Clauser and Horne's 1974 Bell test

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There are several different inequalities that have come to be termed Bell's inequalities. Their intended use is to test quantum mechanics (QM) as against "local realism" or "local hidden variable theories", testing experimentally for quantum entanglement, a feature of quantum mechanics that had bothered Einstein and others right from the early days of the theory. QM predicts that the Bell inequalities will be violated, whilst under local realist theories this is impossible.

Inequalities used in real experiments

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. In the article they first derive the well-known "two-channel" CHSH inequality then, making additional assumptions that amount to assuming a restricted form of "fair sampling" (they assume that all photons that have passed through a polariser have an equal chance of detection), adapt it to the single-channel situation. Both derivations have since been improved upon, the two-channel one in 1971 by Bell (Bell, 1971) and the single channel one by Clauser and Horne in 1974 (Clauser, 1974). The later derivations show that some unnecessary assumptions had been made at first. The two-channel test does not require Bell's assumption about parallel detectors, though it does (in the currently-used form) need the fair sampling assumption and is consequently vulnerable to the "fair sampling loophole". The single-channel inequality (here termed the "CH74" inequality) does not in fact require any assumption about fair sampling, though it does require "no enhancement".

In Clauser and Horne's 1974 derivation, they apply what they term an "Objective Local Theory" rather than a "hidden variable" one, assuming the hidden variable λ does not necessarily determine outcomes completely, only (when taken in conjunction with the detector setting) their probabilities, e.g. p₁(λ, a). The idea has evolved from one introduced by Bell in 1971 (Bell 1971). The model can be converted to a deterministic one by assuming more components to the hidden variable - components associated with the detector rather than with the source - but there is no real advantage in so doing. As it stands it is a very natural model for use in optical experiments.

Derivation of the CH74 inequality from first principles

The CH74 (single-channel) inequality, as used in, for example, two of Aspect's experiments (Aspect, 1981; Aspect, 1982b), is

(1)    S = {N(a, b) − N(a, b′) + N(a′, b) + N(a′, b′) − N(a′ ∞) − N(∞ b)} / N(∞ ∞) ≤ 0,

where a and a′ are polariser (analyser) settings for side 1, b and b′ for side 2, the symbol ∞ denoting absence of a polariser. Each term N represents a coincidence count from a separate subexperiment.

The inhomogeneous CH74 inequality

The following is taken from page 528 of (Clauser, 1974). Equation numbers have been altered and a little [explanatory text added in brackets]:

... in this section, we derive a consequence of [the factorability assumption] which is experimentally testable without N being known, and which contradicts the quantum-mechanical predictions.

Let a and a′ be two orientations of analyser 1, and let b and b′ be two orientations of analyser 2.

The inequalities

(2)         0 ≤ p₁(λ, a) ≤ 1

0 ≤ p₁(λ, a′) ≤ 1

0 ≤ p₂(λ, b) ≤ 1

0 ≤ p₂(λ, b′) ≤ 1

hold if the probabilities are sensible. These inequalities and the theorem [see below] give

(3)           – 1 ≤ p₁(λ, a) p₂(λ, b) – p₁(λ, a) p₂(λ, b′) + p₁(λ, a′) p₂(λ, b)

+ p₁(λ, a′) p₂(λ, b′) – p₁(λ, a′) – p₂(λ, b) ≤ 0

for each λ. Multiplication by ρ(λ) [the probability of the source being in state λ] and integration over λ gives [assuming factorability]

(4)        – 1 ≤ p₁₂(a, b) – p₁₂(a, b′) + p₁₂(a′, b) + p₁₂(a′, b′) – p₁(a′) – p₂(b) ≤ 0

as a necessary constraint on the statistical predictions of any OLT [Objective Local Theory] ...

... The upper limits in (4) are experimentally testable without N being known. Inequality (4) holds perfectly generally for any systems described by OLT. These are new results not previously presented elsewhere. [My italics]

[End of quoted text.]

The CH74 inequality as used in practice

On page 530, we find the derivation of an inequality of similar structure that can be used with real, low-efficiency, detectors. It employs an assumption rather stronger than (2). This is the "no enhancement" assumption, which can be expressed mathematically in the form:

(5)         0 ≤ p₁(λ, a) ≤ p₁(λ, ∞) ≤ 1,

0 ≤ p₂(λ, b) ≤ p₂(λ, ∞) ≤ 1,

where ∞ denotes absence of the polariser, p₁(λ, ∞) the probability of a count from detector 1 when the polariser is absent and the emission is in state λ,and p₂(λ, ∞) likewise for detector 2.

Using the same arguments as before, we find (4) replaced by:

(6)       p₁₂(∞, ∞) ≤ p₁₂(a, b) – p₁₂(a, b′) + p₁₂(a′, b) + p₁₂(a′, b′) – p₁₂(a′, ∞) – p₁₂(∞, b) ≤ 0

On dividing through by p₁₂(∞, ∞) and replacing all probabilities by their estimates (the corresponding counts divided by N, the number of emitted pairs), the right hand inequality becomes the CH74 inequality (1).

