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A Bell inequality is one that must be obeyed under any local hidden variable theory but is violated under quantum mechanics (QM). The term "Bell inequality" can mean any one of a number of inequalities — in practice, in real experiments, the CHSH or CH74 inequality, not the original one derived by John Bell. It places restrictions on the statistical results of measurements on pairs of particles that are physically separated at the time of measurement but come from a common source and share “hidden variable” values. If the particles have taken part in a quantum interaction, then quantum mechanic predicts that a Bell inequality can be infringed, thus providing an opportunity for an experimental test of QM versus local hidden variable theories, the best known of these being by Alain Aspect and his team in Paris (Aspect, 1981, 1982a,b).

As will be seen from the descriptions below, the various inequalities are not quite equivalent. In practice they involve assumptions that are disputed by somelocal realists, opening the way for loopholes.

Bell's original inequality

The inequality that Bell derived (Bell, 1964) was:

1+\operatorname {C} (b,c)\geq |\operatorname {C} (a,b)-\operatorname {C} (a,c)|,

where C is the "quantum correlation" (the expectation value of the product of the outcomes) of the particle pairs and a, b and c settings of the apparatus (in practice, orientations of the analysing polariser) in a Bell test experiment. Outcomes are coded as +1 for the + channel, -1 for the - channel.

This inequality is not used in practice. For one thing, it is only true for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and -1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction.

Optical experiments, incidentally, can scarcely be said to have confirmed this last prediction, since even with analysers parallel there are a great number of non-detections due to the inefficiency of the detectors.

The CHSH inequality

In 1969 Clauser, Horne, Shimony and Holt (Clauser, 1969) derived the slightly more general "CHSH" inequality, though in the proof they gave initially it would appear to be valid only for two-outcome setups. It was only later, in 1971, that Bell published an improved derivation, valid even when there are non-detections. This derivation will be found on the "CHSH inequality" page.

The CHSH inequality is:

-2\leq \operatorname {C} (a,b)-\operatorname {C} (a,b')+\operatorname {C} (a',b)+\operatorname {C} (a,b)\leq 2

and C is, as before, the quantum correlation when the analyser settings are a and b. Explicitly in terms of the various observed probabilities of pairs this is:

\operatorname {C} (a,b)=p_{++}(a,b)+p_{--}(a,b)-p_{+-}(a,b)-p_{-+}(a,b)

This inequality is widely used, though it did not come into favour until after Aspect had inaugurated the idea in 1982 (Aspect, 1982). The reason for reluctance to use it at first was that, strictly speaking, each term C, being formed from sums and differences of four probabilities, ought to be estimated using unbiased estimates of those probabilities. The probabilities ought to be estimated as ratios of the observed frequencies of coincidences to the number N of emitted pairs. This number, however, is not known in real experiments and is, in any case, very large compared to the coincidence counts. If it were to be used, no optical experiment could hope to violate the inequality. So in practice the estimate

\operatorname {C} (ab)={\frac {\operatorname {N} _{++}(a,b)+\operatorname {N} _{--}(a,b)-\operatorname {N} _{+-}(a,b)-\operatorname {N} _{-+}(a,b)}{(\operatorname {N} _{++}(a,b)+\operatorname {N} _{--}(a,b)+\operatorname {N} _{+-}(a,b)+\operatorname {N} _{-+}(a,b)}}

is used. This estimate can only be justified if the sample of detected pairs is a fair sample of those emitted. Local realists believe that there is good reason to expect that the sample is not in fact fair, given the low efficiencies of the detectors and the likely characteristics of the apparatus. The search for more efficient detectors or for other ways of achieving a "loophole-free" experiment continues.

The CH74 inequality

The inequality derived by Clauser and Horne in 1974 depends only on ratios of observed counts, with no presupposition that they represent probability estimates. Knowledge of N, the number of pairs emitted, is not necessary (Clauser, 1974; Clauser, 1978), nor is it important to have high efficiency detectors. Just one assumption is needed in addition to "separability". It is assumed that there is "no enhancement" -- that when a polariser is inserted the probability of detection of a photon never increases. The derivation is given on the page "Clauser and Horne's 1974 Bell test". The inequality is

{\frac {\operatorname {N} (a,b)-\operatorname {N} (a,b')+\operatorname {N} (a',b)+\operatorname {N} (a',b')-\operatorname {N} (a',\infty )-\operatorname {N} (\infty ,b)}{\operatorname {N} (\infty ,\infty )}}\leq 0.

The Wigner-d'Espagnat inequality

The Wigner-d'Espagnat inequality is almost equivalent to Bell's original inequality, though it is expressed in terms of probabilities, not expectation values of products of outcomes.

Just three different settings are used for the detectors, which are taken as giving definite either + or - outcomes. The inequality is:

P_{++}\leq P_{++}(a,c)+P_{++}(c,b)

where P is the probability of a coincidence.

Other inequalities

In 1989 Greenberger, Horne, and Zeilinger produced an alternative to the Bell setup, known as the GHZ experiment. It uses three observers and three electrons, and is designed to be able to distinguished a class of hidden variables from quantum mechanics in a single set of observations (Greenberger, 1990, 1993).

In 1993 Hardy proposed a situation where nonlocality can be inferred (in ideal conditions) without using inequalities (Hardy, 1993).

Applications of Bell’s inequalities

See the page Bell test experiments for applications. These experiments are currently believed by most physicists to have provided convincing support for QM and its “nonlocal” prediction of quantum entanglement. has been assumed that the sample of detected pairs is representative of the pairs emitted by the source. That this assumption may not be true comprises the fair sampling loophole. No absolute check on its validity is feasible.

References

Aspect, 1981: A. Aspect et al., Experimental Tests of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460 (1981)
Aspect, 1982a: A. Aspect et al., Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49, 91 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/

Aspect, 1982b: A. Aspect et al., Experimental Test of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
Bell, 1964: J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195 (1964), reproduced in J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press 1987)
Clauser, 1969: J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880-884 (1969), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
Clauser, 1974: J. F. Clauser and M. A. Horne, Experimental consequences of objective local theories, Physical Review D, 10, 526-35 (1974)
Clauser, 1978: J. F. Clauser and A. Shimony, Bell's theorem: experimental tests and implications, Reports on Progress in Physics 41, 1881 (1978)
Greenberger, 1990: Daniel M. Greenberger, Michael A. Horne, Abner Shimony and Anton Zeilinger, Bell's theorem without inequalities, Am. J. Phys. 58 (12), 1131 (1990); and references therein.
Greenberger, 1993: D. M. Greenberger, M. A. Horne and A. Zeilinger, Multiparticle interferometry and the superposition principle, Physics Today, 22-29 (August 1993)

Hardy, 1993: L. Hardy, Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71 (11) 1665-1668 (1993)