1:
INTRODUCTION
Zeilinger and Greenberger, in their widely publicised “Petition to the American Physical Society” of April 10, 2002[1], typify the attitude of the establishment towards the supposed fact of “quantum entanglement”. They not only take it for granted that it really happens but proceed to build on the idea, implying that it is an essential feature of phenomena as diverse as quantum computing, neutron interferometry and Bose-Einstein Condensates. Yet they, as experts in the field, know that there is no firm evidence, no actual experimental result to which they can point and say: “Here we really do see entanglement.” They know that their interpretation of the experimental evidence — the observed infringements of “Bell inequalities” — rests on assumptions that they consider “plausible” but which cannot be justified scientifically. They believe that, in view of the supposed universal success of quantum mechanics (QM) in other areas, some day a perfect, “loophole-free”, experiment will be conducted, and this will prove once and for all that you cannot model separated “entangled particles” realistically — you cannot assign separate real properties (“hidden variables”) to each. It seems clear to me that they have not fully understood just how readily their assumptions can fail, or what an important role they play in the experiments.
Fig. 1: Anne, Bob and the Chaotic Ball.
The letter S is visible while N, opposite to it, is out of sight. a and b are directions in which the assistants are viewing the ball; F the angle between them.
The main purpose of the present paper is to make available to all the means to understand exactly what is involved in the best-known loophole, variously known as the “fair sampling”, “efficiency” or “variable detection probability” one. I do this by means of an analogy that is, like the loophole itself, very straightforward, demanding no knowledge of QM, no mathematics, just geometry and some common sense facts about how probabilities work. Before introducing my “Chaotic Ball” model (see fig. 1), though, I feel it necessary to say something about the effects that belief in QM and the supposed impossibility of local realist models has had on a section of the scientific community.
1.1: The consequences of belief in “entanglement”
Let me make it quite clear from the outset: I do not regard entanglement of separated particles as physically possible. I class it as one of the indications that QM is not a fully logical theory[2]. Where I appear to cast the blame for the present sorry state of affairs on a few individuals, therefore, this is not really so. I recognise that illogical decisions are the inevitable result of trying to conduct science within an illogical framework.
Ever since Alain Aspect’s experiments of 1981-2[3], people who have believed what the scientific press has told them — that we now have experimental evidence for quantum entanglement — have searched in desperation for a means to reconcile “violation of the Bell inequality” with local realism. Everyone, experts included, would like to save local realism if they can. The theorists’ approach is varied. Zeilinger, for instance, says that nothing mysterious really happens in his experiments, as it is just a matter of the natural behaviour of conditional probabilities — the effect of change of information[4]. Some argue that Bell’s logic was wrong, that he should, for example, have used Bayesian instead of traditional statistical methods[5]. Many are confused by inappropriate notation[6]. Others argue that some kind of faster-than-light signals must be being transferred between the two sides of the experiments, though quite how these could produce the observations has never been spelled out[7]. Yet others suggest a real link between the two sides, analogous to a connecting pipe[8].
One would naturally assume that “experts”, at least, have access to the full facts, but the writers of popular accounts certainly do not — which makes the task of the intelligent amateur unfairly difficult. How are they to make a rational assessment of the situation when they have been misinformed about which test has been infringed (note that Bell’s original versions have never been used), or are assured that Aspect, for instance, produced coincidence curves that covered almost the full range from 0.0 to 1.0 (they covered this range only after both subtraction of accidentals and “normalisation”)? Regarding local realist explanations, most accounts suggest just the one possibility — the basic model covering Bohm’s thought-experiment and resulting in a straight line prediction. They are not told of the curve, remarkably similar to the QM prediction, that comes from the standard local realist assumptions for the actual experiments.
The attempt to live with entanglement, combined with belief in the photon as an indivisible particle, appears also to have had grave consequences on scientific method. The QM model for the Bell test experiments does not have — and, indeed, cannot have without drastic change — enough parameters. The result is that the experimenter is free to choose certain key settings of his apparatus, those most relevant to the present article concerning the intensity of light used and the detailed specification of the photodetectors. He can choose both the make and the settings of his photodetectors as he wishes, justifying his choice by the apparently laudable aim of producing one click for each input photon. But he cannot fail to notice that his choice in fact affects the value he obtains for his Bell test statistic. QM tells him that all he is doing by choosing a different setting is changing the “quantum efficiency” — the probability of detection per photon — which ought to have no effect on the result. Rather than challenge the theory, though, the experimenter takes advantage of the resulting flexibility. He publishes just one result, not all the others obtained in preliminary runs with different detector settings or different beam intensities[9]. His aim seems to have become not to search for the best possible explanation for all his observations but to see if he can find particular conditions in which his apparatus seems to obey the QM formula.