Theorem from Clauser and Horne's original Appendix A

The above derivations depend on the following theorem, proved on page 530 of Clauser and Horne's paper: Given six numbers x₁, x₂, y₁, y₂, X and Y such that

0 ≤ x₁ ≤ X,

0 ≤ x₂ ≤ X,

0 ≤ y₁ ≤ Y,

0 ≤ y₂ ≤ Y,

then the function U = x₁ y₁ – x₁ y₂ + x₂ y₁ + x₂ y₂ – Y x₂ – X y₁ is constrained by the inequality

–  XY ≤ U ≤ 0

Derived inequalities

Special cases of the Bell inequalities apply when there is rotational invariance and the detector settings are selected with the differences all equal to either φ or 3φ for some angle φ. They are easily derived from the general forms (Clauser, 1978, pp 1896, 1905-6). The CH74 inequality can be further contracted in Bell tests involving polarisation to a form known as the Freedman inequality, if there is sufficient symmetry in the experiment and φ is chosen to be π/8.

According to Clauser and Horne (Clauser 1974, p 533) the CHSH two-channel inequality can be regarded as a consequence of the CH74 one (Clauser, 1974, p 533). They mean here the ideal version of the CHSH inequality, though, with the number of emitted pairs N known, not the one used in practice. The latter, using the sum of observed coincidence rates instead of N, requires the additional assumption of fair sampling.

"Loopholes" in applications

The CH74 and related tests were used in all Bell test experiments of the 1970s and some later ones, notably two of Alain Aspect's (Aspect, 1981, 1982b). The early experiments, starting with Freedman and Clauser's (Freedman, 1972), though almost all infringing the inequality, were not accepted as convincing evidence for quantum entanglement. This was at least in part because they did not block the "locality" loophole. There may in addition have been other problems, for example with synchronisation.

Discussion

The important feature of the CH74 inequality is, as is stressed both in the 1974 paper and in the 1978 report on the Bell tests by Clauser and Shimony (Clauser, 1978), that it does not require knowledge of N, the number of emitted pairs. It equally does not require the "fair sampling" assumption that is used to circumvent the problem of not knowing N in the currently accepted version of the CHSH test. This is because, being designed for use with "single-channel" polarisers, the question of whether or not the detected light would have been a fair sample of that emitted had we used a two-channel polariser is not relevant. We neither need nor expect the sample from a single channel to be "fair", i.e. to represent all possible λ values with equal probability.

Alain Aspect himself is among those who quote the earlier, indirect, derivation of the CH74 inequality and are consequently are under the impression that it requires the fair sampling assumption. In a comprehensive article describing his three Bell test experiments (Aspect, 2004) he conveys the impression that the CH74 test is generally inferior, partly because it does not follow so closely Bell's original scheme but also because of the strong assumptions that he thinks are needed.

The CH74 test is in fact perfectly valid in its own right. Apart from the standard assumptions of local realism, which allow the multiplication of independent probabilities to give joint ones, the practical (homogeneous) version of the CH74 test requires only the one extra assumption, that of "no enhancement", as expressed by inequalities (5) above. This assumption seems reasonable on physical grounds, so long as obvious causes of bias (polarisation-biased detectors, say) are avoided. Some local realists, though, have objected to it. For one reason or another, it is the CHSH test that has been, ever since Aspect first used it in the second (Aspect, 1982a) of his three famous experiments, accepted as standard.

References

• Aspect, 1981: A. Aspect et al., Phys. Rev. Lett. 47, 460 (1981)
• Aspect, 1982a: A. Aspect et al., Phys. Rev. Lett. 49, 91 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
• Aspect, 1982b: A. Aspect et al., Phys. Rev. Lett. 49, 1804 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
• Aspect, 2004: A. Aspect, Bell’s theorem: the naïve view of an experimentalist (http://arxiv.org/abs/quant-ph/0402001), Text prepared for a talk at a conference in memory of John Bell, held in Vienna in December 2000. Published in Quantum [Un]speakables – From Bell to Quantum information, R. A. Bertlmann and A. Zeilinger (eds.), (Springer, 2002)
• Bell, 1971: J. S. Bell, in Foundations of Quantum Mechanics, Proceedings of the International School of Physics “Enrico Fermi”, Course XLIX, B. d’Espagnat (Ed.) (Academic, New York, 1971), p. 171 and Appendix B. Pages 171-81 are reproduced as Ch. 4, pp 29-39, of 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, Phys. Rev. Lett. 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, Phys. Rev. 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)
• Freedman, 1972: S. J. Freedman and J. F. Clauser, Experimental test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972)