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Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics and the world as described by classical mechanics. This theorem is named after John Stewart Bell.

In its simplest form, Bell's theorem states:[1]






There is a way to escape the inference of superluminal speeds and spooky action at a distance. But it involves absolute determinism in the universe, the complete absence of free will. Suppose the world is super-deterministic, with not just inanimate nature running on behind-the-scenes clockwork, but with our behavior, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined, including the 'decision' by the experimenter to carry out one set of measurements rather than another, the difficulty disappears. There is no need for a faster-than-light signal to tell particle A what measurement has been carried out on particle B, because the universe, including particle A, already 'knows' what that measurement, and its outcome, will be.[5]

Example of simple Bell type inequality and its violation in quantum mechanics. Top: assuming any probability distribution among 8 possibilities for values of 3 binary variables ABC, we always get the above inequality. Bottom: example of its violation using quantum Born rule: probability is normalized square of amplitude.


Historical backgroundEdit

In the early 1930s, the philosophical implications of the current interpretations of quantum theory troubled many prominent physicists of the day, including Albert Einstein. In a well-known 1935 paper, Boris Podolsky and co-authors Einstein and Nathan Rosen (collectively "EPR") sought to demonstrate by the EPR paradox that quantum mechanics was incomplete. This provided hope that a more complete (and less troubling) theory might one day be discovered. But that conclusion rested on the seemingly reasonable assumptions of locality and realism (together called "local realism" or "local hidden variables", often interchangeably). In the vernacular of Einstein: locality meant no instantaneous ("spooky") action at a distance; realism meant the moon is there even when not being observed. These assumptions were hotly debated in the physics community, notably between Einstein and Niels Bohr.


In experimental tests following Bell's example, now using quantum entanglement of photons instead of electrons, John Clauser and Stuart Freedman (1972) and Alain Aspect et al. (1981) demonstrated that the predictions of quantum mechanics are correct in this regard, although relying on additional unverifiable assumptions that open loopholes for local realism.

In October 2015, Hensen and co-workers[7] reported that they performed a loophole-free Bell test which might force one to reject at least one of the principles of locality, realism, or freedom-of-choice (the last "could" lead to alternative superdeterministic theories).[8] Two of these logical possibilities, non-locality and non-realism, correspond to well-developed interpretations of quantum mechanics, and have many supporters; this is not the case for the third logical possibility, non-freedom. Conclusive experimental evidence of the violation of Bell's inequality would drastically reduce the class of acceptable deterministic theories but would not falsify absolute determinism, which was described by Bell himself as "not just inanimate nature running on behind-the-scenes clockwork, but with our behaviour, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined". However, Bell himself considered absolute determinism an implausible solution.



These three key concepts – locality, realism, freedom – are highly technical and much debated. In particular, the concept of realism is now somewhat different from what it was in discussions in the 1930s. It is more precisely called counterfactual definiteness; it means that we may think of outcomes of measurements that were not actually performed as being just as much part of reality as those that were made. Locality is short for local relativistic causality. (Currently accepted quantum field theories are local in the terminology of the Lagrangian formalism and axiomatic approach.) Freedom refers to the physical possibility of determining settings on measurement devices independently of the internal state of the physical system being measured.




The probability of the same result being obtained at the two locations depends on the relative angles at which the two spin measurements are made, and is strictly between zero and one for all relative angles other than perfectly parallel or antiparallel alignments (0° or 180°). Since total angular momentum is conserved, and since the total spin is zero in the singlet state, the probability of the same result with parallel (antiparallel) alignment is 0 (1). This last prediction is true classically as well as quantum mechanically.


Measuring the spin of these entangled particles along anti-parallel directions—i.e., along the same axis but in opposite directions, the set of all results is perfectly correlated. On the other hand, if measurements are performed along parallel directions they always yield opposite results, and the set of measurements shows perfect anti-correlation. This is in accord with the above stated probabilities of measuring the same result in these two cases. Finally, measurement at perpendicular directions has a 50% chance of matching, and the total set of measurements is uncorrelated. These basic cases are illustrated in the table below. Columns should be read as examples of pairs of values that could be recorded by Alice and Bob with time increasing going to the right.

