Setting the Record Straight on Quantum Entanglement
Caroline H Thompson*
(Started March 16, 2004)
Abstract
All too often, in serious as well as popular articles, we see statements to the effect that “quantum
entanglement” of separated particles is an experimentally established fact and the world cannot be
explained using the methods of “local realism”. The experts, however, know full well that there are
“loopholes” in the experiments concerned — the “Bell tests”. What they do not appear to know is just
how serious the loopholes are, how readily they allow for alternative “local realist” explanations for the
actual observations, or how to conduct valid tests for their presence. I set out, with the aid of an intuitive
model, to explain how the “fair sampling” loophole works and introduce briefly some other less well
known ones.
Early attempts at exploiting loopholes in order to produce “realist” explanations for the observations
were mostly in fact unrealistic, created in ignorance of certain important experimental facts. Prestigious
journals have been driven in desperation to reject all papers on the Bell test loopholes as being of no
interest to physics. The situation has now changed, and ought to be of interest to physics, since new
realist work has two vital consequences: it restores the principle of local causality, necessary for the
rational conduct of science, and redeems the wave model of light, which has been quite unnecessarily
replaced by the “photon” model. There is also a practical consideration: further development of
applications that are currently claimed as involving quantum entanglement will be easier if it is admitted
that they do not do so, any “success” they achieve being in fact due to ordinary correlations — shared
values set at the source.
1: INTRODUCTION
Zeilinger and Greenberger, in their widely publicised
“Petition to the American Physical Society” of April 10,
20021, 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;  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.
Email: ch.thompson1@virgin.net; Web site: http://freespace.virgin.net/ch.thompson1/
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Setting the Record Straight
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 theory2. 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 1981-23, 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 information4.
Some argue that Bell’s logic was wrong, that he should,
for example, have used Bayesian instead of traditional
statistical methods5. Many are confused by inappropriate
notation6. Others argue that some kind of faster-thanlight 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 pipe8.
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 thoughtexperiment 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
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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
intensities9. 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 196410. 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 paper11, which had attempted
to clarify the consequences of acceptance of the “nonseparable” formula for separated particles that was
implied by the quantum formalism. The situation Bell
had in mind was that discussed by Bohm12, 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 ones13 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 197414 — the CH74 test) should be used instead, and
until Aspect’s first 1982 experiment this is what was
used. Pearle in 197015 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
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Setting the Record Straight
not appear to have been widely read. It is highly
mathematical, with little appeal to the intuition.
demonstrate more clearly the failure of the fair sampling
assumption for real optical Bell tests.
The main features of the CHSH and CH74 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.
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 happens19.
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 think16) 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
Khrennikov18 suggests a related test, using a
straightforward subsidiary experiment, that should
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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 Research20.
2: THE “CHAOTIC BALL”
Much of the following material has been available
electronically for some time, at http://arxiv.org/abs/quantph/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 correlation21” 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.
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Setting the Record Straight
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 planepolarised “photons” (not, incidentally, circularly
polarised, as frequently reported22) with parallel, not
orthogonal, polarisation directions.
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.
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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.
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 ,
the angle between the settings23. 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 (/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 nondetection 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”, , the vector
from S to N?
4
Setting the Record Straight
difference,  = b – a, which is 45º for three of the terms
and 135º for the fourth. We can therefore immediately
read off the required values from a graph such as that of
fig. 5, where the curve is calculated from the geometry of
fig. 4 (see next section and Appendix B).
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,
E ( a, b) 
NN  SS  NS  SN
,
NN  SS  NS  SN
(1)
where the terms NN etc. stand for counts of coincidences
in a self-evident manner.
The Bell test in question is the CHSH test referred to
above. It takes the form –2  S  2, where the test
statistic is
S  E (a, b)  E (a, b' )  E (a' , b)  E (a' , b' ). (2)
The parameters a, a, b and b are the detector settings: to
evaluate the four terms four separate sub-experiments are
needed. The settings chosen for the Bell test are those
that produce the greatest difference between the QM and
standard local realist predictions, namely a = 0, a = 90º, b
= 45º and b = 135º. Since we are assuming rotational
invariance, the value of E does not depend on the
individual values of the parameters but on their
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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.
When there are no missing bands it is clear that the
numerical value of each term is 0.5 and that they are all
positive. Thus with no missing bands the model shows
that we have exact equality, with S actually equalling 2.
