Bell's Theorem states that a "Bell inequality" 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 experiments on pairs of particles that have taken part in a quantum-mechanical interaction and then separated. A Bell test experiment is one designed to test whether or not the real world obeys a Bell inequality. Such experiments fall into two classes, depending on whether the analyser used has one or two output channels.

The Bell inequalities

The following section consists of an exposition of inequalities often referred to as Bell inequalities.

Bell's original inequality

The inequality that Bell derived in 1964 was:

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

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:

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

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:

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

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

<math> \operatorname{C}(a b)= \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)} <math>

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

<math>

\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.<math>

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:

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

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).

Conduct of Bell test experiments

A typical CHSH (two-channel) experiment

Scheme of a "two-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a two-channel polariser whose orientation can be set by the experimenter. Emerging signals from each channel are detected and coincidences counted by the coincidence monitor CM.

In practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons, rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments (Aspect, 1981, 1982a,b), the polarisation direction, though other properties can be used. The diagram shows a typical optical experiment of the two-channel kind for which Alain Aspect set a precedent in 1982 (Aspect, 1982a). Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.

Four separate subexperiments are conducted, corresponding to the four terms E(a, b) in the test statistic S ((2) below). The settings a, a′, b and b′ are generally in practice chosen to be 0, 45°, 22.5° and 67.5° respectively — the "Bell test angles" — these being the ones for which the QM formula gives the greatest violation of the inequality.

For each selected value of a and b, the numbers of coincidences in each category (N₊₊, N_--, N_+- and N_-+) are recorded. The experimental estimate for E(a, b) is then calculated as:

(1)        E = (N₊₊ + N_-- − N_+- − N_-+)/(N₊₊ + N_-- + N_+- + N_-+).

Once all four E’s have been estimated, an experimental estimate of the test statistic

(2)       S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′)

can be found. If S is numerically greater than 2 it has infringed the CHSH inequality. The experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories.

A strong assumption has had to be made, however, to justify use of expression (2). It 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.

The derivation of the inequality is given in the CHSH Bell test page.

A typical CH74 (single-channel) experiment

Setup for a "single-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each photon encounters a single channel (e.g. "pile of plates") polariser whose orientation can be set by the experimenter. Emerging signals are detected and coincidences counted by the coincidence monitor CM.

Prior to 1982 all actual Bell tests used "single-channel" polarisers and variations on an inequality designed for this setup. The latter is described in Clauser, Horne, Shimony and Holt's much-cited 1969 article (Clauser, 1969) as being the one suitable for practical use. As with the CHSH test, there are four subexperiments in which each polariser takes one of two possible settings, but in addition there are other subexperiments in which one or other polariser or both are absent. Counts are taken as before and used to estimate the test statistic

(3)       S = (N(a, b) − N(a, b′) + N(a′, b) + N(a′, b′) − N(a′, ∞) − N(∞, b)) / N(∞, ∞),

where the symbol ∞ indicates absence of a polariser.

If S exceeds 0 then the experiment is declared to have infringed Bell's inequality and hence to have "refuted local realism".

The only theoretical assumption (other than Bell's basic ones of the existence of local hidden variables) that has been made in deriving (3) is that when a polariser is inserted the probability of detection of any given photon is never increased: there is "no enhancement". The derivation of this inequality is given in the page on Clauser and Horne's 1974 Bell test.

Experimental assumptions

In addition to the theoretical assumptions made, there are practical ones. There may, for example, be a number of "accidental coincidences" in addition to those of interest. It is assumed that no bias is introduced by subtracting their estimated number before calculating S, but that this is so is not considered by some to be obvious. There may be synchronisation problems — ambiguity in recognising pairs due to the fact that in practice they will not be detected at exactly the same time.

Nevertheless, despite all these deficiences of the actual experiments, one striking fact emerges: the results are, to a very good approximation, what quantum mechanics predicts. If imperfect experiments give us such excellent overlap with quantum predictions, most working quantum physicists would agree with John Bell in expecting that, when a perfect Bell test is done, the Bell inequalities will still be violated. This attitude has lead to the emergence of a new sub-field of physics which is now known as quantum information theory. One of the main achievements of this new branch of physics is showning that violation of Bell's inequalities leads to the possiblity of a secure information transfer, which utilizes the so-called quantum cryptography (involving entangled states of pairs of particles).

Notable experiments

Over the past thirty or so years, a great number of Bell test experiments have now been conducted. These experiments have (subject to a few assumptions, considered by most to be reasonable) confirmed quantum theory and shown results that cannot be explained under local hidden variable theories. Advancements in technology have led to significant improvement in efficencies, as well as a greater variety to methods to test the Bell Theorem. Some of the best known:

Freedman and Clauser, 1972

This was the first actual Bell test, using Freedman's inequality, a variant on the CH74 inequality.

Aspect, 1981-2

Aspect and his team at Orsay, Paris, conducted three Bell tests using calcium cascade sources. The first and last used the CH74 inequality. The second was the first application of the CHSH inequality, the third the famous one (originally suggested by John Bell) in which the choice between the two settings on each side was made during the flight of the photons.

Tittel and the Geneva group, 1998

The Geneva 1998 Bell test experiments showed that distance did not destroy the "entanglement". Light was sent in fibre optic cables over distances of several kilometers before it was analysed. As with almost all Bell tests since about 1985, a "parametric down-conversion" (PDC) source was used.

Weihs' experiment under "strict Einstein locality" conditions

In 1998 Gregor Weihs and a team at Innsbruck, lead by Anton Zeilinger, conducted an ingenious experiment that closed the "locality" loophole, improving on Aspect's of 1982. The choice of detector was made using a quantum process to ensure that is was random. This test violated the CHSH inequality by over 30 standard deviations, the coincidence curves agreeing with those predicted by quantum theory.

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/
• Clauser, 1969: J. F. Clauser, M.A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880-884 (1969), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
• Freedman, 1972: S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972)
• Tittel, 1998: W. Tittel et al., Experimental demonstration of quantum-correlations over more than 10 kilometers (http://arxiv.org/abs/quant-ph/9707042), Physical Review A 57, 3229 (1998); Violation of Bell inequalities by photons more than 10 km apart (http://arxiv.org/abs/quant-ph/9806043), Physical Review Letters 81, 3563 (1998)
• Weihs, 1998: G. Weihs, et al., Violation of Bell’s inequality under strict Einstein locality conditions (http://arXiv.org/abs/quant-ph/9910080), Phys. Rev. Lett. 81, 5039 (1998)