Bell’s theorem is a fundamental result in quantum mechanics: it discriminates between quantum mechanics and all theories where probabilities in measurement results arise from the ignorance of pre-existing local properties. We give an extremely simple proof of Bell's inequality; a single figure suffices. This simplicity may be useful in the unending debate over what exactly the Bell inequality means, because the hypotheses underlying the proof become transparent. It is also a useful didactic tool, as the Bell inequality can be explained in a single intuitive lecture.
REFERENCES
1.
A.
Einstein
, B.
Podolsky
, and N.
Rosen
, “Can quantum-mechanical description of physical reality be considered complete?
,” Phys. Rev.
47
, 777
–780
(1935
).2.
A.
Einstein
, “Reply to criticisms
,” in A. Einstein, Philosopher-Scientist
, edited by P. A.
Schilpp
(Library of Living Philosophers
, Evanston
, 1949
), p. 671
.3.
J. S.
Bell
, “On the Einstein Podolsky Rosen Paradox
,” Physics
1
, 195
–200
(1964
);J. S.
Bell
, “On the problem of hidden variables in quantum mechanics
,” Rev. Mod. Phys.
38
, 447
–452
(1966
).4.
J. S.
Bell
, Speakable and Unspeakable in Quantum Mechanics
(Cambridge U.P.
, Cambridge
, 1987
).5.
N. D.
Mermin
, “Hidden variables and the two theorems of John Bell
,” Rev. Mod. Phys.
65
, 803
–815
(1993
).6.
R.
Werner
, “Bell’s telephone
,” in Quantum Information Theory—an Invitation
, Springer Tracts in Modern Physics 173
, 14
–57
(2001
). Also available at <http://arxiv.org/pdf/quant-ph/0101061>.7.
A.
Peres
, “Unperformed experiments have no results
,” Am. J. Phys.
46
, 745
–747
(1978
).8.
N. D.
Mermin
, “Bringing home the atomic world: Quantum, mysteries for anybody
,” Am. J. Phys.
49
, 940
–943
(1981
);N. D.
Mermin
, “Is the moon there when nobody looks? Reality and the quantum theory
,” Phys. Today
38
(4
), 38
–47
(1985
).9.
J.
Preskill
, “Lecture notes for Physics 229: Quantum Information and Computation
” (1998
), <http://www.theory.caltech.edu/people/preskill/ph229>.10.
G.
Auletta
, Foundations and Interpretation of Quantum Mechanics
(World Scientific
, Singapore
, 2000
).11.
P. G.
Kwiat
and L.
Hardy
, “The mystery of the quantum cakes
,” Am. J. Phys.
68
, 33
–36
(2000
).12.
Kurt
Jacobs
and Howard M.
Wiseman
, “An entangled web of crime: Bell's theorem as a short story
,” Am. J. Phys.
73
, 932
–937
(2005
).13.
A. L.
Macdonald
, “Comment on Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes
,” Phys. Rev. Lett.
49
, 1215
(1982
).14.
A.
Peres
, “Existence of ‘Free will’ as a problem of Physics
,” Found. Phys.
16
, 573
–584
(1986
).15.
H. P.
Stapp
, “S-Matrix Interpretation of Quantum Theory
,” Phys. Rev. D
3
, 1303
–1320
(1971
);H. P.
Stapp
, “Are superluminal connections necessary?
,” Nuovo Cimento
40B
, 191
(1977
);H. P.
Stapp
, “Locality and reality
,” Found. Phys.
10
, 767
–795
(1980
);W.
De Baere
, “On Some Consequences of the Breakdown of Counterfactual Definiteness in the Quantum World
,” Fortschr. Phys.
46
, 843
–847
(1998
).16.
Y.
Aharonov
, A.
Botero
, and M.
Scully
, “Locality or non-locality in quantum mechanics: Hidden variables without spooky action-at-a-distance
,” Z. Natureforsh. A
56
, 5
–15
(2001
).17.
A.
Aspect
, P.
Grangier
, and G.
Roger
, “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities
,” Phys. Rev. Lett.
49
, 91
–94
(1982
).18.
P. G.
Kwiat
, E.
Waks
, A. G.
White
, I.
Appelbaum
, and P. H.
Eberhard
, “Ultrabright source of polarization-entangled photons
,” Phys. Rev. A
60
, R773
–R776
(1999
).19.
M.
Giustina
et al, “Bell violation with entangled photons, free of the fair-sampling assumption
,” Nature
497
, 227
–230
(2013
).20.
P. M.
Pearle
, “Hidden-Variable Example Based upon Data Rejection
,” Phys. Rev. D
2
, 1418
–1425
(1970
).21.
J. S.
Bell
, “Atomic-cascade Photons and Quantum-mechanical nonlocality
,” Comments Atom. Mol. Phys.
9
, 121
–126
(1980
); reprinted as Chap. 13 of Ref. 4.22.
G.
Ghirardi
, “On a recent proof of nonlocality without inequalities
,” Found. Phys.
41
, 1309
–1317
(2011
); arXiv:1101.5252.23.
T.
Norsen
, “Bell locality and the nonlocal character of nature
,” Found. Phys. Lett.
19
, 633
–655
(2006
).© 2013 American Association of Physics Teachers.
2013
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