Bell Test Loopholes Introduction Practical Loopholes

There are two main avenues of escape from the logic of " Bell's Theorem" in quantum mechanics with its implications of " quantum entanglement" or " nonlocality". One can either argue that Bell's logic was wrong and his inequality can be infringed without implying nonlocal effects; or one may challenge the experimental evidence and the correctness of the quantum-mechanical prediction. This article takes the second approach.

The "Bell inequalities" tested in actual " Bell test experiments" are not quite the original ones that Bell proposed , nor do experimental conditions necessarily comply with the assumptions on which the inequalities depend. There are thus "loopholes" that might open the way for alternative, " local hidden variable", explanations for the observed violations of the inequalities. Some of these are described below, the best known being the "fair sampling loophole", associated with the fact that in real experiments, which have almost all used photons and not the spin-1/2 particles Bell had in mind, the detectors are not 100% efficient. They detect only a small fraction of the photons that reach them.

The various different Bell inequalities do not all involve the same assumptions and, in consequence, sometimes lead to different loopholes. It is well known that the CHSH inequality requires fair sampling, while it is perhaps less well known (though it is clear from the 1974 derivation) that the CH74 one does not. The majority of recent experiments (almost all since about 1985) have used the CHSH or related inequalities, amongst which is classed, for this purpose, the "visibility" test. The CH74 inequality requires "no enhancement" but otherwise no special assumptions. All inequalities share certain basic assumptions, for instance that paired detections can be unambiguously distinguished from unpaired ones — that there is no problem with synchronisation.

The loopholes discussed below are those that might be present in real Bell tests. That does not mean to say that in any given experiment they are actually present though, equally, it is possible that more than one may apply.

2 Practical loopholes

2.1 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. It applies to the CHSH and visibility tests unless detection efficiencies are higher than is currently feasible. It is possible to test experimentally for the sample not being fair by checking the constancy of the total coincidence counts, but the variations expected here are small. The loophole may be widespread, especially in recent tests. The principle behind it can be understood intuitively by means of the Chaotic Ball model devised by Caroline Thompson (Thompson, 1996).

The first application in which, due to use of the CHSH test, the fair sampling loophole was relevant, was Aspect's second experiment (Aspect, 1982a).

In 2001 some experiments were conducted that used detectors that were almost 100% efficient, thus avoiding this loophole (Rowe, 2001; Kielpinski, 2001). They were claimed to have demonstrated entanglement between two ions in the same linear laser trap. Though it would be satisfying to provide physical explanations for the violation of Bell tests here, the urgency is not so great: the ions were so close together that Bell's basic assumption — that the two measurements should be made on particles that are too far apart to interact by normal methods — is not met (Vaidman, 2001). All Bell tests depend on the assumption that the detections are, once the "hidden variable" values and detector settings are given, conducted independently on the two particles.

2.2 Enhancement

The CH74 and related tests are subject to the assumption that there is "no enhancement", i.e. that there is no hidden variable value for which the presence of a polariser increases the probability of detection. This assumption is considered suspect by some authors, but in practice, in the few instances in which the CH74 inequality has been used, the test has been invalidated by other more evident loopholes such as the subtraction of accidentals.

2.3 Subtraction of "accidentals"

Adjustment of the data by subtraction of "accidentals", though standard practice in many applications, can bias Bell tests in favour of quantum theory. After a period in which this fact has been ignored by some experimenters, it is now once again accepted . The reader should be aware, though, that it invalidates many published results. Notable examples in which there were large numbers of accidentals are Aspect's experiments (Aspect, 1981, 1982a,b) and the early "long-distance" Bell tests conducted in Geneva (Tittel, 1997). These experiments are discussed in (Thompson, 2003).

2.4 Failure of rotational invariance

The source is said to be "rotationally invariant" if all possible hidden variable values (describing the states of the emitted pairs) are equally likely. The general form of a Bell test does not assume rotational invariance, but a number of experiments have been analysed using a simplified formula that depends upon it. It is possible that there has not always been adequate testing to justify this. Even where, as is usually the case, the actual test applied is general, if the hidden variables are not rotationally invariant this can result in misleading descriptions of the results. Graphs may be presented, for example, of coincidence rate against the difference between the settings a and b, but if a more comprehensive set of experiments had been done it might have become clear that the rate depended on a and b separately (Thompson, 1999). Cases in point may be Weihs’ experiment (Weihs, 1998), presented as having closed the "locality" loophole, and Kwiat’s demonstration of entanglement using an "ultrabright photon source" (Kwiat, 1999).