Alice, 0° + + +
Bob, 180° + + +
(100% identical)
Parallel Pair 1 Pair 2 Pair 3 Pair 4 Pair n
Alice, 0° + + +
Bob, 0° or 360° + +
(100% opposite)
Orthogonal Pair 1 Pair 2 Pair 3 Pair 4 Pair n
Alice, 0° + +
Bob, 90° or 270° + +
(50% identical, 50% opposite)

With the measurements oriented at intermediate angles between these basic cases, the existence of local hidden variables could agree with/would be consistent with a linear dependence of the correlation in the angle but, according to Bell's inequality (see below), could not agree with the dependence predicted by quantum mechanical theory, namely, that the correlation is the negative cosine of the angle. Experimental results match the curve predicted by quantum mechanics.[1]

Over the years, Bell's theorem has undergone a wide variety of experimental tests. However, various common deficiencies in the testing of the theorem have been identified, including the detection loophole[10] and the communication loophole.[10] Over the years experiments have been gradually improved to better address these loopholes. In 2015, the first experiment to simultaneously address all of the loopholes was performed.[7]



Bell's theorem, derived in his seminal 1964 paper titled On the Einstein Podolsky Rosen paradox,[4] has been called, on the assumption that the theory is correct, "the most profound in science".[13] Perhaps of equal importance is Bell's deliberate effort to encourage and bring legitimacy to work on the completeness issues, which had fallen into disrepute.[14] Later in his life, Bell expressed his hope that such work would "continue to inspire those who suspect that what is proved by the impossibility proofs is lack of imagination."[14]


In two respects Bell's 1964 paper was a step forward compared to the EPR paper: firstly, it considered more hidden variables than merely the element of physical reality in the EPR paper; and Bell's inequality was, in part, experimentally testable thus raising the possibility of testing the local realism hypothesis. Limitations on such tests to date are noted below. Whereas Bell's paper deals only with deterministic hidden variable theories, Bell's theorem was later generalized to stochastic theories[16] as well, and it was also realised[17] that the theorem is not so much about hidden variables, as about the outcomes of measurements that could have been taken instead of the one actually taken. Existence of these variables is called the assumption of realism, or the assumption of counterfactual definiteness.



Local realismEdit

The concept of local realism is formalized to state and prove Bell's theorem and generalizations. A common approach is the following:

  2. The values observed by Alice or Bob are functions of the local detector settings and the hidden parameter only. Thus, there are functions A,B : S2 × Λ → {-1, +1} , where a detector setting is modeled as a location on the unit sphere S2, such that
    • The value observed by Alice with detector setting a is A(a, λ)
    • The value observed by Bob with detector setting b is B(b, λ)

Perfect anti-correlation would require B(c, λ) = −A(c, λ), cS2. Implicit in assumption 1) above, the hidden parameter space Λ has a probability measure μ and the expectation of a random variable X on Λ with respect to μ is written


where for accessibility of notation we assume that the probability measure has a probability density p that therefore is nonnegative and integrates to 1. The hidden parameter is often thought of as being associated with the source but it can just as well also contain components associated with the two measurement devices.

Bell inequalitiesEdit

Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. Assuming local realism, certain constraints must hold on the relationships between the correlations between subsequent measurements of the particles under various possible measurement settings. Let A and B be as above. Define for the present purposes three correlation functions:

  • Let Ce(a, b) denote the experimentally measured correlation defined by

where N++ is the number of measurements yielding "spin up" in the direction of a measured by Alice (first subscript +) and "spin up" in the direction of b measured by Bob. The other occurrences of N are analogously defined.

