Bell test experiments
Encyclopedia
The Bell test experiments serve to investigate the validity of the entanglement
effect in quantum mechanics
by using some kind of Bell inequality. John Bell
published the first inequality of this kind in his paper "On the Einstein-Podolsky-Rosen Paradox
".
Bell's Theorem
states that a Bell inequality
must be obeyed under any local hidden variable theory
but can in certain circumstances be violated under quantum mechanics
. The term "Bell inequality" can mean any one of a number of inequalities — in practice, in real experiments, the CHSH or CH74 inequality, not the original one derived by John Bell
. It places restrictions on the statistical results of experiments on sets of particles that have taken part in an interaction and then separated. A Bell test experiment is one designed to test whether or not the real world obeys a Bell inequality. Such experiments fall into two classes, depending on whether the analysers used have one or two output channels. Nevertheless a loophole-free test has not been performed yet.
), rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments, the polarisation direction, though other properties can be used.
The diagram shows a typical optical experiment of the two-channel kind for which Alain Aspect
set a precedent in 1982 (Aspect, 1982a). Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.
Four separate subexperiments are conducted, corresponding to the four terms E(a, b) in the test statistic S ((2) below). The settings a, a′, b and b′ are generally in practice chosen to be 0, 45°, 22.5° and 67.5° respectively — the "Bell test angles" — these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality.
For each selected value of a and b, the numbers of coincidences in each category (N++, N--, N+- and N-+) are recorded. The experimental estimate for E(a, b) is then calculated as:
(1) E = (N++ + N-- − N+- − N-+)/(N++ + N-- + N+- + N-+).
Once all four E’s have been estimated, an experimental estimate of the test statistic
(2) S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′)
can be found. If S is numerically greater than 2 it has infringed the CHSH inequality. The experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories.
A strong assumption has had to be made, however, to justify use of expression (2). It has been assumed that the sample of detected pairs is representative of the pairs emitted by the source. That this assumption may not be true comprises the fair sampling loophole.
The derivation of the inequality is given in the CHSH Bell test page.
(3) S = (N(a, b) − N(a, b′) + N(a′, b) + N(a′, b′) − N(a′, ∞) − N(∞, b)) / N(∞, ∞),
where the symbol ∞ indicates absence of a polariser.
If S exceeds 0 then the experiment is declared to have infringed Bell's inequality and hence to have "refuted local realism".
The only theoretical assumption (other than Bell's basic ones of the existence of local hidden variables) that has been made in deriving (3) is that when a polariser is inserted the probability of detection of any given photon is never increased: there is "no enhancement". The derivation of this inequality is given in the page on Clauser and Horne's 1974 Bell test.
Nevertheless, despite all these deficiencies of the actual experiments, one striking fact emerges: the results are, to a very good approximation, what quantum mechanics predicts. If imperfect experiments give us such excellent overlap with quantum predictions, most working quantum physicists would agree with John Bell
in expecting that, when a perfect Bell test is done, the Bell inequalities will still be violated. This attitude has led to the emergence of a new sub-field of physics which is now known as quantum information theory. One of the main achievements of this new branch of physics is showing that violation of Bell's inequalities leads to the possibility of a secure information transfer, which utilizes the so-called quantum cryptography
(involving entangled states of pairs of particles).
). Advancements in technology have led to significant improvement in efficiencies, as well as a greater variety of methods to test the Bell Theorem.
Some of the best known:
, conducted an ingenious experiment that closed the "locality" loophole, improving on Aspect's of 1982. The choice of detector was made using a quantum process to ensure that it was random. This test violated the CHSH inequality
by over 30 standard deviations, the coincidence curves agreeing with those predicted by quantum theory.
uses the so-called GHZ
state of three particles;
it is reported in Nature (2000)
hidden variable theory
consistent with quantum mechanics must be highly counterintuitive.
s (the experimental result) by letting detection be dependent on a combination of local hidden variables and detector setting.
Experimenters have repeatedly stated that loophole-free tests can be expected in the near future (García-Patrón, 2004). On the other hand, some researchers point out that it is a logical possibility that quantum physics itself prevents a loophole-free test from ever being implemented (Gill, 2003; Santos, 2006).
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
effect in quantum mechanics
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...
by using some kind of Bell inequality. John 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 :...
published the first inequality of this kind in his paper "On the Einstein-Podolsky-Rosen Paradox
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
".
