# Loopholes in Bell test experiments

In Bell test experiments, there may be experimental problems that affect the validity of the experimental findings. The term "Loopholes" is frequently used to denote these problems. See the page on Bell's theorem for the theoretical background to these experimental efforts (see also J. S. Bell 1928-1990). The purpose of the experiment is to test whether nature is best described using a Local hidden variable theory or by the quantum entanglement hypothesis of Quantum mechanics.

The "fair sampling" or "efficiency" problem is the most prominent problem, and affects all experiments performed to date save one (Rowe et al, 2001). This problem was noted first by Pearle in 1970, and Clauser and Horne (1974) devised another result intended to take care of this. Some results were also obtained in the 1980s but the subject has undergone significant research in recent years. The many experiments affected by this problem deal with it without exception by using the "fair sampling" assumption. More on this below.

In some experiments there also may be other possibilities that make "local realist" explanations of Bell test violations possible, these are briefly described below. Each needs to be checked for and screened out before an experiment can be said to rule out local realism, and at least in modern setups, the experimenters do their best to reduce these problems to a minimum.

Many modern experiments are directed at detecting quantum entanglement rather than ruling out Local hidden variable theories, and that task is different since one accepts quantum mechanics at the outset (no entanglement without Quantum mechanics). This is regularly done using Bell's theorem, but in this situation the theorem is used as an Entanglement witness, a dividing line between entangled quantum states and separable quantum states and is then, as such, not as sensitive to the problems described here.

ources of error in (optical) Bell test experiments

In the case of Bell test experiments, if there are sources of error (that are not accounted for by the experimentalists) that might be of enough importance to explain why a particular experiment gives results in favor of quantum entanglement as opposed to local realism, they are called "loopholes." Here some examples of existing and hypothetical experimental errors are explained. There are of course sources of error in all physical experiments. Whether or not any of those presented here have been found important enough to be called loopholes, in general or because of possible mistakes by the performers of some known experiment found in literature, is discussed in the subsequent sections. There are also non-optical Bell test experiments, which are not discussed here.

Example of typical experiment

As a basis for our description of experimental errors let us consider a typical experiment of CHSH type (see picture to the right). In the experiment the source is assumed to emit light in the form of pairs of particle-like photons with each photon sent off in opposite directions. When photons are detected simultaneously (in reality during the same short time interval) at both sides of the "coincidence monitor" a coincident detection is counted. On each side of the "coincidence monitor" there are two inputs that are here named the "+" and the "-" input. The individual photons must (according to quantum mechanics) make a choice and go one way or the other at a two-channel polarizer. For each pair emitted at the source ideally either the "+" or the "-" input on both sides will detect a photon. The four possibilities can be categorized as '++', '+−', '−+' and '−−' and the number of simultaneous detections of all four types (N++, N+-, N-+, N--) is counted over a timespan covering a number of emissions from the source. Then the following is calculated:

(1) E = (N++ + N-- − N+- − N-+)/(N++ + N-- + N+- + N-+).

This is done with polarizer a rotated into two positions that we could call "a" and "a'" and polarizer b into two positions that we could call "b" and "b'" so we get E(a,b),E(a,b'),E(a',b) and E(a',b'). Then the following is calculated:

(2) S = E(a, b) − E(a, b′) + E(a′, b) + E(a′ b′)

Entanglement and local realism give different predicted values on S, thus the experiment (if there are no substantial sources of error) gives an indication to which of the two theories better correspond to reality.

ources of error in the light source

The principal possible errors in the light source are:
*Failure of rotational invariance: The light from the source might have a preferred polarization direction, in which case it is not rotationally invariant.
*Multiple emissions: The light source might emit several pairs at the same time or within a short timespan causing error at detection.

ources of error in the optical polarizer

*Imperfections in the polarizer: The polarizer might influence the relative amplitude or other aspects of reflected and transmitted light in various ways.

ources of error in the detector or detector settings

*The experiment may be set up as not being able to detect photons simultaneously in the "+" and "-" input on the same side of the experiment. If the source may emit more than one pair of photons at any one instant in time or close in time after one another, for example, this could cause errors in the detection.
*Imperfections in the detector: failing to detect some photons or detecting photons even when the light source is turned off (noise).

Detection efficiency loophole and the fair sampling assumption

In Bell test experiments one problem is that detection efficiency may be less than 100%, and this is always the case in optical experiments. This changes the inequalities to be used, for example the CHSH inequality:

$-2 le E\left(a,b\right)-E\left(a,b\text{'}\right)+E\left(a\text{'},b\right)+E\left(a\text{'},b\text{'}\right)le 2$

When data from an experiment is used in the inequality one needs to condition on that a "coincidence" occurred, that a detection occurred in both wings of the experiment. This will change the inequality into

In this formula, the $eta$ denotes the efficiency of the experiment, formally the minimum probability of a coincidence given a detection on one side (Garg & Mermin, 1987; Larsson 1998). In Quantum mechanics, the left-hand side reaches $2sqrt\left\{2\right\}$, which is greater than two, but for a non-100% efficiency the latter formula has a larger right-hand side. And at low efficiency (below $2\left(sqrt\left\{2\right\}-1\right)$≈82%), the inequality is no longer violated.

