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In quantum mechanics,
- Bell's
Theorem states
- that a Bell inequality must be obeyed
under any local hidden variable
theory
- but can in certain circumstance be violated
under quantum mechanics (QM).
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 pairs 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 analyser used has one or two output
channels.
Conduct of Bell test experiments
A typical CHSH (two-channel) experiment
Scheme of a "two-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each
photon encounters a two-channel polariser whose orientation can be set by the
experimenter. Emerging signals from each channel are detected and coincidences counted by
the coincidence monitor CM.
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 conversion ), rather than the atoms that Bell originally had in mind.
The property of interest is, in the best known experiments (Aspect, 1981, 1982a,b), 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 QM 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
Setup for a "single-channel" Bell test
The source S produces pairs of "photons", sent in opposite directions. Each
photon encounters a single channel (e.g. "pile of plates") polariser whose
orientation can be set by the experimenter. Emerging signals are detected and coincidences
counted by the coincidence monitor CM.
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', 8) - N(8, b)) / N(8, 8),
where the symbol 8 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 so 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
in expecting that, when a perfect Bell test is done, the Bell inequalities will still be
violated. This attitude has lead 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).
Notable experiments
Over the past thirty or so years, a great number of Bell
test experiments have now been conducted. These experiments have (subject to a few
assumptions, considered by most to be reasonable) confirmed quantum theory and shown
results that cannot be explained under local hidden variable theories. 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
- Aspect 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 the
famous one (originally suggested by John Bell) in which the choice
between the two settings on each side was made during the flight of the photons.
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, lead by Anton
Zeilinger, 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.
Loopholes
The series of increasingly sophisticated Bell test
experiments has narrowed to a small group the critics who question results by pointing to
loopholes (some hypothetical, others acknowledged), some of which bias the experimental
results in favor of quantum mechanics. An overview of such loopholes can be found in Loopholes
in optical Bell test experiments. So far no Bell test result has been reported
that was free of known loopholes, but such tests are foreseen in the nearby future
(García-Patrón, 2004).
References
- Aspect, 1981: A. Aspect et al., Experimental Tests
of Realistic Local Theories via Bell's Theorem, Phys. Rev. Lett. 47, 460 (1981)
- Aspect, 1982a: A. Aspect et al., Experimental
Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell's
Inequalities, Phys. Rev. Lett. 49, 91 (1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
- Aspect, 1982b: A. Aspect et al., Experimental Test
of Bell's Inequalities Using Time-Varying Analyzers, Phys. Rev. Lett. 49, 1804
(1982), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
- Barrett, 2002 Quantum
Nonlocality, Bell Inequalities and the Memory Loophole quant-ph/0205016 (2002).
- Bell, 1987: J. S. Bell, Speakable and Unspeakable
in Quantum Mechanics, (Cambridge University Press 1987)
- Clauser, 1969: J. F. Clauser, M.A. Horne, A. Shimony
and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys.
Rev. Lett. 23, 880-884 (1969), available at http://fangio.magnet.fsu.edu/~vlad/pr100/
- Clauser, 1974: J. F. Clauser and M. A. Horne, Experimental
consequences of objective local theories, Phys. Rev. D 10, 526-35 (1974)
- Freedman, 1972: S. J. Freedman and J. F. Clauser, Experimental
test of local hidden-variable theories, Phys. Rev. Lett. 28, 938 (1972)
- García-Patrón, 2004: R. García-Patrón, J.
Fiurácek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph. Grangier, Proposal for a Loophole-Free Bell Test
Using Homodyne Detection, Phys. Rev. Lett. 93, 130409 (2004)
- Gill, 2001: R.D. Gill, Accardi
contra Bell (cum mundi): The Impossible Coupling, IMS Lecture Notes - Monograph
Series, 42, 133-154 (2003)
- Kielpinski: D. Kielpinski et al., Recent Results in Trapped-Ion Quantum
Computing (2001)
- Kwiat, 1999: P.G. Kwiat, et al., Ultrabright source of
polarization-entangled photons, Physical Review A 60 (2), R773-R776 (1999)
- Rowe, 2001: M. Rowe et al., Experimental
violation of a Bell’s inequality with efficient detection, Nature 409,
791 (2001)
- Tittel, 1997: W. Tittel et al., Experimental demonstration of
quantum-correlations over more than 10 kilometers, Phys. Rev. A, 57, 3229
(1997)
- Tittel, 1998: W. Tittel et al., Experimental demonstration of
quantum-correlations over more than 10 kilometers, Physical Review A 57,
3229 (1998); Violation of Bell inequalities by photons
more than 10 km apart, Physical Review Letters 81, 3563 (1998)
- Weihs, 1998: G. Weihs, et al., Violation of Bell’s inequality under
strict Einstein locality conditions, Phys. Rev. Lett. 81, 5039 (1998)