  • Let Cq(a, b) denote the correlation as predicted by quantum mechanics. This is given by the expression
where A is the antisymmetric spin wave function. This value is calculated to be
  • Let Ch(a, b) denote the correlation as predicted by any hidden variable theory. In the formalization of above, this is
Details on calculation of Cq(a, b)

The two-particle spin space is the tensor product of the two-dimensional spin Hilbert spaces of the individual particles. Each individual space is an irreducible representation space of the rotation group SO(3). The product space decomposes as a direct sum of irreducible representations with definite total spins 0 and 1 of dimensions 1 and 3 respectively. Full details may be found in Clebsch—Gordan decompostion. The total spin zero subspace is spanned by the singlet state in the product space, a vector explicitly given by


with adjoint in this representation


The way single particle operators act on the product space is exemplified below by the example at hand; one defines the tensor product of operators, where the factors are single particle operators, thus if Π, Ω are single particle operators,




etc., where the superscript in parentheses indicates on which Hilbert space in the tensor product space the action is intended and the action is defined by the right hand side. The singlet state has total spin 0 as may be verified by application of the operator of total spin J · J = (J1 + J2) ⋅ (J1 + J2) by a calculation similar to that presented below.

The expectation value of the operator


in the singlet state can be calculated straightforwardly. One has, by definition of the Pauli matrices,


Upon left application of this on |A one obtains


Likewise, application (to the left) of the operator corresponding to b on A| yields


The inner products on the tensor product space is defined by


Given this, the expectation value reduces to


With this notation, a concise summary of what follows can be made.

  • Theoretically, there exists a, b such that
whatever are the particular characteristics of the hidden variable theory as long as it abides to the rules of local realism as defined above. That is to say, no local hidden variable theory can make the same predictions as quantum mechanics.
  • Experimentally, instances of
have been found (whatever the hidden variable theory), but
has never been found. That is to say, predictions of quantum mechanics have never been falsified by experiment. These experiments include such that can rule out local hidden variable theories. But see below on possible loopholes.

Original Bell's inequalityEdit

The inequality that Bell derived can then be written as:[4]



This simple form has an intuitive explanation, however. It is equivalent to the following elementary result from probability theory. Consider three (highly correlated, and possibly biased) coin-flips X, Y, and Z, with the property that:





Compare this with the predictions from quantum mechanics for the singlet state. For a small angle θ, measured in radians, the chance of a different outcome is approximately   as explained by small-angle approximation. At two times this small angle, the chance of a mismatch is therefore about 4 times larger, since  . But we just argued that it cannot be more than 2 times as large.


CHSH inequalityEdit

Generalizing Bell's original inequality,[4] John Clauser, Michael Horne, Abner Shimony and R. A. Holt introduced the CHSH inequality,[18] which puts classical limits on the set of four correlations in Alice and Bob's experiment, without any assumption of perfect correlations (or anti-correlations) at equal settings


Making the special choice  , denoting  , and assuming perfect anti-correlation at equal settings, perfect correlation at opposite settings, therefore   and  , the CHSH inequality reduces to the original Bell inequality. Nowadays, (1) is also often simply called "the Bell inequality", but sometimes more completely "the Bell-CHSH inequality".


With abbreviated notation


the CHSH inequality can be derived as follows. Each of the four quantities is ±1 and each depends on λ. It follows that for any λ ∈ Λ, one of B + B and BB is zero, and the other is ±2. From this it follows that


and therefore


At the heart of this derivation is a simple algebraic inequality concerning four variables, A, A′, B, B, which take the values ±1 only:




Bell inequalities are violated by quantum mechanical predictionsEdit











so that



Testing by practical experimentsEdit

Scheme of a "two-channel" Bell test







Bell test experiments to date overwhelmingly violate Bell's inequality.

