The CHSH inequality is concerned with measurements obtained by a pair of observers, A and B, who each can detect one signal at a time in one of two distinct own channels or outcomes: for instance A detecting and counting a signal either as (A↑) or (A↓), and B detecting and counting a signal either as (B «), or (B »).
Signals are to be considered and counted only if A and B detect them trial-by-trial together; i.e. for any one signal which has been detected by A in one particular trial, B must have detected precisely one signal in the same trial, and vice versa.
For any one particular trial it may be consequently distinguished and counted whether
P_{(A↑) (B «)}( J ) = _{{ j = first of J} Σ ^{last of J}}(n_{j }(A↑) - n_{j }(A↓)) (n_{j }(B «) - n_{j }(B »)) / (_{{ j = first of J} Σ ^{last of J}} 1),
i.e. the correlation between the channels or outcomes in which A and B individually detected the signals in the trials of set J; where -1 ≤ P_{(A↑) (B «)}( J ) ≤ 1.
Based on suggestions by John Stewart Bell, CHSH now characterize particular sets of trials through ajustable apparatus parameters, or settings; for instance the set J referring to trials which are characterized by A's setting a_{A }, and B's setting b_{B }, another set K referring to trials characterized by A's setting a_{A }, and B's setting c_{B }, and so on. As far as B's settings b_{B } and c_{B } are distinct from each other, the sets of trials J and K are disjoint. Correspondingly, the correlation number P_{(A↑) (B «)}( J ) is written as P_{(A↑) (B «)}( a_{A }, b_{B } ), and P_{(A↑) (B «)}( K ) as P_{(A↑) (B «)}( a_{A }, c_{B } ), etc.
The notion of local hidden variables is now introduced by considering the following question:
Can the individual detection outcomes and corresponding counts as obtained by any one observer, e.g. the numbers (n_{j }(A↑) - n_{j }(A↓)), be expressed as a function A( a_{A }, λ ) (which necessarily assumes the values +1 or -1), i.e. as a function only of the setting of this observer in this trial, and of one other hidden parameter λ, but without an explicit dependence on settings or outcomes concerning the other observer (who is considered far away)?
Or at least: can all correlation numbers such as P_{(A↑) (B «)}( a_{A }, b_{B } ), which may in principle be experimentally determined, be also expressed as in terms of such independent functions, A( a_{A }, λ ), and B( b_{B }, λ ) as an intergral over a hidden variable λ, as
P_{(A↑) (B «)}( a_{A }, b_{B } ) = ∫_{Γ} dλ ρ( λ ) A( a_{A }, λ ) B( b_{B }, λ ),
with a suitable density ρ( λ ) and a suitable integration domain Γ ?
Comparison with the sum which defined P_{(A↑) (B «)}( J ) explicitly above, readily suggests to identify
CHSH are consequently able to evaluate for instance
P_{(A↑) (B «)}( a_{A }, b_{B } ) - P_{(A↑) (B «)}( a_{A }, c_{B } ) = ∫_{Γ} dλ ρ( λ ) A( a_{A }, λ ) B( b_{B }, λ ) - A( a_{A }, λ ) B( c_{B }, λ )).
Inserting the active zero term ± A( a_{A }, λ ) A( q_{A }, λ ) B( b_{B }, λ ) B( c_{B }, λ ), which involves yet another setting, q_{A }, it follows
P_{(A↑) (B «)}( a_{A }, b_{B } ) - P_{(A↑) (B «)}( a_{A }, c_{B } ) =
| P_{(A↑) (B «)}( a_{A }, b_{B } ) - P_{(A↑) (B «)}( a_{A }, c_{B } ) | ≤
| P_{(A↑) (B «)}( a_{A }, b_{B } ) - P_{(A↑) (B «)}( a_{A }, c_{B } ) | ≤ 1 - P_{(A↑) (B «)}( q_{A }, c_{B } ) + 1 - P_{(A↑) (B «)}( q_{A }, b_{B } ).
Finally, the Clauser-Horne-Shimony-Holt inequality is obtained as
| P_{(A↑) (B «)}( a_{A }, b_{B } ) - P_{(A↑) (B «)}( a_{A }, c_{B } ) | + P_{(A↑) (B «)}( q_{A }, c_{B } ) + P_{(A↑) (B «)}( q_{A }, b_{B } ) ≤ 2,
in terms of four correlation numbers which refer to four distinct pairs of settings.
For certain settings, the corresponding experimentally determined correlation numbers which are necessarily obtained from counts in four disjoint sets of trials, can be found to fail the CHSH inequality with considerable significance; as demonstrated for instance by Aspect et al..
Therefore the assumptions based on which the CHSH inequality is derived are collectively unsuitable to represent all experimental results; namely
the assumption of local hidden variables from one and the same constant total domain Γ in all sets of trials.
It is probably worth mentioning that the assumption of local hidden variables which vary between disjoint sets of trials, such as a trial index itself, does generally not allow the derivation of an inequality similar to that of CHSH.
A related inequality has been discussed by Wigner, d'Espagnat, and Bell.