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The EPR paradox, Bell’s inequality, and the question of locality
1.John S. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. 38, 447–452 (1966).http://dx.doi.org/10.1103/RevModPhys.38.447
2.A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev. 47, 777–780 (1935).http://dx.doi.org/10.1103/PhysRev.47.777
3.Simple examples of such correlated systems might be two particles produced from the decay of a parent at rest so that their momenta are opposite or two decay products from a spin singlet parent so that their spins are opposite.
4.John S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics 1, 195–200 (1965).
4.Reprinted in John S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge U. P., Cambridge, 1987).
5.Stuart J. Freedman and John F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28, 938–941 (1972);http://dx.doi.org/10.1103/PhysRevLett.28.938
5.G. Faraci, S. Gutowski, S. Notarrigo, and A. R. Pennisi, “An experimental test of the EPR paradox,” Lett. Nuovo Cimento 9, 607–611 (1974);http://dx.doi.org/10.1007/BF02763124
5.L. R. Kasday, J. D. Ullman, and C. S. Wu, “Angular correlation of Compton-scattered annihilation photons and hidden variables,” Nuovo Cimento 25, 633–661 (1975);http://dx.doi.org/10.1007/BF02724742
5.John F. Clauser, “Experimental investigation of a polarization correlation anomaly,” Phys. Rev. Lett. 36, 1223–1226 (1976);http://dx.doi.org/10.1103/PhysRevLett.36.1223
5.Edward S. Fry and Randall C. Thompson, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 37, 465–468 (1976);http://dx.doi.org/10.1103/PhysRevLett.37.465
5.M. Bruno, M. D’Agostino, and C. Maroni, “Measurement of linear polarization of positron annihilation photons,” Nuovo Cimento 40, 143–152 (1977);http://dx.doi.org/10.1007/BF02739186
5.M. Lamehi-Rachti and W. Mittig, “Quantum mechanics and hidden variables: A test of Bell’s inequality by the measurement of the spin correlation in low-energy proton-proton scattering,” Phys. Rev. D 14, 2543–2555 (1976);http://dx.doi.org/10.1103/PhysRevD.14.2543
5.Alain Aspect, Philippe Grangier, and Gérard Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47, 460–463 (1981);http://dx.doi.org/10.1103/PhysRevLett.47.460
5.Alain Aspect, Philippe Grangier, and Gérard Roger, “Experimental realization of Einstein–Podolsky–Rosen–Bohm Gedankenexperiment: A new violation of Bell’s inequalities,” Phys. Rev. Lett. 49, 91–94 (1982);http://dx.doi.org/10.1103/PhysRevLett.49.91
5.Alain Aspect, Jean Dalibard, and Gérard Roger, “Experimental test of Bell’s inequalities using time-varying analyzers,” Phys. Rev. Lett. 49, 1804–1807 (1982);http://dx.doi.org/10.1103/PhysRevLett.49.1804
5.Y. H. Shih and C. O. Alley, “New type of Einstein–Podolsky–Rosen–Bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988);http://dx.doi.org/10.1103/PhysRevLett.61.2921
5.Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988);http://dx.doi.org/10.1103/PhysRevLett.61.50
5.J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495–2498 (1990);http://dx.doi.org/10.1103/PhysRevLett.64.2495
5.Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein–Podolsky–Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992);http://dx.doi.org/10.1103/PhysRevLett.68.3663
5.P. R. Tapster, J. G. Rarity, and P. C. M. Owens, “Violation of Bell’s inequality over 4 km of optical fiber,” Phys. Rev. Lett. 73, 1923–1926 (1994);http://dx.doi.org/10.1103/PhysRevLett.73.1923
