In physics, the Clauser–Horne–Shimony–Holt (CHSH) inequality can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics cannot be reproduced by local hidden-variable theories. Experimental verification of the inequality being violated is seen as confirmation that nature cannot be described by such theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969. They derived the CHSH inequality, which, as with John Stewart Bell's original inequality, is a constraint—on the statistical occurrence of "coincidences" in a Bell test—which is necessarily true if an underlying local hidden-variable theory exists. In practice, the inequality is routinely violated by modern experiments in quantum mechanics.

The usual form of the CHSH inequality is

where

${\displaystyle a}$ and ${\displaystyle a'}$ are detector settings on side ${\displaystyle A}$, ${\displaystyle b}$ and ${\displaystyle b'}$ on side ${\displaystyle B}$, the four combinations being tested in separate subexperiments. The terms ${\displaystyle E(a,b)}$ etc. are the 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 ${\displaystyle A(a)\times B(b)}$, where ${\displaystyle A,B}$ are the separate outcomes, using the coding +1 for the '+' channel and −1 for the '−' channel. Clauser et al.'s 1969 derivation was oriented 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 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 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.

The mathematical formalism of quantum mechanics predicts that the value of ${\displaystyle S}$ exceeds 2 for systems prepared in suitable entangled states and the appropriate choice of measurement settings (see below). The maximum violation predicted by quantum mechanics is ${\displaystyle 2{\sqrt {2}}}$ (Tsirelson's bound) and can be obtained from a maximal entangled Bell state.

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.

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, 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 ${\displaystyle E(a,b)}$ in the test statistic S (, above). The settings a = 0°, a′ = 45°, b = 22.5°, and b′ = 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.