Editing CHSH inequality (section)

== Experiments ==
Many Bell tests conducted subsequent to [[Alain Aspect]]'s second experiment in 1982 have used the CHSH inequality, estimating the terms using (3) and assuming fair sampling. Some dramatic violations of the inequality have been reported.<ref>{{Cite journal |last1=Hensen |first1=B. |last2=Bernien |first2=H. |last3=Dréau |first3=A. E. |last4=Reiserer |first4=A. |last5=Kalb |first5=N. |last6=Blok |first6=M. S. |last7=Ruitenberg |first7=J. |last8=Vermeulen |first8=R. F. L. |last9=Schouten |first9=R. N. |last10=Abellán |first10=C. |last11=Amaya |first11=W. |year=2015 |title=Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres |journal=Nature |volume=526 |issue=7575 |pages=682–686 |arxiv=1508.05949 |bibcode=2015Natur.526..682H |doi=10.1038/nature15759 |pmid=26503041 |last13=Mitchell |first13=M. W. |last14=Markham |first14=M. |last15=Twitchen |first15=D. J. |last12=Pruneri |first16=D. |last17=Wehner |first17=S. |last18=Taminiau |first18=T. H. |last19=Hanson |first19=R. |first12=V. |last16=Elkouss|s2cid=205246446 }}</ref>[[Image:Two channel bell test.svg|300px|thumb|right|'''Schematic 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 (''a'' and ''b'') 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 rather than the electrons that Bell originally had in mind.  The property of interest is, in the best known experiments,<ref name="Aspect-1981">{{citation |author1=Alain Aspect |author2=Philippe Grangier |author3=Gérard Roger |year=1981 |title=Experimental Tests of Realistic Local Theories via Bell's Theorem |journal=Phys. Rev. Lett. |volume=47 |issue=7 |pages=460–3 |doi=10.1103/PhysRevLett.47.460 |bibcode=1981PhRvL..47..460A|doi-access=free }}</ref><ref name="Aspect-1982a">{{citation |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 |page=91 |doi=10.1103/PhysRevLett.49.91 |bibcode=1982PhRvL..49...91A|doi-access=free }}</ref><ref name="Aspect-1982b">{{citation |author1=Alain Aspect |author2=Jean Dalibard |author3=Gérard Roger |year=1982 |title=Experimental Test of Bell's Inequalities Using Time-Varying Analyzers |journal=Phys. Rev. Lett. |volume=49 |issue=25 |pages=1804–7 |doi=10.1103/PhysRevLett.49.1804 |bibcode=1982PhRvL..49.1804A|doi-access=free }}</ref> 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 '++', '+&minus;', '&minus;+' or '&minus;&minus;' and corresponding counts accumulated.

Four separate subexperiments are conducted, corresponding to the four terms <math>E(a, b)</math> in the test statistic ''S'' ({{EquationRef|2}}, above).  The settings {{math|1=''a'' = 0°}}, {{math|1=''a''&prime; = 45°}}, {{math|1=''b'' = 22.5°}}, and {{math|1=''b''&prime; = 67.5°}} are generally in practice chosen—the "Bell test angles"—these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality.

For each selected value of <math>a,b</math>, the numbers of coincidences in each category <math>\left\{ N_{++}, N_{--}, N_{+-}, N_{-+} \right\}</math> are recorded.  The experimental estimate for <math>E(a, b)</math> is then calculated as:
{{NumBlk||<math display="block"> E(a,b) = \frac {N_{++}  - N_{+-} - N_{-+} + N_{--}} {N_{++}  + N_{+-} + N_{-+}+ N_{--}}</math>|{{EquationRef|3}}}}

Once all the {{mvar|E}}'s have been estimated, an experimental estimate of ''S'' (Eq. {{EquationNote|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 quantum mechanics prediction and ruled out all local hidden-variable theories.

The CHSH paper lists many preconditions (or "reasonable and/or presumable assumptions") to derive the simplified theorem and formula. For example, for the method to be valid, it has to be assumed that the detected pairs are a fair sample of those emitted. In actual experiments, detectors are never 100% efficient, so that only a sample of the emitted pairs are detected. A subtle, related requirement is that the hidden variables do not influence or determine detection probability in a way that would lead to different samples at each arm of the experiment.

The CHSH inequality has been violated with [[photon]] pairs, [[beryllium]] ion pairs, [[ytterbium]] ion pairs, [[rubidium]] atom pairs, whole rubidium-atom cloud pairs, [[Nitrogen-vacancy center|nitrogen vacancies]] in [[diamond]]s, and [[Josephson phase]] [[phase qubit|qubits]].<ref>{{cite web|url=https://scitechdaily.com/first-experimental-proof-that-quantum-entanglement-is-real/amp/|publisher=[[SciTech (magazine)|SciTech Daily]]|title=First Experimental Proof That Quantum Entanglement Is Real|date=9 October 2022 |access-date=October 10, 2022}}</ref>