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.
An “imperfection” has increased the correlation, in
contradiction to the opinion, voiced among others by Bell
himself, that imperfections are unlikely ever to do this.
It is not hard to imagine real situations, especially in the
optical experiments, in which something like “missing
bands” will occur, biasing this version of Bell’s test in
favour of quantum mechanics. Note that the “visibility”
test used in recent experiments such as Tittel’s longdistance Bell tests24 is equally unsatisfactory, biased from
this same cause. The realist upper limit on the standard
test statistic when there is imperfect detection is 4, not 2,
well above the quantum-mechanical one of 22  2.8.
The visibility can be as high as 1, not limited to the
maximum of 0.5 that follows from the commonlyaccepted assumptions.
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
 = 75° = 5/12 in the notation of the appendix.
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Setting the Record Straight
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. 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. 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
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,
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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 ) 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.
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Setting the Record Straight
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.
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:
P(a, b)   d  ( ) p A (a,  ) p B (b,  )
(3)
where the integration is performed over the complete
“hidden variable” space spanned by , the weighting
factor, , represents the relative frequencies of the
different “states”, , of the source. pA and pB are the
probabilities of detection, given the detector setting (a or
b) and .
The simplest assumption is that pA and pB are cos2 (a – )
and cos2 (b – ) 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.  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.
2.3: Generalising the ball model
As it stands, the 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,
D:\106742796.doc
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
7
Setting the Record Straight
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 accepted27. 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 , 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 separately29. Cases in point may be Weihs’
experiment, presented as having closed the “locality”
loophole30, and Kwiat’s demonstration of entanglement
using an “ultrabright photon source31”.
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 involved32. These include one of historical
importance — that of Freedman and Clauser, in 197233
— 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
D:\106742796.doc
Aspect’s experiments) increases the options for local
realist models34.
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 Unspeakable35) 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.
Perhaps too much emphasis has been placed on Bell tests
at the expense of ordinary scientific method, which would
have led to comprehensive investigation of the relative
merits of the QM versus local realist models. As
suggested above, there may be deviations from Malus’
Law that become noticeable when the light is weaker or
the detectors less efficient. Should not the intensity of the
light and characteristics of the detectors be parameters of
a genuine physical model? Contrary to the expressed
opinion of such authorities as Bell (page 109 of Speakable
and Unspeakble) and Clauser and Shimony36, the Chaotic
Ball model tells us that imperfections do not always
decrease quantum correlations. Less efficient detectors
8
Setting the Record Straight
are likely to result in wider effective missing bands and
hence stronger estimated correlations.
Hopefully, I have convinced readers that not all local
realist models are contrived, or are as weird as quantum
theory. They do not, as expressed for example by Prof
Laloë37, require conspiracies between the detectors. Nor
are they complicated. The basic formula (equation (3)
above), valid in the absence of complicating factors such
as “accidentals” or synchronisation problems, has been
known all along. It is a straightforward consequence of
the assumption that the observed correlations originate
from shared properties acquired at the source, and the
individual detection events are independent.
A frequent objection is that local realism cannot match
quantum theory when it comes to accurate quantatitive
predictions. True, it cannot easily match exactly the
quantum-mechanical coincidence formulae (the ball
model, illustrating principles only, does not even attempt
to), but what is required is surely a match with
experimental results, not with the quantum theory
predictions. Though the quantum-mechanical predictions
have generally been presented as being correct, do they
remain correct when the experimental conditions are
slightly altered? Do they correctly predict the whole
observed coincidence curve? The QM formula can be —
and frequently has been — altered to allow for a few
changes in conditions, but such adaptations require
considerable expertise in the formalism. Adapting the
local realist model, on the other hand, requires no special
training in any formalism, only understanding of chains of
cause and effect.
“Any theory will account for some facts; but only
the true explanation will satisfy all the conditions
of the problem …” (William Crookes, 187539)
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.
The realist model for the optical experiments seems to
require that we model light as a wave 38, assuming the
energy of each individual light pulse (“photon”) to be
split at the polariser — something no photon can do. The
intensity of the emerging pulse then influences
statistically the probability of the detector firing. Despite
the success of the photon model of light in many
applications of quantum optics, this success is at the price
of recognised conceptual difficulties. The possibility that
wave models that allow for the idiosyncrasies of the
apparatus used — deviations from Malus’ Law for
example — may be able to account in a much more
straightforward manner for all “quantum optical” effects
is the subject of ongoing research.