Two classes of Bell inequalitiesEdit

The fair sampling problem was faced openly in the 1970s. In early designs of their 1973 experiment, Freedman and Clauser[19] used fair sampling in the form of the Clauser–Horne–Shimony–Holt (CHSH[18]) hypothesis. However, shortly afterwards Clauser and Horne[16] made the important distinction between inhomogeneous (IBI) and homogeneous (HBI) Bell inequalities. Testing an IBI requires that we compare certain coincidence rates in two separated detectors with the singles rates of the two detectors. Nobody needed to perform the experiment, because singles rates with all detectors in the 1970s were at least ten times all the coincidence rates. So, taking into account this low detector efficiency, the QM prediction actually satisfied the IBI. To arrive at an experimental design in which the QM prediction violates IBI we require detectors whose efficiency exceeds 82.8% for singlet states,[20] but have very low dark rate and short dead and resolving times. This is now within reach.

Practical challengesEdit

Because, at that time, even the best detectors didn't detect a large fraction of all photons, Clauser and Horne[16] recognized that testing Bell's inequality required some extra assumptions. They introduced the No Enhancement Hypothesis (NEH):



The experiment was performed by Freedman and Clauser,[19] who found that the Bell's inequality was violated. So the no-enhancement hypothesis cannot be true in a local hidden variables model.


Metaphysical aspectsEdit








There have also been repeated claims that Bell's arguments are irrelevant because they depend on hidden assumptions that, in fact, are questionable. For example, E. T. Jaynes[26] claimed in 1989 that there are two hidden assumptions in Bell's theorem that could limit its generality. According to him:

  1. Bell interpreted conditional probability P(X|Y) as a causal inference, i.e. Y exerted a causal inference on X in reality. However, P(X|Y) actually only means logical inference (induction). Causes cannot travel faster than light or backward in time, but deduction can.


A recent flurry of activity about implications for determinism arose with the paper: The Free Will Theorem[28] which stated "the response of a spin 1 particle to a triple experiment is free—that is to say, is not a function of properties of that part of the universe that is earlier than this response with respect to any given inertial frame."[29] This theorem raised awareness of a tension between determinism fully governing an experiment (on the one hand) and Alice and Bob being free to choose any settings they like for their observations (on the other).[30][31] The philosopher David Hodgson supports this theorem as showing that determinism is unscientific, and that quantum mechanics allows observers (at least in some instances) the freedom to make observations of their choosing, thereby leaving the door open for free will.[32]

General remarksEdit

The violations of Bell's inequalities, due to quantum entanglement, provide near definitive demonstrations of something that was already strongly suspected: that quantum physics cannot be represented by any version of the classical picture of physics.[33] Some earlier elements that had seemed incompatible with classical pictures included complementarity and wavefunction collapse. The Bell violations show that no resolution of such issues can avoid the ultimate strangeness of quantum behavior.[34]



