidden-variable theories are usually imagined to be deterministic in the sense that the hidden variables evolve according to deterministic equations and therefore could be used to predict experimental results. This idea may be what EPR had in mind when they talked of a “complete” physical theory. Strictly, however, a hidden-variable theory could be nondeterministic; the hidden variables could evolve randomly (possibly even discontinuously) so that their values at one instant do not specify their values at the next instant. Bell referred to this possibility in
J. S.
Bell
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9.
Many physicists seem to believe that Bell’s theorem rests on the assumptions of locality and realism. This perspective is found in such notable works as those of
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Clauser
and
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In addition, the following papers from Ref. 5 promote the Bell theorem as a test of local realism: Clauser (
1976
), Lamehi-Rachti and Mittig, Aspect et al. (
1981
), Aspect et al. (
1982
), Rarity and Tapster, Ou et al. (
1992
), Weihs et al., and Rowe et al.
10.
Some discussions of Bell’s theorem focus solely on the locality assumption, although some of these authors may have in mind a different definition of locality than what we employ in this paper (see Ref. 16). For example, see
Nathan
Rosen
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Bell’s theorem and quantum mechanics
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62
,
109
110
(
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Tim
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);
Malcolm
Browne
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Far apart, 2 particles respond faster than light
,” New York Times, July 22, C1–C2 (
1997
);
Charles
Seife
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‘Spooky action’ passes a relativistic test
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Science
287
,
1909
1010
(
2000
);
Detlef
Dürr
,
Sheldon
Goldstein
,
Roderich
Tumulka
, and
Nino
Zanghí
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John Bell and Bell’s theorem
,” in
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, 2nd ed. (
Macmillan
,
Detroit, MI
,
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);
Travis
Norsen
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Bell locality and the nonlocal character of nature
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19
,
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David Z.
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300
(
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Bell’s theorem without hidden variables
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Asher
Peres
,
Quantum Theory: Concepts and Methods
(
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,
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,
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Frank J.
Tipler
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Does quantum nonlocality exist? Bell’s theorem and the many-worlds interpretation
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12.
Henry Pierce
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13.
David
Bohm
,
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(
Prentice-Hall
,
Englewood Cliffs, NJ
,
1951
), Sec. 16.
14.
In practice, there is a problem with measuring photon polarizations using polarizing filters. Only one of the two possible polarization states passes through the filter and is detectable; the orthogonal polarization is absorbed by the filter. For that reason, real experiments use polarization detectors that give a definite signal for both polarization states, such as birefringent crystals that direct the two polarizations in different directions. In this paper, we stick to polarizing filters only because they are more familiar to most readers, and we will pretend that we can positively identify both polarization states.
15.
There are several physical processes that produce twin-state photons: Parametric down-conversion in which a single high energy photon is converted into a pair of lower energy correlated photons in a nonlinear crystal, certain transitions of atomic states (SPS cascades) that emit two photons as the atom decays to the ground state, and annihilation of spin-zero particle states into two gamma rays. It is easy to verify that all of these sources produce photons that satisfy our description of the twin state.
16.
Our definition of locality based on pointlike local interactions is not the only definition used in the literature. For a review, see
P. H.
Eberhard
, “
Bell’s theorem and the different concepts of locality
,”
Nuovo Cimento Soc. Ital. Fis., B
46
,
392
418
(
1978
)
Note that Eberhard’s first three “locality properties” all implicitly assume counterfactual definiteness in addition to an absence of nonlocal effects (see especially Eberhard’s discussion of this issue on p.
402
)
Therefore each of these definitions can be used by itself to derive Bell’s inequality. Bell rederived his inequality using a more general approach than local hidden variables in later work [
J. S.
Bell
, “
The theory of local beables
,”
Speakable and Unspeakable in Quantum Mechanics
(
Cambridge U. P.
,
Cambridge
,
1987
)] based solely on a concept of “local causality.” However, Bell defined local causality in terms of single-valued “beables” that implicitly obey counterfactual definiteness. This condition is violated by any theory that includes superpositions (such as orthodox quantum mechanics or many-worlds quantum mechanics), in which there are multiple possibilities for a measurement result.
17.
If a faster-than-light signal is identified in one reference frame, there is guaranteed to be another reference frame in which that signal travels backward in time, thus violating causality.
18.
Bell actually described measurements of spin 1/2 particles, but these measurements are logically equivalent to the photon polarization measurements we use in our example.
19.
Which measurement is identified as the first measurement depends on the reference frame. Here, we assume we are making measurements in the frame in which the two-photon system is described by Eq. (1). In any other reference frame, the spin entanglement is partially or completely transformed into momentum entanglement
See
Robert M.
Gingrich
and
Christoph
Adami
, “
Quantum entanglement of moving bodies
,”
Phys. Rev. Lett.
89
,
270402
1
(
2002
).
[PubMed]
20.
A very similar example of a Bell experiment is given in this eloquent book:
Nick
Herbert
,
Quantum Reality: Beyond the New Physics
(
Anchor Books
,
New York
,
1985
). However, Herbert uses this example only to discuss locality and does not offer an analysis of any other assumptions in the logic.