The spin-offs from experiments related to attempted
applications of quantum entanglement — improved
technology, for example, in the areas of optical and nanoscale communications and computing — justify continued
research in this area. The technology stands in its own
right. The theory behind it, though, remains an open
question. It is likely to remain so for some time to come,
since, by the indiscriminate rejection of all papers on the
Bell tests loopholes (see Appendix C), the prestigious
journals are inadvertently suppressing physically
plausible local realist explanations.
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9
Setting the Record Straight
APPENDIX A: COMPARISON OF CHSH AND CH74 TESTS
Source
CHSH
CH74
Attributed to CHSH 1969 paper. Never in fact
supported by authors.
Best derivation: appendix to
(reproduced in quant-ph/9903066).
Two-channel40:
Single-channel:
–2S2
S<0
CH74
paper
Experimental
design
where
where
Formula
S  E ( a, b)  E ( a, b' )  E ( a ' , b)  E ( a ' , b' )
and
E ( a, b) 
N    N   N    N 
N    N   N    N 
S  P ( a, b)  P ( a, b' )  P ( a ' , b)  P ( a ' , b' )
 P ( a ' ,  )  P ( , b )
N ( a , b)
and P(a, b) 
,
N (, )
the symbol  indicating absence of polariser41
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:
N  N  N  N  N
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Assumes “no enhancement”
Hard to violate.
10
Setting the Record Straight
realism. Of course nobody proposed a local
realistic theory that would reproduce quantitative
predictions of quantum theory (energy levels,
transition rates, etc.).
APPENDIX B: CALCULATED PREDICTIONS OF
THE CHAOTIC BALL MODEL
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”.
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. Holstein42, is:
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.
1 tan 
sin
PSS ( ,  )  1 (cos 1 ( sin
 )  cos ( tan  ) cos  ),
(B1)
where  = /2 and  is the half-angle defining the
proportion of the surface for which each assistant makes a
definite reading (zero corresponds to none; /2 to the
whole surface). PSS achieves a maximum of ½ (1 – cos )
when  = 0, which is less than the QM prediction of 0.5
unless  is /2. When   , 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
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A. Zeilinger and D. Greenberg, “Petition to the
American Physical Society for the Creation of a Topical
Group on Quantum Information, Concepts, and
Computation (Quicc)”, New York, April 10, 2002,
http://www.sci.ccny.cuny.edu/~greenbgr/letter.html
1
2
If QM were logical, would we find both Bohr and
Feynman telling us that nobody understands it?
3
A. Aspect, et al., Phys. Rev. Lett. 47, 460 (1981); 49,
91 (1982) and 49, 1804 (1982). The two 1982 papers are
available electronically at
http://fangio.magnet.fsu.edu/~vlad/pr100/
4
A. Zeilinger et al., Physics Today, February 1999, pp
11-15 and 89-92, correspondence re Goldstein’s article,
Physics Today, March 1998, pp 42-46. What Zeilinger
and others do not seem to realise is that, if the results
really can be fully explained as the consequences of
change of information, then this amounts to admission
that the Bell inequality being used is not a genuine one. If
it were, then no such simple explanation would be
possible.
5
A. F. Kracklauer (private communication) argues that
Bell “misused the chain rule” of probability theory. [To
me, his arguments amount to evidence that he has not
understood the role of hidden variables.] E. T. Jaynes
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Setting the Record Straight
argues that Bell should have used Bayesian methods. See
his article: “Clearing up the mysteries (the original goal)”,
pp. 1-27 of Maximum Entropy and Bayesian Methods, J.
Skilling, Editor, Kluwer Academic Publishers, Dordrecht,
Holland (1989),
http://bayes.wustl.edu/etj/articles/cmystery.pdf
[Much as I admire Jaynes, he is wrong here. Indeed, his
simple example of balls in a “Bernouilli urn” is not
appropriate, since in the real experiments we effectively
have sampling with replacement, not without.]
