GHZ experiment
Encyclopedia
GHZ experiments are a class of physics experiments that may be used to generate starkly contrasting predictions from local hidden variable theory
and quantum mechanical theory
, and permit immediate comparison with actual experimental results. A GHZ experiment is similar to a test of Bell's inequality
, except using three or more entangled particles
, rather than two. With specific settings of GHZ experiments, it is possible to demonstrate absolute contradictions between the predictions of local hidden variable theory and those of quantum mechanics, whereas tests of Bell's inequality only demonstrate contradictions of a statistical nature. The results of actual GHZ experiments agree with the predictions of quantum mechanics.
The GHZ experiments are named for Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger
(GHZ) who first analyzed certain measurements involving four observers. and who subsequently (together with Abner Shimony
, upon a suggestion by David Mermin
) applied their arguments to certain measurements involving three observers.
s in an entangled
state, with the photons being in a superposition
of being all horizontally polarized (HHH) or all vertically polarized (VVV), with respect to some coordinate system
. Prior to any measurements being made, the polarizations of the photons are indeterminate; If a measurement is made on one of the photons using a two-channel polarizer
aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50% probability for each orientation, and the other two photons immediately assume the identical polarization.
In a GHZ experiment regarding photon polarization, however, a set of measurements is performed on the three entangled photons using two-channel polarizers set to various orientations relative to the coordinate system. For specific combinations of orientations, perfect (rather than statistical) correlations between the three polarizations are predicted by both local hidden variable theory (aka "local realism") and by quantum mechanical theory, and the predictions may be contradictory. For instance, if the polarization of two of the photons are measured and determined to be rotated +45° from horizontal, then local hidden variable theory predicts that the polarization of the third photon will also be +45° from horizontal. However, quantum mechanical theory predicts that it will be +45° from vertical.
The results of actual experiments agree with the predictions of quantum mechanics, not those of local realism.
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
similarly, it can be distinguished and counted whether
and correspondingly, it can be distinguished and counted whether
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 ) = (nj (A↑) - nj (A↓)) (nj (B «) - nj (B »)) (nj (C ◊) - nj (C ♦))
can be evaluated in each trial.
Following an argument by John Stewart Bell
, each trial is now characterized by particular individual adjustable 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:
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 (nj (A↑) - nj (A↓)), be expressed as a function A( ax , 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 ◊)( ax , bx , cx ), be expressed as a product of such independent functions, A( ax , λ ), B( bx , λ ) and C( cx , 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
where j denotes any one trial which is characterized by the specific settings
ax , bx , and cx , 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:
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
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.
Because we have no control over the hidden variables, the contradiction derived above cannot be directly tested in an experiment.
Also, Λic is the complement
of Λi.
Now, equation (1) can only be true if at least one of the other three is false. Therefore
Λ1 ⊆ Λ2c ∪ Λ3c ∪ Λ4c.
In terms of probability,
p(Λ1) ≤ p(Λ2c ∪ Λ3c ∪ Λ4c).
By the rules of probability theory, it follows that
p(Λ1) ≤ p(Λ2c) + p(Λ3c) + p(Λ4c).
This inequality allows for an experimental test.
GHSZ also show that the fair sampling assumption can be dispensed with if the detector efficiencies are at least 90.8%.
Local hidden variable theory
In quantum mechanics, a local hidden variable theory is one in which distant events are assumed to have no instantaneous effect on local ones....
and quantum mechanical theory
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, and permit immediate comparison with actual experimental results. A GHZ experiment is similar to a test of Bell's inequality
Bell test experiments
The Bell test experiments serve to investigate the validity of the entanglement effect in quantum mechanics by using some kind of Bell inequality...
, except using three or more entangled particles
Subatomic particle
In physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...
, rather than two. With specific settings of GHZ experiments, it is possible to demonstrate absolute contradictions between the predictions of local hidden variable theory and those of quantum mechanics, whereas tests of Bell's inequality only demonstrate contradictions of a statistical nature. The results of actual GHZ experiments agree with the predictions of quantum mechanics.
The GHZ experiments are named for Daniel M. Greenberger, Michael A. Horne, and Anton Zeilinger
Anton Zeilinger
Anton Zeilinger is an Austrian quantum physicist. He is currently professor of physics at the University of Vienna, previously University of Innsbruck. He is also the director of the Vienna branch of the Institute for Quantum Optics and Quantum Information IQOQI at the Austrian Academy of Sciences...
(GHZ) who first analyzed certain measurements involving four observers. and who subsequently (together with Abner Shimony
Abner Shimony
Abner Shimony is an American physicist and philosopher of science specializing in quantum theory.-Career:Shimony obtained his BA in Mathematics and Philosophy from Yale University in 1948, and an MA in Philosophy from the University of Chicago in 1950. He obtained his Ph.D...
, upon a suggestion by David Mermin
David Mermin
Nathaniel David Mermin is a solid-state physicist at Cornell University best known for the eponymous Mermin-Wagner theorem and his application of the term "Boojum" to superfluidity, and for the quote "Shut up and calculate!"Together with Neil W...
) applied their arguments to certain measurements involving three observers.
Summary description and example
A GHZ experiment is performed using a quantum system in a GHZ state. An example of a GHZ state is three photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s in an entangled
Entanglement
Entanglement may refer to:* Quantum entanglement* Orientation entanglement* Entanglement * Entanglement of polymer chains, see Reptation* Wire entanglement...
state, with the photons being in a superposition
Quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
of being all horizontally polarized (HHH) or all vertically polarized (VVV), with respect to some coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
. Prior to any measurements being made, the polarizations of the photons are indeterminate; If a measurement is made on one of the photons using a two-channel polarizer
Polarizer
A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular...
aligned with the axes of the coordinate system, the photon assumes either horizontal or vertical polarization, with 50% probability for each orientation, and the other two photons immediately assume the identical polarization.