2.5 Synchronisation problems

There is reason to think that in a few experiments bias could be caused when the coincidence window is shorter than some of the light pulses involved (Thompson, 1997). Experiments that might be affected include one of historical importance — that of Freedman and Clauser (Freedman, 1972) — which used the Freedman test (related to the CHSH one), and might not have been sullied by any of the above possibilities.

2.6 Double detections

In many experiments (for example (Weihs, 1998)) the electronics is such that simultaneous '+' and '-' counts from both outputs of a polariser can never occur, only one or the other being recorded. Under quantum mechanics, they will not occur anyway, but under a wave theory the suppression of these counts will cause even the basic realist prediction (see the " local hidden variable theory" page) to yield "unfair sampling". The effect is negligible, however, if the detection efficiencies are low.

3 A theoretical loophole: "locality"

A loophole that is notably absent from the above section is the so-called "locality" or "light-cone" one, whereby some unspecified mechanism is taken as conveying additional information between the two detectors so as to increase their correlation above the classical limit. In the view of many realists, this has never been a serious contender. Its properties would have to be quite extraordinary, as it is required to explain entanglement in a great variety of geometrical setups, including over a distance of several kilometers in the Geneva experiments of 1997-8 (Tittel, 1997-8).

John Bell supported Aspect’s investigation of it (see page 109 of (Bell, 1987)) and had some active involvement with the work, being on the examining board for his PhD. Aspect's most famous experiment (Aspect, 1982b) almost closed this loophole, though it has been criticised. The idea was to leave the choice of settings of the polarisers until after the photons had left the source, but the method used might have involved some periodicity. It was felt desirable to eliminate periodicity, which Weihs later succeeded in doing, using a random quantum process to control the choice (Weihs, 1998). (Weihs, incidentally, used the CHSH test, so his Bell test violation was subject to the fair sampling loophole.)

4 Conclusion

Apart from the fair sampling one, loopholes gain on the whole little attention, not being in themselves of any great theoretical interest. Most of the loopholes mentioned in this article are associated with practical difficulties brought up by Clauser and Horne as mere "endnotes" to their seminal 1974 paper (Clauser, 1974). Little else has been published even on the well known loopholes, however, the local realist point of view being currently out of favour. What little there is (for example ( Marshall, 1983)) is far outweighed by the literature on experimental evidence for quantum entanglement, discussions of its consequences on the assumption that experiments have proved that it really happens, and papers reporting the beginnings of practical applications.

Experimenters, however, are aware of the loopholes, and the experimental violations of Bell's inequalities would not have been so widely accepted as supporting quantum theory had it not been understood that they had made every effort to eliminate them. Nevertheless, for a realist searching for alternative explanations of the facts, or even for someone trying to apply quantum entanglement in quantum computing or quantum cryptography, it might be worth their while to ask if some loopholes might not have been as thoroughly closed as the experimenters thought — if, perhaps, quantum theory had suggested they could not be open whereas alternative theories said otherwise. The remarkable correlations achieved might yet prove to be due to local hidden variables, ordinary shared values carried from the source.

5 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/
Bell, 1987: J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, (Cambridge University Press 1987)
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, Phys. Rev. Lett. 28, 938 (1972)
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)
Thompson, 1996: C. H. Thompson, The Chaotic Ball: An Intuitive Analogy for EPR Experiments, Found. Phys. Lett. 9, 357 (1996)
Thompson, 1997: C. H. Thompson, Timing, ‘accidentals’ and other artifacts in EPR Experiments (1997)
Thompson, 1999: C. H. Thompson, Rotational invariance, phase relationships and the quantum entanglement illusion (1999)
Thompson, 2003: C. H. Thompson, Subtraction of ‘accidentals’ and the validity of Bell tests, Galilean Electrodynamics 14 (3), 43-50 (2003)
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., Long-distance Bell-type tests using energy-time entangled photons, (1998)
Vaidman: L. Vaidman, Bell's inequality: more tests are needed (2001)
Weihs, 1998: G. Weihs, et al., Violation of Bell’s inequality under strict Einstein locality conditions, Phys. Rev. Lett. 81, 5039 (1998) and private correspondence

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1 Introduction