A further publication problem, common to the whole scientific endeavor, is the failure to publish both null results and anomalous ones that do not reach “statistical significance”. There are occasions on which it is the anomalies that are critical. Clearly these should be investigated, increasing replication if necessary. A few are to be found in the PhD thesis of the most famous of the Bell test experimenters, Alain Aspect, of which more in the Section 2.2 below.
1.2: Background to the Bell tests, the “fair sampling” assumption and the Chaotic Ball model
Bell devised the original test in 1964[10]. His inequality was designed to settle experimentally a dispute that had been going on since the 1930’s, centered around Einstein, Podolsky and Rosen’s 1935 paper[11], which had attempted to clarify the consequences of acceptance of the “non-separable” formula for separated particles that was implied by the quantum formalism. The situation Bell had in mind was that discussed by Bohm[12], in which atoms that had previously been part of the same molecule separated and were detected after passage between pairs of “Stern-Gerlach” magnets. It seemed reasonable for him to assume that every atom was detected. It would be categorised either as “spin up” or “spin down” according as to which way it had been deflected.
When it came to real experiments, however, it was found that the only practical ones[13] were those involving pairs of light signals, treated as “photons”. It was recognised that these were not all detected: photodetectors only register a proportion of the input photons. There was a problem: it was not known in advance how many pairs of photons were produced by the source. Clauser, Horne, Shimony and Holt published in 1969 a paper that has been interpreted as proposing what has now become the standard test — the CHSH test. The authors did not in fact recommend it, saying that in practice a different test (effectively the version published by Clauser and Horne in 1974[14] — the CH74 test) should be used instead, and until Aspect’s first 1982 experiment this is what was used. Pearle in 1970[15] had explained just why the CHSH test was unsatisfactory: unless the detection efficiency was very high, it was possible there could be “variable detection probabilities” and these could cause a local realist model to violate it. Pearle’s paper, however, does not appear to have been widely read. It is highly mathematical, with little appeal to the intuition.
The main features of the two tests are set out in Appendix A. The critical difference is that the CHSH test relies on “fair sampling”, whilst the CH74 one does not. The latter requires only the relatively innocuous assumption of “no enhancement” — the presence of a polariser does not, for any hidden variable value, increase the probability of detection.
One may well ask why the community reverted after 1982 to the CHSH test, having previously rejected it. Aspect considered it to be closer to Bell’s original than the CH74 one, the latter using only one of the two possible outputs from the polarisers and requiring extra experimental runs with polarisers absent, but this is not sufficient reason. The rejected test can be derived independently, is equally valid and is, in view of its non-dependency on fair sampling, in most situations superior. A possible reason, however, has recently come to my attention. There exist alternative derivations of the CH74 inequality. It can be derived in a way that parallels that of the CHSH one, copying a whole group of assumptions that depend on fair sampling. Aspect had seen one of these, and perhaps thought it the only one. He thought (and apparently continues to think[16]) that the CH74 test is not only a departure from Bell’s intentions but, if possible, subject to even more potential bias than the CHSH one. Where Aspect has led, others have followed.
Again unfortunately so far as the pursuit of truth is concerned, the community has become convinced that it is not possible to test for fair sampling. In fact it is possible to do considerably more in this direction than has become customary. Aspect is among those who did perform the minimum test (see later), looking for constancy of the observed total coincidence count, but after finding (as reported in his PhD thesis[17]) that there were slight variations, he did not decide to abandon the Bell test. Instead he devised a further modification that he thought would correct for any bias. Nobody appears to have checked his assumptions here. His test may not have been correct.
Later workers appear to have followed Aspect’s example without much question, assuming the constancy of the total counts without necessarily fully testing it. As will be shown, the greatest variation is expected to be between detector settings midway between those used in the Bell tests. They are frequently not even investigated. A test of constancy for just the “Bell test angles” is not a test of fair sampling at all: the coincidence rates are expected in local realist models as well as quantum theory to all be equal, by symmetry.