6
Bell in his original paper (ref 10 below) used Boolean
notation, A for ‘+’ outcome, Ā for ‘–’. This precludes a
zero or null outcome or (a possiblity rarely even
mentioned) the simultaneous registering of a ‘+’ and a ‘–’
from the two output ports of the same polariser. Though
it is possible to adapt Bell’s notation to cover zero’s, it is
better by far to abandon it completely when dealing with
the optical experiments and switch to Clauser and Horne’s
1974 approach. C & H concentrate on just the ‘+’
outcomes and use the notation p(, a) for the probability
of a detection of a “photon” with hidden variable  by an
analyser set at angle a.
7
M. Wolff, Exploring the Physics of the Unknown
Universe, Technotran Press, California 1990.
D. Aerts et al., “The Violation of Bell Inequalities in the
Macroworld”, http://arxiv.org/abs/quant-ph/0007044 or
the original (1982) exposition of Aerts’ linked-vessel
analogy, item 11 at
http://www.vub.ac.be/CLEA/aerts/publications/chronolog
ical.html
8
9
It should be noted that, under a wave model of light,
there is more than one way to alter a beam intensity. The
number of light pulses per second can be altered, or the
intensity per pulse, or both at once. If all that is altered is
the number per second, keeping the intensity per pulse
fixed, then QM is quite correct: this should have no effect
on the Bell test. If, however, the apparatus is manipulated
(by means of focusing, filters or whatever) so that the
intensity per pulse is changed, this can affect the result,
since real photodetectors are not quite as “linear” as they
should be.
Bell, John S, “On the Einstein-Podolsky-Rosen
paradox”, Physics 1, 195 (1964), reproduced as Ch. 2, pp
14-21, of J. S. Bell, Speakable and Unspeakable in
Quantum Mechanics, (Cambridge University Press 1987).
10
Einstein, A., B. Podolsky, and N. Rosen, “Can
Quantum-Mechanical Description of Physical Reality be
Considered Complete?”, Phys. Rev. 47, 77 (1935).
11
12
Bohm, D., Quantum Mechanics, Prentice-Hall 1951
13
There have been several attempts at Bell tests using
particles but none has been satisfactory. They are much
more difficult both to conduct and to interpret, the
interpretation invariably depending strongly on theory.
See for example M. Lamehi-Rachti and W Mittig,
“Quantum Mechanics and hidden variables: a test of
Bell’s inequality by the measurement of the spin
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correlation in low-energy proton scattering”, Phys. Rev. D
14, 2543-2555 (1976).
J. F. Clauser and M. A. Horne, “Experimental
consequences of objective local theories”, Phys. Rev. D
10, 526-35 (1974).
14
Pearle, P, “Hidden-Variable Example Based upon Data
Rejection”, Phys. Rev. D 2, 1418-25 (1970).
15
Alain Aspect, “Bell’s theorem: the naïve view of an
experimentalist”, Text prepared for a talk at a conference
in memory of John Bell, held in Vienna in December
2000. Published in "Quantum [Un]speakables – From
Bell to Quantum information", edited by R. A. Bertlmann
and
A.
Zeilinger,
Springer
(2002);
http://arxiv.org/abs/quant-ph/0402001
16
17
A. Aspect, Trois tests expérimentaux des inégalités de
Bell par mesure de corrélation de polarisation de
photons, PhD thesis No. 2674, Université de Paris-Sud,
Centre D’Orsay, (1983).
G. Adenier and A. Khrennikov, “Testing the Fair
Sampling Assumption for EPR-Bell Experiments with
Polarizer Beamsplitters”,
http://arXiv.org/abs/quantph/0306045
18
19
Note that it is possible to model any real Bell test on a
computer by taking each event as it happens: pairs of light
signals are generated, with correlated (equal?)
polarisation directions; the intensity of each is reduced by
passage through a polariser; the resulting signal interacts
with a detector and is either detected or not, at a time that
is partly random; coincidence circuitry tests whether or
not the detection times of the two signals are within a
chosen time window. This procedure, as the author
admits (private correspondence) is not carried out in
Kracklauer’s reported realist “simulation” in “Betting on
Bell”, http://arxiv.org/abs/quant-ph/0302113 . Given his
method, it comes as no surprise that he comes to a false
conclusion: that a local realist model that obeys Malus’
Law exactly can reproduce the QM formula.