21.
Our example of Bell’s inequality derives from comparison of measurements at 0° and ±30°. A more general result can be derived for correlations between any arbitrary three angles. Take C(θ1,θ2) to represent the comparison between measurements of the two photons at angles θ1 and θ2, with C=1 identifying a mismatch (one transmission and one absorption) and C=1 identifying a coincidence (both absorbed or both transmitted). We can verify that in general 1+C(θ1,θ3)C(θ1,θ2)+C(θ2,θ3). If C(θ1,θ2)=C(θ2,θ3)=1 (measurements at the three angles are either all absorbed or all transmitted), then C(θ1,θ3)=1 as well, and both sides of the inequality equal 2. If either C(θ1,θ2)=1 or C(θ2,θ3)=1, then the left side of the inequality is at most zero, and the right side of the inequality is at least zero. By taking averages over many measurements, we have 1+C(θ1,θ3)C(θ1,θ2)+C(θ2,θ3). In the special case where C(θ1,θ2)=C(θ2,θ3)=0.5 (that is, 25% mismatch), we get C(θ1,θ3)?0 (that is, greater than 50% coincidence), which agrees with our example. There is a sign difference between this result and Bell’s original inequality (see Ref. 4) only because his result was derived for singlet state fermions, while our result applies to twin state photons.
22.
The case α=30° gives the maximum difference between quantum mechanics and the inequality limit from Bell’s theorem.
23.
In fact, there are locality assumptions throughout the argument. Initially, we considered measurements with θ1=θ2=0 to verify perfect coincidence between the two-photon measurements. Next we rotated filter 2 to angle α to verify (assuming we did not affect the sequence at filter 1) that the θ2=α sequence coincides with the θ1=0 sequence at the 75% level. Alternatively, we rotated filter 1 to θ1=α to verify (assuming we did not affect sequence at filter 2) that the θ1=α sequence coincides with the θ2=0 sequence at the 75% level. Every step in our example involved an implicit locality assumption.
24.
The case against superluminal signaling also rests on the assumption that Bob’s entangled photon cannot be copied. This feature of quantum mechanics, known as the “no-cloning theorem,” was proven in the work of
W. K.
Wootters
and
W. H.
Zurek
, “
A single quantum cannot be cloned
,”
Nature (London)
299
,
802
803
(
1982
).
25.
Bell emphasized that determinism was not a critical assumption when he published a more general proof of his inequality based on assumptions of local distributions in hidden variables. See
J. S.
Bell
, “
Introduction to the hidden-variable question
,” in
Foundations of Quantum Mechanics
, edited by
B.
d’Espagnat
(
Academic
,
New York
,
1971
), pp.
171
181
.
26.
An example of counterfactual reasoning is a statement of the form: “If we had made a certain alternative measurement (rather than the one we did make), we would have obtained such-and-such result.” Counterfactual definiteness implies that a statement such as the former has a definite truth value (it is either true or false).
27.
Stapp claimed to circumvent counterfactual definiteness in the work of
Henry P.
Stapp
, “
Nonlocal character of quantum theory
,”
Am. J. Phys.
65
,
300
304
(
1997
)
However, he was repeatedly challenged on this claim. See, for instance:
N. David
Mermin
, “
Nonlocal character of quantum theory?
Am. J. Phys.
66
,
920
924
(
1998
);
W.
Unruh
, “
Nonlocality, counterfactuals, and quantum mechanics
,”
Phys. Rev. A
59
,
126
130
(
1999
);
Lev
Vaidman
, “
Time-symmetrized counterfactuals in quantum theory
,”
Found. Phys.
29
,
755
765
(
1999
);
A.
Shimony
and
H.
Stein
, “
Comment on ‘Nonlocal character of quantum theory’ by Henry P. Stapp
,”
Am. J. Phys.
69
,
848
853
(
2001
).
28.
Hugh
Everett
 III
, “
Relative state formulation of quantum mechanics
,”
Rev. Mod. Phys.
29
,
454
462
(
1957
).
29.
Bryce S.
DeWitt
and
Neill
Graham
,
The Many-Worlds Interpretation of Quantum Mechanics
(
Princeton U. P.
,
Princeton
,
1973
).
30.
For a thorough review of the debate, see
Maximilian
Schlosshauer
,
Decoherence and the Quantum-to-Classical Transition
(
Springer
,
New York
,
2007
).
31.
David
Deutsch
, “
Quantum theory of probability and decisions
,”
Proc. R. Soc. London, Ser. A
455
,
3129
3137
(
1999
).
32.
David
Wallace
, “
Quantum probability from subjective likelihood: Improving on Deutsch’s proof of the probability rule
,” e-print arXiv:quant-ph/0312157, http://lanl.arxiv.org/abs/quant-ph/0312157.
33.
For a recent critique of this approach, see
Huw
Price
, “
Decisions, decisions, decisions: Can Savage salvage Everettian probability?
,” e-print arXiv:quant-ph/0802.1390, http://lanl.arxiv.org/abs/0802.1390.
