Bell-Test

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

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. No absolute check on its validity is feasible.

The derivation of the inequality is given in the CHSH Bell test page.

A typical CH74 (single-channel) experiment

Datei:Single-channel Bell test.png

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′, ∞) − 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 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.

Possible Loopholes

There exists a small group of critics of the various Bell experiments, which includes the statistician Caroline Thompson as well as some physicists. They emphasize the existence of loopholes (some hypothetical, others acknowledged) and point out that various considerations bias the experimental results in favor of quantum mechanics. According to them, the results to date are inconclusive. Generally, these arguments have not been taken as being substantive by the physics community but to be fair, a few of their arguments are mentioned below.

Fair sampling

The "fair sampling assumption" states that the sample of detected pairs is representative of the pairs emitted. The possibility of this not being true comprises the fair sampling, detection, efficiency or variable detection probability loophole (Pearle, 1970). In 2001 an experiment was conducted that used detection methods that were almost 100% efficient, thus avoiding this loophole (Rowe, 2001; Kielpinski, 2001). This experiment demonstrated violation of the CHSH inequality using two trapped ions. The setup, however, did not satisfy one of the essential criteria for Bell inequalities to apply: the two sides of the experiment were not sufficiently far separated (Vaidman, 2001). Many physicists, however, share Bell's opinion when he wrote that "it is hard for me to believe that quantum mechanics works so nicely for inefficient practical set-ups and is yet going to fail badly when sufficient refinements are made."

Enhancement

The CH74 inequality, and some others, are derived using the assumption that the presence of the analyzers never increases the probabilities of certain outcomes. If an experiment shows that these inequalities are violated then we do not have a general disproof for all local hidden variable theories, just those for which the analyzers cause no enhancement.

Subtraction of "Accidentals"

No experiment done so far has ever had its equipment absolutely perfect, as in the explanation of the set up above. In a real situation it can be that for some reason entangled photons will not reach their respective detectors at the same time, or that 'noise' in the form of stray photons will interfere. These imperfections lead to so called 'accidentals', where coincidences are detected, even though they are not an entangled pair. To try and limit the effects of these accidentals an experimenter will define a 'coincidence window', a time in which they expect coincidences to occur. Only coincidences that occur inside this window are considered, those that occur outside it will be ignored as accidentals. While this seems good experimental conduct, it has been shown (Thompson, 2003) that this approach actually unfairly biases the data in favour of quantum mechanical predictions, giving proponents of local hidden variable theories cause to complain.

Rotational Invariance

It is assumed in Bell test experiments that it is only the angles between different detector orientations, not the actual angles themselves, that are significant in the data, meaning that rotating all detectors by the same angle will have no effect. If the photons involved in the experiments prefer some polarization direction then the test is not rotationally invariant, as a rotation of both detectors will have an effect. Rotational invariance is not an assumption required to derive Bell's inequalities, so this is not a theoretical problem, but the assumption is often used to analyze data in experiments and so could cause data to suggest violations of the inequalities when really they may not.

Double Detections

If local hidden variable theories are true then it is theoretically possible that in some situations a detector could have cause to measure both +1 and -1 simultaneously. This is not possible in quantum mechanics and is also not possible due to the electronics of the detectors used. This means that the physical constraints of the detectors may bias the data towards quantum physics.

The Memory Loophole

Local hidden variable theories could be constructed that violate the inequalities if the particles involved had memory, that is if the measurement of the nth pair was affected by the n-1 pairs that preceded it. A related loophole, the simultaneous measurement loophole, states that local hidden variable theories could be constructed that would violate the inequalities if all pairs are measured simultaneously. There has been some research in the area of these loopholes (Barrett, 2002) that show how the data can be analyzed so as not to be biased towards quantum mechanics. When analyzed in this way, however, the data still suggests that quantum mechanics is correct.

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)
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)
Thompson, 2003 C.H. Thompson., Subtraction of “Accidentals” and the Validity of Bell Tests, Galilean Electrodynamics, 14, 43-50, (2003).
Weihs, 1998: G. Weihs, et al., Violation of Bell’s inequality under strict Einstein locality conditions, Phys. Rev. Lett. 81, 5039 (1998)