5.Paul G. Kwiat, Klaus Mattle, Harald Weinfurter, and Anton Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995);http://dx.doi.org/10.1103/PhysRevLett.75.4337
5.W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Violation of Bell inequalities by photons more than 10 km apart,” Phys. Rev. Lett. 81, 3563–3566 (1998);http://dx.doi.org/10.1103/PhysRevLett.81.3563
5.Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, and Anton Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998);http://dx.doi.org/10.1103/PhysRevLett.81.5039
5.M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, “Experimental violation of a Bell’s inequality with efficient detection,” Nature (London) 409, 791–794 (2001).http://dx.doi.org/10.1038/35057215
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.http://dx.doi.org/10.1038/35057415
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, “Quantum mechanical ideas,” Science 177, 880–881 (1972).http://dx.doi.org/10.1126/science.177.4052.880
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 John F. Clauser and Abner Shimony, “Bell’s theorem: Experimental tests and implications,” Rep. Prog. Phys. 41, 1881–1927 (1978);http://dx.doi.org/10.1088/0034-4885/41/12/002
9.Bernard d’Espagnat, “The quantum theory and reality,” Sci. Am. 241(5), 158–181 (1979);
9.Jon P. Jarrett, “On the physical significance of the locality conditions in the Bell arguments,” Nous 18, 569–589 (1984);http://dx.doi.org/10.2307/2214878
9.M. Ferrero, T. W. Marshall, and E. Santos, “Bell’s theorem: Local realism versus quantum mechanics,” Am. J. Phys. 58, 683–688 (1990);http://dx.doi.org/10.1119/1.16400
9.Raymond Y. Chiao and John C. Garrison, “Realism or locality: Which should we abandon?” Found. Phys. 29, 553–560 (1999);http://dx.doi.org/10.1023/A:1018860125021
9.Tsubasa Ichikawa, Satoshi Tamura, and Izumi Tsutsui, “Testing the EPR locality using B-mesons,” Phys. Lett. A 373, 39–44 (2008);http://dx.doi.org/10.1016/j.physleta.2008.10.013
9.Abner Shimony, “Bell’s theorem,” in The Stanford Encyclopedia of Philosophy, Fall 2008 ed., edited by Edward N. Zalta, ⟨plato.stanford.edu/archives/fall2008/entries/bell-theorem/⟩
9.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, “Bell’s theorem and quantum mechanics,” Am. J. Phys. 62, 109–110 (1994);http://dx.doi.org/10.1119/1.17626
10.Tim Maudlin, Quantum Non-locality and Relativity (Blackwell, Cambridge, 1995);
10.Malcolm Browne, “Far apart, 2 particles respond faster than light,” New York Times, July 22, C1–C2 (1997);
10.Charles Seife, “‘Spooky action’ passes a relativistic test,” Science 287, 1909–1010 (2000);http://dx.doi.org/10.1126/science.287.5460.1909
10.Detlef Dürr, Sheldon Goldstein, Roderich Tumulka, and Nino Zanghí, “John Bell and Bell’s theorem,” in The Encyclopedia of Philosophy, 2nd ed. (Macmillan, Detroit, MI, 2006);
10.Travis Norsen, “Bell locality and the nonlocal character of nature,” Found. Phys. Lett. 19, 633–655 (2006);http://dx.doi.org/10.1007/s10702-006-1055-9
10.David Z. Albert and Rivka Galchen, “Was Einstein wrong?: A quantum threat to special relativity,” Sci. Am. 300(3), 32–39 (2009)
10.In addition, the papers by Tapster et al., Kwiat et al., and Tittel et al. in Ref. 5 discuss Bell’s experiments only as a demonstration of nonlocality.