Bell's Theorem
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
states that a Bell inequality
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
must be obeyed under any local hidden variable theory
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....
but can in certain circumstances be violated under quantum mechanics
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...
. The term "Bell inequality" can mean any one of a number of inequalities — in practice, in real experiments, the CHSH or CH74 inequality, not the original one derived by John 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 :...
. It places restrictions on the statistical results of experiments on sets of particles that have taken part in an interaction and then separated. A Bell test experiment is one designed to test whether or not the real world obeys a Bell inequality. Such experiments fall into two classes, depending on whether the analysers used have one or two output channels. Nevertheless a loophole-free test has not been performed yet.
Conduct of optical Bell test experiments
In practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons (produced by atomic cascade or spontaneous parametric down conversionSpontaneous parametric down conversion
Spontaneous parametric down-conversion is an important process in quantum optics, used especially as a source of entangled photon pairs, and of single photons.-Basic process:...
), rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments, the polarisation direction, though other properties can be used.
A typical CHSH (two-channel) experiment
Alain Aspect
Alain Aspect is a French physicist noted for his experimental work on quantum entanglement....
set a precedent in 1982 (Aspect, 1982a). Coincidences (simultaneous detections) are recorded, the results being categorised as '++', '+−', '−+' or '−−' and corresponding counts accumulated.
Four separate subexperiments are conducted, corresponding to the four terms E(a, b) in the test statistic S ((2) below). The settings a, a′, b and b′ are generally in practice chosen to be 0, 45°, 22.5° and 67.5° respectively — the "Bell test angles" — these being the ones for which the quantum mechanical formula gives the greatest violation of the inequality.
For each selected value of a and b, the numbers of coincidences in each category (N++, N--, N+- and N-+) are recorded. The experimental estimate for E(a, b) is then calculated as:
(1) E = (N++ + N-- − N+- − N-+)/(N++ + N-- + N+- + N-+).
Once all four E’s have been estimated, an experimental estimate of the test statistic
(2) S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′)
can be found. If S is numerically greater than 2 it has infringed the CHSH inequality. The experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories.
A strong assumption has had to be made, however, to justify use of expression (2). It has been assumed that the sample of detected pairs is representative of the pairs emitted by the source. That this assumption may not be true comprises the fair sampling loophole.
The derivation of the inequality is given in the CHSH Bell test page.
A typical CH74 (single-channel) experiment
Prior to 1982 all actual Bell tests used "single-channel" polarisers and variations on an inequality designed for this setup. The latter is described in Clauser, Horne, Shimony and Holt's much-cited 1969 article (Clauser, 1969) as being the one suitable for practical use. As with the CHSH test, there are four subexperiments in which each polariser takes one of two possible settings, but in addition there are other subexperiments in which one or other polariser or both are absent. Counts are taken as before and used to estimate the test statistic.(3) S = (N(a, b) − N(a, b′) + N(a′, b) + N(a′, b′) − N(a′, ∞) − N(∞, b)) / N(∞, ∞),
where the symbol ∞ indicates absence of a polariser.
If S exceeds 0 then the experiment is declared to have infringed Bell's inequality and hence to have "refuted local realism".
The only theoretical assumption (other than Bell's basic ones of the existence of local hidden variables) that has been made in deriving (3) is that when a polariser is inserted the probability of detection of any given photon is never increased: there is "no enhancement". The derivation of this inequality is given in the page on Clauser and Horne's 1974 Bell test.
Experimental assumptions
In addition to the theoretical assumptions made, there are practical ones. There may, for example, be a number of "accidental coincidences" in addition to those of interest. It is assumed that no bias is introduced by subtracting their estimated number before calculating S, but that this is true is not considered by some to be obvious. There may be synchronisation problems — ambiguity in recognising pairs due to the fact that in practice they will not be detected at exactly the same time.Nevertheless, despite all these deficiencies of the actual experiments, one striking fact emerges: the results are, to a very good approximation, what quantum mechanics predicts. If imperfect experiments give us such excellent overlap with quantum predictions, most working quantum physicists would agree with John 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 :...
in expecting that, when a perfect Bell test is done, the Bell inequalities will still be violated. This attitude has led to the emergence of a new sub-field of physics which is now known as quantum information theory. One of the main achievements of this new branch of physics is showing that violation of Bell's inequalities leads to the possibility of a secure information transfer, which utilizes the so-called quantum cryptography
Quantum cryptography
Quantum key distribution uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages...
(involving entangled states of pairs of particles).
