CHSH Inequality - Statement of The Inequality

Statement of The Inequality

The usual form of the CHSH inequality is:

(1) − 2 ≤ S ≤ 2,

where

(2) S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′).

a and a′ are detector settings on side A, b and b′ on side B, the four combinations being tested in separate subexperiments. The terms 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 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 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 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.

The mathematical formalism of quantum mechanics predicts a maximum value for S of, which is greater than 2, and CHSH violations are therefore predicted by the theory of quantum mechanics.