In a GHZ experiment regarding photon polarization, however, a set of measurements is performed on the three entangled photons using two-channel polarizers set to various orientations relative to the coordinate system. For specific combinations of orientations, perfect (rather than statistical) correlations between the three polarizations are predicted by both local hidden variable theory (aka "local realism") and by quantum mechanical theory, and the predictions may be contradictory. For instance, if the polarization of two of the photons are measured and determined to be rotated +45° from horizontal, then local hidden variable theory predicts that the polarization of the third photon will also be +45° from horizontal. However, quantum mechanical theory predicts that it will be +45° from vertical.
The results of actual experiments agree with the predictions of quantum mechanics, not those of local realism.
Preliminary considerations
Frequently considered cases of GHZ experiments are concerned with observations obtained by three measurements, A, B, and C, each of which detects one signal at a time in one of two distinct mutually exclusive 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 nt (A↑) = 1 and nt (A↓) = 0, in this particular trial t, or
- A detected a signal as (A↓) and not as (A↑), with corresponding counts nf (A↑) = 0 and nf (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 ng (B «) = 1 and ng (B ») = 0, in this particular trial g, or
- B detected a signal as (B ») and not as (B «), with corresponding counts nh (B «) = 0 and nh (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 nm(C ◊) = 0 and nm(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 ) = (nj (A↑) - nj (A↓)) (nj (B «) - nj (B »)) (nj (C ◊) - nj (C ♦))
can be evaluated in each trial.
Following an argument by John Stewart Bell
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...
, each trial is now characterized by particular individual adjustable 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 ) = (ns (A↑) - ns (A↓)) (ns (B «) - ns (B »)) (ns (C ◊) - ns (C ♦)) = -1,
- trial u with settings a2 , b1 , and c1 , and with detector counts such that
- p(A↑) (B «) (C ◊)( u ) = (nu (A↑) - nu (A↓)) (nu (B «) - nu (B »)) (nu (C ◊) - nu (C ♦)) = 1,
- trial v with settings a1 , b2 , and c1 , and with detector counts such that
- p(A↑) (B «) (C ◊)( v ) = (nv (A↑) - nv (A↓)) (nv (B «) - nv (B »)) (nv (C ◊) - nv (C ♦)) = 1, and
- trial w with settings a1 , b1 , and c2 , and with detector counts such that
- p(A↑) (B «) (C ◊)( w ) = (nw (A↑) - nw (A↓)) (nw (B «) - nw (B »)) (nw (C ◊) - nw (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 (nj (A↑) - nj (A↓)), be expressed as a function A( ax , 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 ◊)( ax , bx , cx ), be expressed as a product of such independent functions, A( ax , λ ), B( bx , λ ) and C( cx , 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( ax , j ) → (nj (A↑) - nj (A↓)),
- B( bx , j ) → (nj (B «) - nj (B »)), and
- C( cx , j ) → (nj (C ◊) - nj (C ♦)),
where j denotes any one trial which is characterized by the specific settings
ax , bx , and cx , 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:
- A( a2 , λ ) B( b2 , λ ) C( c2 , λ ) = -1,
- A( a2 , λ ) B( b1 , λ ) C( c1 , λ ) = 1,
- A( a1 , λ ) B( b2 , λ ) C( c1 , λ ) = 1, and
- 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.
Because we have no control over the hidden variables, the contradiction derived above cannot be directly tested in an experiment.
Deriving an Inequality
Since equations (1) through (4) above cannot be satisfied simultaneously when the hidden variable, λ, takes the same value in each equation, GHSZ proceed by allowing λ to take different values in each equation. They define- Λ1= the set of all λ's such that equation (1) holds,
- Λ2= the set of all λ's such that equation (2) holds,
- Λ3= the set of all λ's such that equation (3) holds,
- Λ4= the set of all λ's such that equation (4) holds.
Also, Λic is the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of Λi.
Now, equation (1) can only be true if at least one of the other three is false. Therefore
Λ1 ⊆ Λ2c ∪ Λ3c ∪ Λ4c.
In terms of probability,
p(Λ1) ≤ p(Λ2c ∪ Λ3c ∪ Λ4c).
By the rules of probability theory, it follows that
p(Λ1) ≤ p(Λ2c) + p(Λ3c) + p(Λ4c).
This inequality allows for an experimental test.
Testing the inequality
To test the inequality just derived, GHSZ need to make one more assumption, the "fair sampling" assumption. Because of inefficiencies in real detectors, in some trials of the experiment only one or two particles of the triple will be detected. Fair sampling assumes that these inefficiencies are unrelated to the hidden variables; in other words, the number of triples actually detected in any run of the experiment is proportional to the number that would have been detected if the apparatus had no inefficiencies - with the same constant of proportionality for all possible settings of the apparatus. With this assumption, p(Λ1) can be determined by choosing the apparatus settings a2 , b2 , and c2 , counting the number of triples for which the outcome is -1, and dividing by the total number of triples observed at that setting. The other probabilities can be determined in a similar manner, allowing a direct experimental test of the inequality.GHSZ also show that the fair sampling assumption can be dispensed with if the detector efficiencies are at least 90.8%.