GHZ experiment

GHZ experiments are a class of experiments which arise 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; and with particular relevance to the EPR experiment. The GHZ experiments are distinguished by involving more than two observers or detectors. They are named for Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger (GHZ) who first analyzed certain measurements involving four observers [1], and who subsequently (together with A. Shimony, upon a suggestion by N. D. Mermin) applied their arguments to certain measurements involving three observers [2].

Frequently considered cases of GHZ experiments are concerned with measurements obtained by three observers, A, B, and C, 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 as (A↓), B detecting and counting a signal either as (B «) or as (B »), and C detecting and counting a signal either as (C ◊) or as (C ♦).

Signals are to be considered and counted only if A, B, and C 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 C 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;

and correspondingly, it can be distinguished and counted whether

• C detected a signal as (C ◊) and not as (C ♦), with corresponding counts n _l(C ◊) = 1 and n _l(C ♦) = 0, in this particular trial l, or
• C detected a signal as (C ♦) and not as (C ◊), with corresponding counts n_m(C ◊) = 0 and n_m(C ♦) = 1, in this particular trial m, where trials l and m are evidently distinct.

For any one trial j it may be consequently distinguished in which particular channels signals were detected and counted by A, B, and C together, in this particular trial j; and correlation numbers such as

p_{(A↑) (B «) (C ◊)}( j ) = (n_j (A↑) - n_j (A↓)) (n_j (B «) - n_j (B »)) (n_j (C ◊) - n_j (C ♦)) can be evaluated in each trial.

Following an argument by John Stewart Bell, each trial is now characterized by particular individual ajustable apparatus parameters, or settings of the observers involved. There are (at least) two distinguishable settings being considered for each, namely A's settings a₁ , and a₂ , B's settings b₁ , and b₂ , and C's settings c₁ , and c₂ .

Trial s for instance would be characterized by A's setting a₂ , B's setting b₂ , and C's settings c₂ ; another trial, r, would be characterized by A's setting a₂ , B's setting b₂ , and C's settings c₁ , and so on. (Since C's settings are distinct between trials r and s, therefore these two trials are distinct.) Correspondingly, the correlation number p_{(A↑) (B «) (C ◊)}( s ) is written as p_{(A↑) (B «) (C ◊)}( a₂ , b₂ , c₂ ), the correlation number p_{(A↑) (B «) (C ◊)}( r ) is written as p_{(A↑) (B «) (C ◊)}( a₂ , b₂ , c₁ ) and so on.

Further, as GHZ and collaborators demonstrate in detail, the following four distinct trials, with their various separate detector counts and with suitably identified settings, may be considered and be found experimentally:

• trial s as shown above, characterized by the settings a₂ , b₂ , and c₂ , and with detector counts such that

p_{(A↑) (B «) (C ◊)}( s ) = (n_s (A↑) - n_s (A↓)) (n_s (B «) - n_s (B »)) (n_s (C ◊) - n_s (C ♦)) = -1,

• trial u with settings a₂ , b₁ , and c₁ , and with detector counts such that

p_{(A↑) (B «) (C ◊)}( u ) = (n_u (A↑) - n_u (A↓)) (n_u (B «) - n_u (B »)) (n_u (C ◊) - n_u (C ♦)) = 1,

• trial v with settings a₁ , b₂ , and c₁ , and with detector counts such that

p_{(A↑) (B «) (C ◊)}( v ) = (n_v (A↑) - n_v (A↓)) (n_v (B «) - n_v (B »)) (n_v (C ◊) - n_v (C ♦)) = 1, and

• trial w with settings a₁ , b₁ , and c₂ , and with detector counts such that

p_{(A↑) (B «) (C ◊)}( w ) = (n_w (A↑) - n_w (A↓)) (n_w (B «) - n_w (B »)) (n_w (C ◊) - n_w (C ♦)) = 1.

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_x , 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 observers (who are considered far away)?

Therefore: can the correlation numbers such as p_{(A↑) (B «) (C ◊)}( a_x , b_x , c_x ), be expressed as a product of such independent functions, A( a_x , λ ), B( b_x , λ ) and C( c_x , λ ), for all trials and all settings, with a suitable hidden variable value λ?

Comparison with the product which defined p_{(A↑) (B «) (C ◊)}( j ) explicitly above, readily suggests to identify

• λ → j,
• A( a_x , j ) → (n_j (A↑) - n_j (A↓)),
• B( b_x , j ) → (n_j (B «) - n_j (B »)), and
• C( c_x , j ) → (n_j (C ◊) - n_j (C ♦)),

where j denotes any one trial which is characterized by the specific settings a_x , b_x , and c_x , of A, B, and of C, respectively.

However, GHZ and collaborators also require that the hidden variable argument to functions A(), B(), and C() may take the same value, λ, even in distinct trials, being characterized by distinct settings.

Consequently, substituting these functions into the consistent conditions on four distinct trials, u, v, w, and s shown above, they are able to obtain the following four equations concerning one and the same value λ:

1. A( a₂ , λ ) B( b₂ , λ ) C( c₂ , λ ) = -1,
2. A( a₂ , λ ) B( b₁ , λ ) C( c₁ , λ ) = 1,
3. A( a₁ , λ ) B( b₂ , λ ) C( c₁ , λ ) = 1, and
4. A( a₁ , λ ) B( b₁ , λ ) C( c₂ , λ ) = 1.

Taking the product of the last three equations, and noting that A( a₁ , λ ) A( a₁ , λ ) = 1, B( b₁ , λ ) B( b₁ , λ ) = 1, and C( c₁ , λ ) C( c₁ , λ ) = 1, yields

A( a₂ , λ ) B( b₂ , λ ) C( c₂ , λ ) = 1

in contradiction to the first equation; 1 ≠ -1.

Given that the four trials under consideration can indeed be consistently considered and experimentally realized, the assumptions concerning hidden variables which lead to the indicated mathematical contradiction are therefore collectively unsuitable to represent all experimental results; namely
the assumption of local hidden variables which occur equally in distinct trials.

It is probably worth mentioning that the assumption of local hidden variables which vary between distinct trials, such as a trial index itself, does generally not allow to derive a mathematical contradiction as indicated by GHZ.

References

• Daniel M. Greenberger, Michael A. Horne, Abner Shimony, Anton Zeilinger, Bell's theorem without inequalities, Am. J. Phys. 58 (12), 1131 (1990); and references therein.