Though variations in total counts are small and so perhaps difficult to establish, a recent paper by Adenier and Khrennikov[18] suggests a related test, using a straightforward subsidiary experiment, that should demonstrate more clearly the failure of the fair sampling assumption for real optical Bell tests.
The Chaotic Ball model presented here was designed in 1994 in order to move the above discussion to an intuitive level. The model does not pretend to represent any real experiment, only the principle involved. For the real experiments, it is better to work with algebraic models, introduced briefly later (see Section 2.3). This is because of the geometrical differences between spins (for which it is natural to assume each represented by a vector in three dimensions) and polarisation, which is defined in just two dimensions, with the directions diametrically opposite being equivalent. For the full generality of a local realist model it is best to use computer simulation, modelling each event as it happens[19].
1.3: Other Bell test loopholes
Other loopholes are covered briefly below, but for more information the reader is referred to papers available at http://arxiv.org/abs/quant-ph/ (9711044, 9903066, 9912082 and 0210150). The second of these covers the matter of “subtraction of accidentals”, which can be shown to be of crucial importance in certain experiments. The background to this is covered informally in a paper published in Accountability in Research[20].
2: THE “CHAOTIC BALL”
Much of the following material has been available electronically for some time, at http://arxiv.org/abs/quant-ph/0210150. The reader is reminded that the ball model as it stands corresponds to experiments that have never actually been done. It illustrates a principle only.
Let
us consider Bohm’s thought experiment, commonly taken as the standard example
of the entanglement conundrum that Einstein, Podolsky and Rosen discussed in
their seminal 1935 paper. A molecule is
assumed to split into two atoms, A and B, of opposite spin, that
separate in opposite directions. They
are sent to pairs of “Stern-Gerlach” magnets, whose orientations can be chosen
by the experimenter, and counts taken of the various “coincidences” of spin
“up” and spin “down”. The obvious
“realist” assumption is that each atom leaves the source with its own well-defined
spin (a vector pointing in any direction), and it is the fact that the spins
are opposite that accounts for the observed coincidence pattern. (The realist notion of spin cannot be the
same as the quantum theory one, since in quantum theory “up” and “down” are
concepts defined with respect to the magnet orientations, which can be
varied. Under quantum mechanics, the
particles exist in a superposition of up and down states until measured.)
Bell’s
original inequality was designed to apply to the estimated “quantum correlation[21]”
between the particles. He proved that
the realist assumption, based on the premise that the detection events for a
given pair of particles are independent, leads to statistical limits on this
correlation that are exceeded by the QM prediction. However, as mentioned above, his inequality depended on the
assumption that all particles were detected.
When
detection is perfect there is no problem, but when it is not, the “detection
loophole” creeps in. What assumptions
can we reasonably make? Under quantum
theory, the most natural one is that all emitted particles have an equal chance
of non-detection (the sample detected is “fair”, not varying with the settings
of the detectors). The realist picture,
however, is different.
Let
us replace the detectors by two assistants, Anne (A) and Bob (B), the source of
particles by a large ball on which are marked, at opposite points on the
surface, an N and an S (fig. 1).
The assistants look at the ball, which turns randomly about its centre
(the term “chaotic”, though bearing little relation to the modern use of the
term, is retained for historical reasons).
They record, at agreed times, whether they see an N or an S. When sufficient records have been made they
get together and compile a list of the coincidences — the numbers of
occurrences of NN, SS, NS and SN, where the first
letter is Anne’s and the second Bob’s observation.
The
astute reader will notice that, if the vector from S to N
corresponds to the “spin” of the atom, the model covers the case in which the
spins on the A and B sides are identical, not
opposite. Anne and Bob are looking at
identical copies of the ball, which can conveniently be represented as a single
one. This simplification aids visualisation whilst having no significant effect
on the logic. The difference
mathematically is just a matter of change of sign, with no effect on numerical
values. In point of fact, the
assumption of identical spins makes the model better suited to some of the
actual optical experiments. Aspect’s,
for example, involved plane-polarised “photons” (not, incidentally, circularly
polarised, as frequently reported[22])
with parallel, not orthogonal, polarisation directions.
Fig. 2: The registered coincidences: Chaotic Ball with perfect detectors.
The first letter of each pair denotes what Anne records, the second Bob, when the S is in the region indicated.
