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ORIGINAL PAPER
‘Unconventional’ experiments in biology
and medicine with optimized design
based on quantum-like correlations
Francis Beauvais*
91, Grande Rue, 92310 S
evres, France
In previous articles, a description of ‘unconventional’ experiments (e.g. in vitro or clinical
studies based on high dilutions, ‘memory of water’ or homeopathy) using quantum-like
probability was proposed. Because the mathematical formulations of quantum logic are
frequently an obstacle for physicians and biologists, a modified modeling that rests on
classical probability is described in the present article. This modeling is inspired from
a relational interpretation of quantum physics that applies not only to microscopic ob-
jects, but also to macroscopic structures, including experimental devices and observers.
In this framework, any outcome of an experiment is not an absolute property of the
observed system as usually considered but is expressed relatively to an observer. A
team of interacting observers is thus described from an external view point based on
two principles: the outcomes of experiments are expressed relatively to each observer
and the observers agree on outcomes when they interact with each other. If probability
fluctuations are also taken into account, correlations between ‘expected’ and observed
outcomes emerge. Moreover, quantum-like correlations are predicted in experiments
with local blind design but not with centralized blind design. No assumption on ‘mem-
ory’ or other physical modification of water is necessary in the present description
although such hypotheses cannot be formally discarded.
In conclusion, a simple modeling of ‘unconventional’ experiments based on classical
probability is now available and its predictions can be tested. The underlying concepts
are sufficiently intuitive to be spread into the homeopathy community and beyond. It
is hoped that this modeling will encourage new studies with optimized designs for
in vitro experiments and clinical trials. Homeopathy (2017) 106,55e66.
Keywords: Randomized clinical trials; Memory of water; Quantum-like probabilities
Introduction
In 2017, despite several decades of clinical trials and
in vitro studies, the scientific community remains highly
sceptic about homeopathy and high dilutions.
1,2
In
particular, many scientists remain unconvinced by
randomized controlled blind trials and meta-analyses
with homeopathy medicines.
3,4
The recent systematic
review of randomized clinical trials and meta-analysis of
Mathie et al. concluded that, despite the small number of
trials with reliable evidence, homeopathy might have small
effect.
5
Hahn et al. performed a review of meta-analyses in
homeopathy and reported that clinical trials of homeopath-
ic remedies were most often superior to placebo.
6
They
noted also that different meta-analyses could have opposite
conclusions even though they were based on practically the
same data. As pointed out by Hahn et al., the heterogeneity
of the trials and their various quality levels encourage inter-
pretation and personal bias (for or against homeopathy)
during the selection process of the data to be pooled.
The absence of rationale for diluting active compounds
beyond Avogadro’s limit is also a frequent argument to
*Correspondence: Francis Beauvais, 91, Grande Rue, 92310,
S
evres, France.
E-mail: beauvais@netcourrier.com
Received 16 August 2016; revised 10 December 2016; accepted 6
January 2017
Homeopathy (2017) 106, 55e66
Ó2017 The Faculty of Homeopathy. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.homp.2017.01.002, available online at http://www.sciencedirect.com
disprove homeopathy.
7
Explanations for the persistence of
a biological or therapeutic efficacy in the absence of the
active molecules have been developed, which can be clas-
sified in two categories: local and non-local hypotheses.
Historically, local hypotheses have been prevailing and
continue to be explored. To put it in simple terms, local hy-
potheses propose that the efficacy of homeopathy is related
to physical agents that are present in medicines or test sam-
ples. Because the initial molecules have been eliminated in
the highly diluted samples, it has been proposed that the
‘memory’ of the starting molecule is nevertheless kept in
water despite the apparent unstructured character of the
liquid element. This hypothesis has been popularized under
the well-known expression ‘memory of water’ after ‘Ben-
veniste’s affair’.
8
In favor of the role of water, one can cite the initial
studies of Demangeat who reported physical changes in
high dilutions using nuclear magnetic resonance; more
recently these changes have been related to the formation
of nanostructures and nanobubbles during the diluting pro-
cess.
