CHSH inequality

The Clauser-Horne-Shimony-Holt inequality (CHSH), arises in quantum mechanics, in discussion and experimental determination of whether local hidden variables are required for, or even compatible with, the representation of experimental results; with particular relevance to the EPR thought experiment. It is named for the authors, John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, of the paper [1.] in which this inequality was first derived, and where its utility for experimental test was emphasized.

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

• A detected a signal as (A↑) and not as (A↓), with corresponding counts n_t (A↑) = 1 and n_t (A↓) = 0, in this particular trial t, or
• A detected a signal as (A↓) and not as (A↑), with corresponding counts n_f (A↑) = 0 and n_f (A↓) = 1, in this particular trial f, where trials f and t are evidently distinct.

Similarly, it can be distinguished and counted whether

• B detected a signal as (B «) and not as (B »), with corresponding counts n_g (B «) = 1 and n_g (B ») = 0, in this particular trial g, or
• B detected a signal as (B ») and not as (B «), with corresponding counts n_h (B «) = 0 and n_h (B ») = 1, in this particular trial h, where trials g and h are evidently distinct.

Further, for any one trial j it may be consequently distinguished and counted whether

• (A↑), and (B «) were detected together in this particular trial j, or
• (A↑), and (B ») were detected together, or
• (A↓), and (B «) were detected together, or
• (A↓), and (B ») were detected together in this trial.

Summing the counts over all trials j of a given set J of trials, one can evaluate for instance

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

• λ → j,
• Γ → J,
• ∫_Γ dλ → _{{ j = first of J} Σ ^{last of J}},
• A( a_A , j ) → (n_j (A↑) - n_j (A↓)),
• B( b_B , j ) → (n_j (B «) - n_j (B »)), and
• ρ( j ) = 1 / ∫_Γ dλ → 1/(_{{ j = first of J} Σ ^{last of J}} 1).

However, with Bell, CHSH consider and require the integration domain Γ being total, constant and equal in all instances; both correlation numbers P_{(A↑) (B «)}( a_A , b_B ) and P_{(A↑) (B «)}( a_A , c_B ) are being expressed as integrals over one and the same integration domain Γ, and with the same density ρ( λ ), even though these two correlation numbers refer to disjoint sets of trials, J vs. K.

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 ) =

∫_Γ dλ ρ( λ ) (A( a_A , λ ) B( b_B , λ ))

(1 - A( q_A , λ ) B( c_B , λ )) - (A( a_A , λ ) B( c_B , λ )) (1 - A( q_A , λ ) B( b_B , λ )).

Noting that -1 ≤ (A( a_A , λ ) B( b_B , λ )) ≤ 1, and -1 ≤ (A( a_A , λ ) B( c_B , λ )) ≤ 1, therefore

| P_{(A↑) (B «)}( a_A , b_B ) - P_{(A↑) (B «)}( a_A , c_B ) | ≤

∫_Γ dλ ρ( λ ) (1 - A( q_A , λ ) B( c_B , λ )) + ∫_Γ dλ ρ( λ ) (1 - A( q_A , λ ) B( b_B , λ )).

Identifying the integrals (as above, over one and the same integration domain) as correlation numbers for corresponding settings, and using the definition of ρ( λ ) as a density, results in

| 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.

Reference

• John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, p. 880 (1969).