With
this simplification, geometry dictates that if the ball takes up all possible
orientations with equal frequency (there is “rotational invariance”) then the
relative frequencies of the four different coincidence types will correspond to
four areas on the surface of an abstract fixed sphere as shown in fig. 2.
Anne’s
observations correspond to two hemispheres, Bob’s to a different pair, the
dividing circles being determined by the positions of the assistants. We conduct a series of experiments, each
with fixed lines of sight (“detector settings”) a and b. It can readily be verified that the model
will reproduce the standard “deterministic local realist” prediction, with
linear relationship between the number of coincidences and F, the angle between the settings[23]. This is shown in fig. 3, which also shows
the quantum mechanical prediction, a sine curve.
Fig. 3: Predicted coincidence
curves.
The
straight line gives the local realist prediction for the probability that both
Anne and Bob see an S, if there are no missing bands; the curve is the QM
prediction, ½ cos2 (F/2).
What
happens, though, if the assistants do not both make a record at every agreed
time? If the only reason they miss a
record is that they are very easily distracted, this poses little problem. So long as the probability of non-detection
can be taken to be random, the expected pattern of coincidences will remain
unaltered. What if the reason for the
missing record varies with the orientation of the a ball, though — with the
“hidden variable”, l, the vector from S to N?
Fig. 4: Chaotic Ball with missing bands.
There is no coincidence unless both assistants make a record, so some data is thrown away.
Suppose
the ball is so large that the assistants cannot see the whole of the hemisphere
nearest to them. The picture changes to
that shown in fig. 4, in which the shaded areas represent the regions in which,
when occupied by the S, coincidences will be recorded as indicated. The ratios between the areas, which are what
matter in Bell tests, change — indeed, some areas may disappear
altogether. If the bands are very
large, there will be certain positions of the assistants for which the
estimated quantum correlation (E, equation (1) below) is not even
defined, since there are no coincidences.
New
decisions are required. Whereas before
it was clear that if we wanted to normalise our coincidence rates we would
divide by the total number of observations, which would correspond to the area
of the whole surface, there is now a temptation to divide instead by the total
shaded area. The former is correct if
we want the proportion of coincidences to emitted pairs, but it is,
regrettably, the latter that has been chosen in actual Bell test
experiments. It is easily shown that
the model will now inevitably, for a range of parameter choices, infringe the
relevant Bell test if our estimates of “quantum correlation” are the usual
ones, namely,
, (1)
(2)
Fig. 5: Predicted quantum correlation, E, versus angle.
The
curve corresponds to (moderate-sized) missing bands, the straight line to
none. See Appendix B for the formula for the central section of
the curve.
If
we do have missing bands, however, although the four terms are still all equal
and all positive, each will have increased!
The Bell test will be infringed.
2. 1: Detailed Predictions for the basic
model
For
the case of “hard-edged” symmetrical missing bands, the predictions can be
given exactly for any choice of missing band width and angle between
detectors. The formula for the
coincidence rate PSS (fig. 6) is given in Appendix B, though
qualitative predictions can be made just by inspection of diagrams such as fig.
4. The graphs below show the results
when the missing bands subtend an angle of 30º at the centre of the ball,
corresponding to b = 75° = 5p/12 in the notation of the
appendix.
Fig.
6: Predicted coincidence rate, PSS.
Note
that the curve is actually zero for certain angles. This has the interesting consequence that under certain
conditions the quantum correlation is not even defined. These correspond to cases in which Anne and
Bob are very close to the ball so that each sees only a small circle. Their circles may not overlap: they may
score no coincidences. In real
experiments this would never quite be seen to happen, since there are always
“dark counts”, but no useful Bell test could be conducted: the variance of the
statistic would be too large.
Fig. 7: Total coincidence rate, Tobs/N.
The
total coincidence rate, Tobs /N = (NN + SS + NS + SN)/N
, for this model is illustrated in fig. 7.
The fact that it is not constant provides a useful, though not quite
conclusive, test for unfair sampling — the presence of something equivalent to
our missing bands. In real situations,
in which there are no hard edges to the bands but a gradation from white to
black, the curve will be smoother and the contrast between maximum and minimum
perhaps not so great, but if the “detection loophole” is in operation and is
causing infringements of the CHSH inequality, some difference between the total
at 0º and that at 90º should be present and detectable. It is important to notice, though, that no
difference is to be expected between the “Bell test angles”, 45º and 135º —
a fact that can be deduced from Pearle’s paper of 1970 but which seems now to
have been forgotten. (See also Section
2.2 below.)
