Editing Bell test (section)

== Conduct of optical Bell test experiments ==
In practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons (produced by [[atomic cascade]] or [[spontaneous parametric down conversion]]), rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments, the [[polarisation (waves)|polarisation]] direction, though other properties can be used. Such experiments fall into two classes, depending on whether the analysers used have one or two output channels.

=== A typical CHSH (two-channel) experiment ===
{{Main|CHSH inequality}}
[[File:Two channel bell test.svg|300px|thumb|right|'''Scheme of a "two-channel" Bell test'''<br>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.]]

The diagram shows a typical optical experiment of the two-channel kind for which [[Alain Aspect]] set a precedent in 1982.<ref name="Aspect-1982a">{{cite journal |author1=Alain Aspect |author2=Philippe Grangier |author3=Gérard Roger |year=1982 |title=Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's Inequalities |journal=Phys. Rev. Lett. |volume=49 |issue=2 |pages=91–4 |doi=10.1103/PhysRevLett.49.91|bibcode = 1982PhRvL..49...91A|doi-access=free }}</ref> Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+&minus;', '&minus;+' or '&minus;&minus;' and corresponding counts accumulated.

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

For each selected value of ''a'' and ''b'', the numbers of coincidences in each category (''N''<sub>++</sub>, ''N''<sub>−−</sub>, ''N''<sub>+−</sub> and ''N''<sub>−+</sub>) are recorded. The experimental estimate for ''E''(''a'', ''b'') is then calculated as:

{{NumBlk|:|<math>E = \frac {N_{++}  - N_{+-} - N_{-+} + N_{--}} {N_{++}  + N_{+-} + N_{-+}+ N_{--}}</math>|{{EquationRef|1}}}}

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

{{NumBlk|:|<math>S = E(a, b) - E\left(a, b'\right) + E\left(a', b\right) + E\left(a', b'\right).</math>|{{EquationRef|2}}}}

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), namely, that the sample of detected pairs is representative of the pairs emitted by the source. Denial of this assumption is called the [[Loopholes in Bell test experiments|fair sampling loophole]].

=== A typical CH74 (single-channel) experiment ===
[[File:Single-channel Bell test.svg|300px|thumb|right|'''Setup for a "single-channel" Bell test'''<br>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 as being the one suitable for practical use.<ref name="Clauser-1969"/> 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.

{{NumBlk|:|<math>S = \frac{N(a, b) - N(a, b') + N(a', b) + N(a', b') - N(a', \infty) - N(\infty, b)}{N(\infty, \infty)},</math>|{{EquationRef|3}}}}

where the symbol ∞ indicates absence of a polariser.

If ''S'' exceeds 0 then the experiment is declared to have infringed the CH inequality and hence to have refuted local hidden-variables. This inequality is known as CH inequality instead of CHSH as it was also derived in a 1974 article by Clauser and Horne more rigorously and under weaker assumptions.<ref name="Clauser-1974">{{cite journal |author1=J.F. Clauser |author2=M.A. Horne |year=1974 |title=Experimental consequences of objective local theories |journal=Phys. Rev. D |volume=10 |issue=2 |pages=526–35 |doi=10.1103/PhysRevD.10.526|bibcode = 1974PhRvD..10..526C}}</ref>