See alsoEdit


  2. ^ Mermin, David (April 1985). "Is the moon there when nobody looks? Reality and the quantum theory" (PDF). Physics Today. 38: 38–47. Bibcode:1985PhT....38d..38M. doi:10.1063/1.880968. 
  3. ^ Stapp, Henry P. (1975). "Bell's Theorem and World Process". Nuovo Cimento. 29B (2): 270–276. Bibcode:1975NCimB..29..270S. doi:10.1007/BF02728310.  (Quote on p. 271)
  4. ^ a b c d e f g Bell, John (1964). "On the Einstein Podolsky Rosen Paradox" (PDF). Physics. 1 (3): 195–200. 
  8. ^ Zeeya Merali (2015-08-27). "Quantum 'spookiness' passes toughest test yet". Nature News. Retrieved 2017-06-03. 
  10. ^ a b Article on Bell's Theorem by Abner Shimony in the Stanford Encyclopedia of Philosophy, (2004).
  13. ^ Stapp 1975
  14. ^ a b Bell, JS (1982). "On the impossible pilot wave" (PDF). Foundations of Physics. 12: 989–99. Bibcode:1982FoPh...12..989B. doi:10.1007/bf01889272.  Reprinted in Speakable and unspeakable in quantum mechanics: collected papers on quantum philosophy. CUP, 2004, p. 160.
  16. ^ a b c Clauser, John F. (1974). "Experimental consequences of objective local theories" (PDF). Physical Review D. 10 (2): 526–535. Bibcode:1974PhRvD..10..526C. doi:10.1103/PhysRevD.10.526. Archived from the original (PDF) on 2013-12-25. 
  17. ^ Eberhard, P. H. (1977). "Bell's theorem without hidden variables" (PDF). Nuovo Cimento B. 38: 75–80. Bibcode:1977NCimB..38...75E. doi:10.1007/BF02726212. 
  18. ^ a b Clauser, John; Horne, Michael; Shimony, Abner; Holt, Richard (1969). "Proposed Experiment to Test Local Hidden-Variable Theories". Physical Review Letters. 23 (15): 880–884. Bibcode:1969PhRvL..23..880C. doi:10.1103/PhysRevLett.23.880. 
  19. ^ a b Freedman, Stuart J.; Clauser, John F. (1972). "Experimental Test of Local Hidden-Variable Theories" (PDF). Physical Review Letters. 28 (14): 938–941. Bibcode:1972PhRvL..28..938F. doi:10.1103/PhysRevLett.28.938. 
  20. ^ Anupam Garg; N.D. Mermin (1987), "Detector inefficiencies in the Einstein-Podolsky-Rosen experiment", Phys. Rev. D, 25 (12): 3831–5, Bibcode:1987PhRvD..35.3831G, doi:10.1103/PhysRevD.35.3831 
  21. ^ Reid, M. D.; Walls, D. F. (1986). "Violations of classical inequalities in quantum optics". Physical Review A. 34: 1260–1276. Bibcode:1986PhRvA..34.1260R. doi:10.1103/PhysRevA.34.1260. 
  23. ^ Cramer, John (1986). "The transactional interpretation of quantum mechanics". Reviews of Modern Physics. 58 (3): 647–687. Bibcode:1986RvMP...58..647C. doi:10.1103/RevModPhys.58.647. 
  26. ^ Jaynes, E. T. (1989). "Clearing up Mysteries—The Original Goal" (PDF). Maximum Entropy and Bayesian Methods: 12. doi:10.1007/978-94-015-7860-8_1. 
  27. ^ Gill, Richard D. (2003). "Time, Finite Statistics, and Bell's Fifth Position". Proc. of "Foundations of Probability and Physics - 2", Ser. Math. Modelling in Phys., Engin., and Cogn. Sc. Växjö Univ. Press. 5/2002: 179–206. arXiv:quant-ph/0301059 . 
  28. ^ Conway, John; Simon Kochen (2006). "The Free Will Theorem". Foundations of Physics. 36 (10): 1441–1473. arXiv:quant-ph/0604079 . Bibcode:2006FoPh...36.1441C. doi:10.1007/s10701-006-9068-6. 
  29. ^ Conway, John H. & Simon Kochen (2009). "The strong free will theorem" (PDF). Notices of the AMS. 56 (2): 226–232. 
  30. ^ Cator, Eric & Klaas Landsman (2014). "Constraints on determinism: Bell versus Conway–Kochen". Foundations of Physics. 44 (7): 781–791. arXiv:1402.1972 . Bibcode:2014FoPh...44..781C. doi:10.1007/s10701-014-9815-z. 
  31. ^ Esfeld, Michael (2015). "Bell's Theorem and the Issue of Determinism and Indeterminism". Foundations of Physics. 45: 471–482. arXiv:1503.00660 . Bibcode:2015FoPh...45..471E. doi:10.1007/s10701-015-9883-8. 
  35. ^ R.G. Lerner; G.L. Trigg (1991). Encyclopaedia of Physics (2nd ed.). VHC publishers. p. 495. ISBN 0-89573-752-3. 


Further readingEdit

The following are intended for general audiences.

  • Goldstein, Sheldon; et al. "Bell's theorem". Scholarpedia. 6 (10): 8378. Bibcode:2011SchpJ...6.8378G. doi:10.4249/scholarpedia.8378. 

External linksEdit