11.P. H. Eberhard, “Bell’s theorem without hidden variables,” Nuovo Cimento Soc. Ital. Fis., B 38, 75–80 (1977);http://dx.doi.org/10.1007/BF02726212
11.Asher Peres, “Unperformed experiments have no results,” Am. J. Phys. 46, 745–747 (1978);http://dx.doi.org/10.1119/1.11393
11.Nick Herbert and Jack Karush, “Generalization of Bell’s theorem,” Found. Phys. 8, 313–317 (1978);http://dx.doi.org/10.1007/BF00715216
11.Brian Skyrms, “Counterfactual definiteness and local causation,” Philos. Sci. 49, 43–50 (1982);http://dx.doi.org/10.1086/289033
11.Michael Redhead, Incompleteness, Nonlocality, and Realism (Oxford U. P., New York, 1987);
11.Asher Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 1995), Chap. 6;
11.Mark A. Rubin, “Locality in the Everett interpretation of Heisenberg-picture quantum mechanics,” Found. Phys. Lett. 14, 301–322 (2001);http://dx.doi.org/10.1023/A:1012357515678
11.W. De Baere, “On the consequences of retaining the general validity of locality in physical theory,” Found. Phys. 35, 33–56 (2005).http://dx.doi.org/10.1007/s10701-004-1912-y
12.Henry Pierce Stapp, “S-matrix interpretation of quantum theory,” Phys. Rev. D 3, 1303–1320 (1971).http://dx.doi.org/10.1103/PhysRevD.3.1303
13.David Bohm, Quantum Theory and Measurement (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)http://dx.doi.org/10.1007/BF02728628
16.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)
16.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
19.See Robert M. Gingrich and Christoph Adami, “Quantum entanglement of moving bodies,” Phys. Rev. Lett. 89, 270402–1 (2002).http://dx.doi.org/10.1103/PhysRevLett.89.270402
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 to represent the comparison between measurements of the two photons at angles and , with identifying a mismatch (one transmission and one absorption) and identifying a coincidence (both absorbed or both transmitted). We can verify that in general . If (measurements at the three angles are either all absorbed or all transmitted), then as well, and both sides of the inequality equal 2. If either or , 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 . In the special case where (that is, 25% mismatch), we get (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 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 to verify perfect coincidence between the two-photon measurements. Next we rotated filter 2 to angle to verify (assuming we did not affect the sequence at filter 1) that the sequence coincides with the sequence at the 75% level. Alternatively, we rotated filter 1 to to verify (assuming we did not affect sequence at filter 2) that the sequence coincides with the 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).http://dx.doi.org/10.1038/299802a0
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)http://dx.doi.org/10.1119/1.18511
27.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);http://dx.doi.org/10.1119/1.18990
27.W. Unruh, “Nonlocality, counterfactuals, and quantum mechanics,” Phys. Rev. A 59, 126–130 (1999);http://dx.doi.org/10.1103/PhysRevA.59.126
27.Lev Vaidman, “Time-symmetrized counterfactuals in quantum theory,” Found. Phys. 29, 755–765 (1999);http://dx.doi.org/10.1023/A:1018826607583
27.A. Shimony and H. Stein, “Comment on ‘Nonlocal character of quantum theory’ by Henry P. Stapp,” Am. J. Phys. 69, 848–853 (2001).http://dx.doi.org/10.1119/1.1374250
28.Hugh Everett III, “Relative state formulation of quantum mechanics,” Rev. Mod. Phys. 29, 454–462 (1957).http://dx.doi.org/10.1103/RevModPhys.29.454
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).http://dx.doi.org/10.1098/rspa.1999.0443
34.W. H. Zurek, “Probabilities from entanglement, Born’s rule from envariance,” Phys. Rev. A 71, 52105–1 (2005)http://dx.doi.org/10.1103/PhysRevA.71.052105
34.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.http://dx.doi.org/10.1007/s10701-004-1941-6
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);http://dx.doi.org/10.1119/1.16243
35.Lucien Hardy, “Nonlocality for 2 particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993);http://dx.doi.org/10.1103/PhysRevLett.71.1665
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),http://dx.doi.org/10.1016/0375-9601(82)90264-X
36.and C. Hewitt-Horsman and V. Vedral, “Entanglement without nonlocality,” Phys. Rev. A 76, 062319–1 (2007).http://dx.doi.org/10.1103/PhysRevA.76.062319
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)http://dx.doi.org/10.1103/PhysRevLett.23.880
41.and John F. Clauser and Michael A. Horne, “Experimental consequences of objective local theories,” Phys. Rev. D 10, 526–535 (1974).http://dx.doi.org/10.1103/PhysRevD.10.526
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).http://dx.doi.org/10.1023/A:1026096313729
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).http://dx.doi.org/10.1038/nature05677
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.http://dx.doi.org/10.1002/(SICI)1521-3978(199811)46:6/8<855::AID-PROP855>3.0.CO;2-Q
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