34.
W. H.
Zurek
, “
Probabilities from entanglement, Born’s rule from envariance
,”
Phys. Rev. A
71
,
52105
1
(
2005
)
For further discussion of this approach, see
M.
Schlosshauer
and
A.
Fine
, “
On Zurek’s derivation of the Born rule
,”
Found. Phys.
35
,
197
213
(
2005
), and references therein.
35.
See, for example,
Daniel A.
Greenberger
,
Michael A.
Horne
,
Abner
Shimony
, and
Anton
Zeilinger
, “
Bell’s theorem without inequalities
,”
Am. J. Phys.
58
,
1131
1143
(
1990
);
Lucien
Hardy
, “
Nonlocality for 2 particles without inequalities for almost all entangled states
,”
Phys. Rev. Lett.
71
,
1665
1668
(
1993
);
[PubMed]
Daniel M.
Greenberger
,
Michael
Horne
, and
Anton
Zeilinger
, “
A Bell theorem without inequalities for two particles, using efficient detectors
,” eprint arXiv:quant-ph0510201, http://lanl.gov/abs/quant-ph/0510201.
36.
Our explanation of the many-worlds interpretation branching in the text follows similar descriptions by
Don N.
Page
, “
The Einstein–Podolsky–Rosen physical reality is completely described by quantum mechanics
,”
Phys. Lett. A
91
,
57
60
(
1982
),
Michael Clive
Price
, “
The Everett FAQ
,” ⟨www.hedweb.com/manworld.htm⟩,
and
C.
Hewitt-Horsman
and
V.
Vedral
, “
Entanglement without nonlocality
,”
Phys. Rev. A
76
,
062319
1
(
2007
).
37.
An important caveat is worth mentioning. We have argued that the measurement process of the many-worlds interpretation, namely, the branching into different components of a superposition through quantum entanglement, occurs at particular spacetime points and therefore represents a local process. Strictly, this argument does not rule out the possibility of including other nonlocal effects in the theory. If, for example, the many-worlds interpretation framework were applied to a relativistic theory that included spacelike propagators (as found in quantum field theories, for instance), we could argue that the resulting theory contains nonlocal effects, even though the macroscopic branching obeys relativistic causality. The point here is that nonlocality is not required to satisfy Bell’s experiment.
38.
J. S.
Bell
, “
Bertlmann’s socks and the nature of reality
,”
Speakable and Unspeakable in Quantum Mechanics
(
Cambridge U. P.
,
Cambridge
,
1987
).
39.
D.
Bohm
and
B. J.
Hiley
,
The Undivided Universe: An Ontological Interpretation of Quantum Theory
(
Routledge
,
New York
,
1993
).
40.
If there were no extra assumption (like counterfactual definiteness) in the definition of EPR’s elements of reality, then the elements of reality would follow directly from an assumption of locality (and also the experimental fact of the EPR correlations, which are undeniable). These in turn could be used to derive Bell’s inequality. Following this reasoning, some scientists insist that Bell’s inequality rests only on the assumption of locality and that counterfactual definiteness, which is implied in the definition of elements of reality, is inferred rather than assumed. This line of thought neglects to realize that the single-reality assumption is already built into the definition of EPR’s elements. Multireality interpretations such as many worlds provide a contrasting viewpoint.
41.
In addition to Ref. 35, see
John F.
Clauser
,
Michael A.
Horne
,
Abner
Shimony
, and
Richard A.
Holt
, “
Proposed experiment to test local hidden-variable theories
,”
Phys. Rev. Lett.
23
,
880
884
(
1969
)
and
John F.
Clauser
and
Michael A.
Horne
, “
Experimental consequences of objective local theories
,”
Phys. Rev. D
10
,
526
535
(
1974
).
42.
It is possible that a nonlocal and/or counterfactually indefinite theory might coincidently satisfy Bell’s inequality (just because it is nonlocal and/or counterfactually indefinite doesn’t mean it must violate the inequality) while violating one of the other constraints. In this case, a theory that passed the Bell test might still be ruled out. However, I know of no particular theory in this category.
43.
A. J.
Leggett
, “
Nonlocal hidden-variable theories and quantum mechanics: An incompatibility theorem
,”
Found. Phys.
33
,
1469
1493
(
2003
).
44.
Simon
Gröblacher
,
Tomasz
Paterek
,
Rainer
Kaltenbaek
,
Caslav
Brukner
,
Marek
Zukowski
,
Markus
Aspelmeyer
, and
Anton
Zeilinger
, “
An experimental test of nonlocal realism
,”
Nature (London)
446
,
871
875
(
2007
).
45.
See
Max
Tegmark
, “
The interpretation of quantum mechanics: Many worlds or many words?
,”
Fortschr. Phys.
46
,
855
862
(
1998
). Unfortunately, this clever approach offers proof of the many-worlds-style reality only to the person that performs the experiment (and even then only to the version of that person who survives the suicide attempt). It offers no proof for the rest of the community or consolation to the family members who lost their beloved experimenter. Moreover, if the many-worlds interpretation were false, it offers no proof at all.
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