 |
|
| ||||||||||||||||||||||||||
|
|
| ||||||||||||||||||||||||||
|
|
|
|
|
| ||||||||
The CHSH Bell test is an application of Bell's theorem, intended to distinguish between quantum mechanics (QM) and local hidden variable theories. It is named after Clauser, Horne, Shimony and Holt, who described it in a much-cited paper published in 1969 (Clauser, 1969). They derived the "CHSH inequality", though not quite in the form we now know it and using rather more restrictive assumptions than are in fact necessary. The inequality, as with John Bell's original one (Bell, 1964), applies to a statistical property of counts of "coincidences" in a quantum correlations of the particle pairs, where the quantum correlation is defined to be the expectation value of the product of the "outcomes" of the experiment, i.e. the statistical average of A(a).B(b), where A and B are the separate outcomes, using the coding +1 for the '+' channel and −1 for the '−' channel. Clauser et al's 1969 derivation was orientated towards the use of "two-channel" detectors, and indeed it is for these that it is generally used, but under their method the only possible outcomes were +1 and −1. In order to adapt it to real situations, which at the time meant the use of polarised light and single-channel polarisers, they had to interpret '−' as meaning "non-detection in the '+' channel", i.e. either '−' or nothing. They did not in the original article discuss how the two-channel inequality could be applied in real experiments with real imperfect detectors, though it was later proved (Bell, 1971) that the inequality itself was equally valid. The occurrence of zero outcomes, though, means it is no longer so obvious how the values of E are to be estimated from the experimental data.
Note that in all actual Bell test experiments it is assumed that the source stays essentially constant, being characterised at any given instant by a state ("hidden variable") λ that has a constant distribution ρ(λ) and is unaffected by the choice of detector setting.
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, 1981-2), the polarisation direction, though other properties can be used. The diagram shows a typical optical experiment. 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) above). 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:
(3) E = (N++ + N-- − N+- − N-+)/(N++ + N-- + N+- + N-+)
Once all the E’s have been estimated, an experimental estimate of S (expression (2)) can be found. If it is numerically greater than 2 it has infringed the CHSH inequality and the experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories.
The original 1969 derivation will not be given here since it is not easy to follow and involves the assumption that the outcomes are all +1 or −1, never zero. Bell's 1971 derivation is more general. He effectively assumes the "Objective Local Theory" later used by Clauser and Horne (Clauser, 1974). It is assumed that any hidden variables associated with the detectors themselves are independent on the two sides and can be averaged out from the start. Another derivation of interest is given in Clauser and Horne's 1974 paper, in which they start from the CH74 inequality.
It would appear from both these later derivations that the only assumptions really needed for the inequality itself (as opposed to the method of estimation of the test statistic) are that the distribution of the possible states of the source remains contant and the detectors on the two sides act independently.
The following is based on page 37 of Bell's Speakable and Unspeakable (Bell, 1971), the main change being to use the symbol ‘E’ instead of ‘P’ for the expected value of the quantum correlation. This avoids any implication that the quantum correlation is itself a probability.
We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ(λ), the integral of which over the complete hidden variable space is 1. We thus assume we can write
(4) E(a, b) = ∫dλρ(λ)A(a, λ)B(b, λ)
where A and B are the average values of the outcomes. Since the possible values of A and B are −1, 0 and +1, it follows that
(5) |A|≤1, |B|≤ 1.
Then, if a, a′, b and b′ are alternative settings for the detectors,
(6) E(a, b) − E(a, b′) = ∫dλρ(λ)[A(a, λ)B(b, λ) − A(a, λ)B(b′, λ)]
Then using (5)
or, using the fact that the integral of ρ(λ) is 1,
which includes the CHSH inequality.
In their 1974 paper (Clauser, 1974), Clauser and Horne show that the CHSH inequality can be derived from the CH74 one. As they tell us, in a two-channel experiment the CH74 Bell test, for which "fair sampling" in this sense is not needed.
Bell's theorem; Bell test experiments; Clauser and Horne's 1974 Bell test; Bell test loopholes; Local hidden variable theory; Quantum entanglement
|
||||
| View Live Article | This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License | |
||