We
can derive the expected value of the ordinary “normalised” quantum correlation
(1) in which division is by NN + SS + NS + SN , with results as shown in
fig. 5, but it is of interest to look also at the “un-normalised” one, (NN +
SS – NS – SN)/N , with expected value:
PNN + PSS –
PNS – PSN ,
plotted
in fig. 8.
Fig. 8: Un-normalised "quantum
correlation".
The
match with the QM prediction is considerably less impressive, the curve not
reaching the maximum of 1 and not having the feature of a zero slope for
parallel detectors. Whilst for the
chosen example (missing bands subtending 30°) the model gives the
CHSH test statistic of S = 3.331 > 2, the un-normalised estimate will never
exceed 2 because the values at the “Bell test angles” will always all be
numerically less than 0.5. Clearly (NN
+ SS – NS – SN)/N is an unbiased estimate of the quantum correlation; the usual
expression, (NN + SS – NS – SN)/(NN + SS + NS + SN) is not. Bell’s inequality assumes the use of
unbiased estimates.
2.2: Discussion
Is
the kind of missing band effect modelled by the Chaotic Ball likely to occur in
real experiments? The answer is,
effectively, “Yes.” If an experiment
using Stern-Gerlach magnets and spin-1/2 particles were ever to be possible,
then perhaps all particles would be detected so the problem would not arise —
though if there were to be any
non-detections, is it not likely that they would occur mostly for those
particles whose spin was almost orthogonal to the direction determined by the
magnets, so that it was not clear in which direction they “should” be
deflected?
The
vast majority of real experiments to date, though, have used light, with the
direction of plane polarisation used in place of spin. In these, the sampling will be biased unless
Malus’ Law (that intensity is proportional to cos2 l)
is obeyed exactly, for all intensities of input signal. (Note that I am assuming that individual
“photons” are really classical pulses of light, and these can vary in intensity.) The situation that gives rise to effectively
missing bands and hence to high values of the CHSH test statistic is one in
which the probabilities of detection are (whether because of the behaviour at
the polariser or at the detector) lower than given by Malus’ Law for angles of
polarisation of around 45º and less.
Little or no effort seems to be made
by most experimenters to check the true operating characteristics of
their apparatus, which is, in practice, not perfect. Whilst in most context imperfections can simply be accepted, in
this case, where they can bias the results, the true characteristics need to be
built into the model.
Is
enough done to test for the presence of missing bands (or regions of the hidden
variable space that have less than the full probability of detection)? The answer would seem to be that, as
mentioned in the introductory sections, in recent experiments understanding of
what is needed has somehow become lost.
Though, for example, Fattal et al.[25]
check that the total coincidence count is constant for their Bell test angles,
they do not report looking at any others.
From fig. 8 above it is immediately clear that this is not enough: the
total for the Bell test angles is not expected to vary. For a test of constancy of total coincidence
count to be of any value, other angles — preferably 0 and 90º (0 and 45º in
polarisation experiments) — must be included.
Note that testing for constancy of the singles rates is by no means
sufficient: it is constant for the ball model with missing bands, yet the total
coincidence count varies.
The
CHSH test must, I think, be taken to be unreliable, probably always biased
towards the quantum theory prediction.
As already mentioned, Clauser and Horne devised a test (the CH74 test,
see Appendix A) that is, so long as there is no “enhancement”, not
intrinsically biased. It is, in my view
unsurprisingly, not so readily violated.
Most early experiments did violate it so some extent, but not by the
margins achieved for the CHSH test, the reasons for violation lying, I believe,
in more subtle loopholes such as synchronisation problems (see below). The fact that this alternative test is
available and less prone to bias is yet another truth that seems to have been
lost. I have already suggested a
possible reason: perhaps Alain Aspect’s belief is typical. He states16 that the CH74 test
depends on the fair sampling assumption.
This is not in fact true, or at least, not in a way that directly causes
bias. It is clear from the referenced paper
that his belief comes from use of a derivation that does depend on fair
sampling. Clauser and Horne’s
derivation, however, does not. It
follows that the test itself does not.