Notable experiments
Over the past thirty or so years, a great number of Bell test experiments have now been conducted. These experiments are subject to assumptions, in particular the ‘no enhancement’ hypothesis of Clauser and Horne (above). The experiments are commonly interpreted to rule out local hidden variable theories, but they could also be said to demonstrate ‘signal enhancement’, which relates to the stochastic resonance phenomenon (Two classes of Bell inequalitiesBell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
). Advancements in technology have led to significant improvement in efficiencies, as well as a greater variety of methods to test the Bell Theorem.
Some of the best known:
Freedman and Clauser, 1972
- This was the first actual Bell test, using Freedman's inequality, a variant on the CH74 inequality.
Aspect, 1981-2
- Alain AspectAlain AspectAlain Aspect is a French physicist noted for his experimental work on quantum entanglement....
and his team at Orsay, Paris, conducted three Bell tests using calcium cascade sources. The first and last used the CH74 inequality. The second was the first application of the CHSH inequality. The third (and most famous) was arranged such that the choice between the two settings on each side was made during the flight of the photons (as originally suggested by John BellJohn Stewart BellJohn 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 :...
).
Tittel and the Geneva group, 1998
- The Geneva 1998 Bell test experiments showed that distance did not destroy the "entanglement". Light was sent in fibre optic cables over distances of several kilometers before it was analysed. As with almost all Bell tests since about 1985, a "parametric down-conversion" (PDC) source was used.
Weihs' experiment under "strict Einstein locality" conditions
In 1998 Gregor Weihs and a team at Innsbruck, led by Anton ZeilingerAnton 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...
, conducted an ingenious experiment that closed the "locality" loophole, improving on Aspect's of 1982. The choice of detector was made using a quantum process to ensure that it was random. This test violated the CHSH inequality
CHSH inequality
In physics, the CHSH Bell test is an application of Bell's theorem, intended to distinguish between the entanglement hypothesis of quantum mechanics and local hidden variable theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony and Richard Holt, who described it in a much-cited...
by over 30 standard deviations, the coincidence curves agreeing with those predicted by quantum theory.
Pan et al.'s experiment on the GHZ state
This is the first of new Bell-type experiments on more than two particles; this oneuses the so-called GHZ
Greenberger-Horne-Zeilinger state
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state which involves at least three subsystems . It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger in 1989...
state of three particles;
it is reported in Nature (2000)
Rowe et al. (2001) are the first to close the detection loophole
The detection loophole was first closed in an experiment with two entangled trapped ions, carried out in the ion storage group of David Wineland at the National Institute of Standards and Technology in Boulder. The experiment had detection efficiencies well over 90%.Gröblacher et al. (2007) test of Leggett-type non-local realist theories
A specific class of non-local theories suggested by Anthony Leggett is ruled out. Based on this, the authors conclude that any possible non-localNonlocality
In Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...
hidden variable theory
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...
consistent with quantum mechanics must be highly counterintuitive.
Salart et al. (2008) Separation in a Bell Test
This experiment filled a loophole by providing an 18 km separation between detectors, which is sufficient to allow the completion of the quantum state measurements before any information could have traveled between the two detectors.Ansmann et al. (2009) Overcoming the detection loophole in solid state
This experiment surmounted the detection loophole using a pair of superconducting qubits in an entangled state. However, the experiment still suffered from the locality loophole because the qubits were only separated by a few millimeters.Loopholes
Though the series of increasingly sophisticated Bell test experiments has convinced the physics community in general that local realism is untenable, there are still critics who point out that the outcome of every single experiment done so far that violates a Bell inequality can, at least theoretically, be explained by faults in the experimental setup, experimental procedure or that the equipment used does not behave as well as it is supposed to. These possibilities are known as "loopholes". The most serious loophole is the detection loophole, which means that particles are not always detected in both wings of the experiment. It is possible to "engineer" quantum correlationQuantum correlation
In Bell test experiments the term quantum correlation has come to mean the expectation value of the product of the outcomes on the two sides. In other words, the expected change in physical characteristics as one quantum system passes through an interaction site...
s (the experimental result) by letting detection be dependent on a combination of local hidden variables and detector setting.
Experimenters have repeatedly stated that loophole-free tests can be expected in the near future (García-Patrón, 2004). On the other hand, some researchers point out that it is a logical possibility that quantum physics itself prevents a loophole-free test from ever being implemented (Gill, 2003; Santos, 2006).