Usually, the "fair sampling assumption" is used in this situation (alternatively, the "no-enhancement assumption"). It states that the sample of detected pairs is representative of the pairs emitted, in which case the right-hand side above is reduced to 2, irrespective of the efficiency. Please note that this comprises a third postulate necessary for violation in low-efficiency experiments, in addition to the (two) postulates of Local Realism. There is unfortunately no way to test experimentally whether a given experiment does fair sampling, so it is really an "assumption" if a very natural one.

There are tests that are not sensitive to this problem, such as the Clauser-Horne test, but these have the same performance as the latter of the two inequalities above; they cannot be violated unless the efficiency exceeds a certain bound. For example, in the Clauser-Horne test, the bound is ⅔≈67% (Eberhard, 199X; Larsson, 2000).

With only one exception, all Bell test experiments to date are affected by this problem, and a typical optical experiment has around 5-30% efficiency. The bounds are actively pursued at the moment (2006). The exception to the rule, the Rowe et al (2001) experiment is performed using two ions rather than photons, and had 100% efficiency. Unfortunately, it has its own problems, see below.Fact|date=May 2008

Other loopholes

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. 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).

Double detections

In many experiments 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 to yield “unfair sampling”. The effect is negligible, however, if the detection efficiencies are low.

Locality

Another problem is the so-called “locality” or “light-cone” loophole. The Bell inequality is motivated by the absence of communication between the two measurement sites. In experiments, this is usually ensured simply by prohibiting "any" light-speed communication by separating the two sites and then ensuring that the measurement duration is shorter than the time it would take for any light-speed signal from one site to the other, or indeed, to the source. An experiment that does not do this cannot test Local Realism, for obvious reasons. Note that the needed mechanism would necessarily be outside Quantum Mechanics, and needs to explain “entanglement” in a great variety of geometrical setups, over distances of several kilometers, and between a variety of systems.

There are, so far, not so many experiments that really rule out the locality loophole. John Bell supported Aspect’s investigation of it (Bell, 1987b, p. 109) and had some active involvement with the work, being on the examining board for Aspect’s PhD. Aspect improved the separation of the sites and did the first attempt on really having independent random detector orientations. Weihs et al improved on this with a distance on the order of a few hundred meters in their experiment in addition to using random settings retrieved from a quantum system. This remains the best attempt to date.

This loophole is more hypothetical than the other possible loopholes in that there are no known physical mechanisms that could cause a problem due to locality.

uperdeterminism

Even if all experimental loopholes are closed, there is still a theoretical loophole that may allow the construction of a local realist theory that agrees with experiment. Bell's Theorem assumes that the polarizer settings can be chosen independently of any local hidden variable that determines the detection probabilities. But if both the polarizer settings and the experimental outcome are determined by a variable in their common past, the observed detection rates could be produced without information travelling faster than light (Bell, 1987a). Bell has referred to this possibility as "superdeterminism" (Bell, 1985).

References

* Bell, 1987a: J. S. Bell, "Free variables and local causality," Epistemological Letters, Feb. 1977. Reprinted as Chapter 12 of J. S. Bell, "Speakable and Unspeakable in Quantum Mechanics", (Cambridge University Press 1987)
* Bell, 1987b: J. S. Bell, "Atomic-cascade photons and quantum-mechanical nonlocality". Reprinted as Chapter 13 of 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)
* Kwiat, 1999: P.G. Kwiat, et al., [http://arXiv.org/abs/quant-ph/9810003 "Ultrabright source of polarization-entangled photons"] , Physical Review A 60 (2), R773-R776 (1999)
* P Pearle, "Hidden-Variable Example Based upon Data Rejection", Phys. Rev. D, 2, 1418-25 (1970)
* Thompson, 1996: C. H. Thompson, [http://arxiv.org/abs/quant-ph/9611037 "The Chaotic Ball: An Intuitive Analogy for EPR Experiments"] , Found. Phys. Lett. 9, 357 (1996)
* Tittel, 1997: W. Tittel et al., [http://arxiv.org/abs/quant-ph/9707042 "Experimental demonstration of quantum-correlations over more than 10 kilometers"] , Phys. Rev. A, 57, 3229 (1997)
* Tittel, 1999: W. Tittel et al., [http://arxiv.org/abs/quant-ph/9809025 "Long-distance Bell-type tests using energy-time entangled photons"] , Phys. Rev. A, 59, 4150 (1999)
* Weihs, 1998: G. Weihs, et al., [http://arXiv.org/abs/quant-ph/9810080 "Violation of Bell’s inequality under strict Einstein locality conditions"] , Phys. Rev. Lett. 81, 5039 (1998)
* To be added properly: Garg & Mermin (PRD 1987?), Larsson (PRA 1998), ...

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