2.3: Generalising the ball model
As
it stands, the ball model explains the fact that Aspect observed slight
variations in the total coincidence count.
It can readily be generalised to explain another “anomaly” mentioned in
his PhD thesis in relation to his two-channel experiment, namely the fact that
his counts equivalent to my NS and SN were not quite
equal. There was a small difference,
not quite reaching “statistical significance”.
This can be explained if we allow for two asymmetries in his actual
setup: the fact that his polarisers did not split exactly 50-50 and the fact
that the “photons” on the two sides were of different wavelengths, requiring
different photodetectors whose characteristics could not be expected to be
identical. The situation thus
corresponds, in principle, to one in which the missing bands for Anne and Bob
are of unequal width and are not centralised.
To
achieve a better fit with the QM prediction, we can make the edges of the ball
“fuzzy”, so that the probability of detection varies gradually from 0 to
1. There is no purpose, however, in
carrying this out in detail, since what is really needed is a model for the
two-dimensional case with equivalence between opposite points that corresponds
to the real optical experiments.
We
turn instead to the general realist model, coming directly from Bell’s
assumptions so long as there are no problems with synchronisation etc.. The coincidence probability is:
( 3 )
where the
integration is performed over the complete “hidden variable” space spanned by l, the
weighting factor, r, represents the relative frequencies
of the different “states”, l, of the source. pA
and pB are the probabilities of detection,
given the detector setting (a or b) and l.
The simplest assumption is that pA and pB are cos2 (a – l) and cos2 (b – l) respectively, reflecting adherence to Malus’ law on passage through the polariser, together with “perfect” detectors, the probability of detection being exactly proportional to the input intensity. r is constant if we have “rotational invariance” of the source. The predicted quantum correlation for this case is shown in fig. 9. Marshall et al.[26] have shown in their article of 1983 how replacing the cosine-squared terms by rather more general expressions can produce realist predictions arbitrarily close to the quantum theory curve.
Fig. 9: Local realist prediction for “quantum correlation” for (perfect) optical Bell tests
The full curve is the realist prediction, the dotted curve the QM one.
For
yet further generality, covering problems such as the presence of accidentals
or matters to do with synchronisation (see next section), recourse to computer
simulation, taking full account of the specific experimental details, is likely
to be needed. (See endnote 19.) The principle is always the same, and always
straightforward. There should be no
need to call in an expert.
3: OTHER LOOPHOLES
The
detection loophole is, at least among professionals, well known, but the fact that
it affects some versions of Bell’s test and not others is perhaps less well
understood. Different loopholes apply
to different versions, for each version comes with its attendant assumptions. Some come very much under the heading of
“experimental detail” and have, as such, little interest to the
theoretician. If we wish to decide on
the value to be placed on a Bell test, however, such details cannot be ignored.
I. 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[27]. The reader should be aware, though, that it
invalidates many published results[28].
II. Failure of
rotational invariance: 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, i.e. if some values are favoured more than others, this can
result in misleading descriptions of the results. Graphs may be presented, for example, of coincidence rate against
j, 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[29]. Cases in point may be Weihs’ experiment,
presented as having closed the “locality” loophole[30],
and Kwiat’s demonstration of entanglement using an “ultrabright photon source[31]”.
III. 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[32]. These include one of historical importance —
that of Freedman and Clauser, in 1972[33]
— which used a test not sullied by either of the above possibilities.
IV. “Enhancement”: Tests such as that used by Freedman and
Clauser (essentially the CH74 test) 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, notably Marshall and Santos, 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.
5. Asymmetry: Whilst not
necessarily invalidating Bell tests, the presence of asymmetry (for instance,
the different frequencies of the light on the two sides of Aspect’s experiments)
increases the options for local realist models[34].
A loophole that is notably absent from the above list is the so-called “timing”, “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. John Bell supported Aspect’s investigation of it (see page 109 of Speakable and Unspeakable[35]) and had some active involvement with the work, being on the examining board for Aspect’s PhD. Weihs improved upon the test in his experiment of 199830, but nobody has ever put forward plausible ideas for the mechanism. 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 1997-824,28.
There
may well be yet more loopholes. For
instance, 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 QM, 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, since the three- or four-fold coincidences
involved (two on one side, one or more on the other) then hardly ever happen.