C. H. Thompson, “The Tangled Methods of Quantum
Entanglement Experiments”, Accountability in Research,
6
(4),
311-332
(1999);
http://freespace.virgin.net/ch.thompson1/Tangled/tangled.
html
20
21
The definition that Bell gave (page 15 of ref 35) for
quantum correlation was the “expectation” value of the
product of the “outcomes” on the two sides, where the
“outcome” is defined to be +1 or –1 according to which of
two possible cases is observed. It is to be assumed that he
was using the word “expectation” in its usual statistical
sense and that an unbiased estimate would be used.
22
See for example Johnjoe McFadden, Quantum
Evolution: Life in the Multiverse, (Flamingo, London,
2000) page 200.
The prediction of a linear relationship for the “perfect”
case is most easily verified by drawing diagrams of the
ball as seen from above. The dividing circles are then
23
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Setting the Record Straight
straight lines through the centre and the areas required are
proportional to the angles between them.
W. Tittel et al., “Experimental demonstration of
quantum-correlations over more than 10 kilometers”,
Phys. Rev. A, 57, 3229 (1997), http://arxiv.org/abs/quantph/9707042
24
D. Fattal et al., “Entanglement formation and violation
of Bell’s inequality with a semiconductor single photon
source”, Phys. Rev. Lett. 92, 037903 (2004),
http://arxiv.org/abs/quant-ph/0305048
25
single-channel experiments. See for example P. G. Kwiat
et al., “Ultrabright source of polarization-entangled
photons”, Phys. Rev. A 60 (2), R773-R776 (1999),
http://arXiv.org/abs/quant-ph/9810003
41
Though the derivation of the CH74 inequality is in
terms of probabilities, the actual test (as Clauser and
Horne recognised) could be conducted on the raw counts,
since the limit is zero. Normalising by division by
N(,) is for convenience when comparing experiments.
42
H. Holstein, private communication, 2002.
T. W. Marshall, E. Santos and F. Selleri: “Local
Realism has not been Refuted by Atomic-Cascade
Experiments”, Phys. Lett. A 98, 5-9 (1983).
26
W. Tittel et al., “Long-distance Bell-type tests using
energy-time
entangled
photons”,
http://arxiv.org/abs/quant-ph/9809025 (1998).
27
C. H. Thompson, “Rotational invariance, phase
relationships and the quantum entanglement illusion”,
http://xxx.lanl.gov/abs/quant-ph/9912082 (1999).
28
C. H. Thompson, “Rotational invariance, phase
relationships and the quantum entanglement illusion”,
http://xxx.lanl.gov/abs/quant-ph/9912082 (1999).
29
G. Weihs, et al., “Violation of Bell’s inequality under
strict Einstein locality conditions”, Phys. Rev. Lett. 81,
5039 (1998) and http://arXiv.org/abs/quant-ph/9910080,
and private correspondence.
30
P.G. Kwiat et al., “Ultrabright source of polarizationentangled photons”, Phys. Rev. A 60 (2), R773-R776
(1999), http://arXiv.org/abs/quant-ph/9810003
31
C. H. Thompson, “Timing, ‘accidentals’ and other
artifacts
in
EPR
Experiments”
(1997),
http://arxiv.org/abs/quant-ph/9711044
32
33
S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28,
938 (1972).
S. Caser, “Objective local theories and the symmetry
between analysers”, Phys. Lett. A 102, 152-8 (1984).
34
35
J. S. Bell, Speakable and Unspeakable in Quantum
Mechanics, (Cambridge University Press 1987).
J. F. Clauser and A. Shimony, “Bell’s theorem:
experimental tests and implications”, Reports on Progress
in Physics 41, 1881 (1978).
36
F. Laloë, “Do we really understand quantum
mechanics? Strange correlations, paradoxes and
theorems”, Am. J. Phys., 69(6), 655-701, (June 2001).
37
38
Adenier and Krennikov (ref 16 above) have devised a
photon model with variable detection probabilities, but it
does not make realistic assumptions about the behaviour
at polarisers. It assumes that the light that emerges has a
wide spread of possible polarisation directions, which is
known experimentally not to be the case.
W. Crookes, “The Mechanical Action of Light”,
Quarterly Journal of Science VI, 337-352 (July 1875).
39
40
Though intended for use with two-channel detectors,
the CHSH test can, with a little ingenuity, be used for
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13