Editing Bell test (section)

=== 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]].