4: CONCLUSION
The “Chaotic Ball” models a hypothetical Bell test experiment in a manner that encourages the use of intuition and realism. It illustrates the fact, well known to those working in the field, that if not all particles are detected there is risk of bias in the standard tests used, which are no longer able to discriminate between the nonseparable quantum-mechanical model and local realism. Knowledge of an alternative test, and the fact that this test does not suffer from the same bias, appears to have been lost, as has understanding of a reasonable check that could at least indicate when the observed coincidences are not a fair sample.
“Any theory will account for some facts; but only the true explanation will satisfy all the conditions of the problem …” (William Crookes, 1875[39])
ACKNOWLEDGEMENTS
Thanks are due to Franck Laloë for encouragement to air again the Chaotic Ball model. The work would not have been completed without the moral support of David Falla and Horst Holstein of the University of Wales, Aberystwyth, and of the many who have expressed appreciation of my web site or contributions to Internet discussions. My use, from 1993 to 2003, of the computer and library facilities at Aberystwyth was by courtesy of the Department of Computer Science, of which I was an associate member.
APPENDIX A: COMPARISON OF CHSH AND CH74 TESTS
|
|
CHSH |
CH74 |
|
Source |
Attributed to CHSH 1969 paper. Never in fact supported by authors. |
Best derivation: appendix to CH74 paper (reproduced in
quant-ph/9903066). |
|
Experimental design |
Two-channel[40]:
|
Single-channel:
|
|
|
– 2 £ S £ 2 |
S < 0 |
|
Formula |
where
and
|
where
and
the symbol ¥ indicating absence of polariser[41] |
|
Used |
Variants of this and the related “visibility” test have been used in the majority of experiments since 1982. |
Variants were used in all experiments up to 1982. |
|
Advantages |
Relatively easy to violate. |
Does not depend on fair sampling. |
|
Disadvantages |
Depends on the fair sampling assumption, which implies among other things:
|
Assumes “no enhancement” Hard to violate. |
, 
APPENDIX B: CALCULATED PREDICTIONS OF THE CHAOTIC BALL MODEL
Fig. B1: Definition of
angles used in equation (B1).
The
main formula for the proportion PSS of “like” coincidences such as SS with respect to the
number of emitted pairs N comes from the area of overlap of two circles
on the surface of a sphere (see figs. 4 and B1). The result, as calculated by H. Holstein[42],
is:
(B1)
where
a = j/2 and b is
the half-angle defining the proportion of the surface for which each assistant
makes a definite reading (zero corresponds to none; p/2
to the whole surface). PSS
achieves a maximum of ½ (1 – cos b) when a = 0,
which is less than the QM prediction of 0.5 unless b is p/2. When a ³ b, it is zero (see fig. 6
of main text).
APPENDIX C: POLICY STATEMENT
Attempts at publishing the core of the current paper in American Physical Society journals have failed due to application of the following editorial policy statement:
In 1964, John Bell proved that local realistic theories led to an upper bound on correlations between distant events (Bell's inequality) and that quantum mechanics had predictions that violated that inequality. Ten years later, experimenters started to test in the laboratory the violation of Bell's inequality (or similar predictions of local realism). No experiment is perfect, and various authors invented "loopholes" such that the experiments were still compatible with local realism. Of course nobody proposed a local realistic theory that would reproduce quantitative predictions of quantum theory (energy levels, transition rates, etc.).
This loophole hunting has no interest whatsoever in physics. It tells us nothing on the properties of nature. It makes no prediction that can be tested in new experiments. Therefore I recommend not to publish such papers in Physical Review A. Perhaps they might be suitable for a journal on the philosophy of science.
The above attitude has also caused failure of another important paper, that on the Subtraction of Accidentals”.
Despite my protestations that the loopholes are there to be discovered, not “invented”; that it is unreasonable to expect a paper that explains the Bell test results — essentially a matter of logic and experimental method — also to discuss energy levels and transition rates; that my ideas do lead to new physics (in that they give new reason to replace the photon model of light by a wave model); that they do make testable predictions; and that it is not philosophers of science who need to know about them but experimenters and theorists, there seems never to have been any chance of acceptance of my submissions.
I am not alone in this experience. Though I should be the first to admit that most so-called “realist” papers on the subject fully deserve rejection, to hold rigidly to the above policy statement cannot be in the long-term interests of physics.