9
Other recent results suggested that the supramolecu-
lar chemistry of solvatochromic dyes was modified by a
homeopathic medicine and could allow to detect high dilu-
tions.
10
The role of supposed modifications of water what-
soever for carrying specific biological activity remain
however to be established. Benveniste suggested that
diluted molecules emitted an electromagnetic ‘signature’
that could be captured by a copper coil and transmitted
to samples of ‘na€
ıve’ water that acquired the biological
properties of the initial molecules as a magnetic tape
does.
11,12
The physicists Del Giudice and Preparata proposed that
long-range ‘quantum-coherent domains’ could be a sup-
port for ‘memory of water’, but how these domains might
create a specific ‘memory’ remained undefined.
13
More-
over, a difficulty arises for applying hypotheses related to
the physical properties of water to homeopathy since the
most frequent mode of administration of homeopathic
medicines is granules made of sugar. What becomes the
role of water in these dry conditions is a question that is
not addressed by the local theories of ‘memory of water’.
Finally, all mechanisms that have been proposed as a sup-
port for the biological activity of high dilutions lack key
experimental data on specificity to be convincing. Indeed,
until now, no correlation has been demonstrated between
specific modifications of the physical properties of water
and the corresponding specific biological changes.
For the sake of completeness on local theories, one
should add that some authors have suggested that low
amounts of the active substance were, in fact, present in
highly diluted samples. Thus, Temgire et al. recently pro-
posed that silicates from glass walls participated in the for-
mation of silica-coated nanostructures that transported the
initial ingredient throughout the dilution process.
14
If true,
this explanation would be however incomplete because it
cannot apply to high dilutions performed in plastic tubes
as it is usual in biology laboratories. Ironically, similar ar-
guments emphasizing ‘contamination’ from tube to tube or
imperfect dilution process have been repeatedly put for-
ward to dismiss the reality of the effects of high dilutions.
8
In all cases, it remains to demonstrate that such tiny traces
of the initial active ingredient are sufficient to trigger a bio-
logical change.
Non-local or quantum-like descriptions
of homeopathy trials
The idea that the blind randomized clinical trial (RCT) is
an inadequate tool for assessing homeopathy is widely
shared in homeopathy community.
15,16
Meanwhile all
homeopathy practitioners agree that the medicines they
use do not act as mere placebos. Local theories are
unable to explain this discrepancy and hypotheses have
been built on some ideas from quantum physics. Thus,
Walach proposed a non-local interpretation of homeopathy
in order to escape the classical relationship between ho-
meopathic remedies and symptoms.
17
Atmanspacher
et al. described a generalized quantum physics (formerly
weak quantum physics) in order to define more precisely
the usage of notions such as complementarity and entan-
glement in domains outside physics.
18
Entanglement is
the property that allows correlations between quantum ob-
jects after they have interacted even if astronomical dis-
tances separate them. These ideas have been developed
more specifically for homeopathy mainly by Walach and
Milgrom in series of articles and also by other authors.
19e24
Although most of these authors refer to entanglement to
explain the action of homeopathy, their versions differ,
particularly on what is entangled (patient, practitioner
and/or homeopathic medication). In 2013, I proposed a
modeling of homeopathy clinical trials using quantum-
like probabilities where the negative effects of blinding
in homeopathy trials were taken into account.
25
This
modeling was an adaptation of a previous model aimed
to describe Benveniste’s in vitro experiments.
26
Most physicians and biologists are admittedly unenthu-
siastic to read articles with mathematical reasoning. The
quantum formalism conveys counterintuitive notions that
are described with unfamiliar mathematical tools (Hilbert’s
space, state vectors, non commutative observables, etc). In
the present article, I propose a more finalized version of the
previous modeling that has the supplementary advantage to
rest on classical probability (a quantum-like logic is never-
theless at work).
Brief review of Benveniste’s
experiments
Because the present modeling of ‘unconventional’ ex-
periments in biology and medicine is the result of reflec-
tions on Benveniste’s experiments, I will briefly
summarize the story of the ‘memory of water’, which is
well known by most readers of Homeopathy.
27e32
I will
not describe the experimental details and results with
high dilutions and ‘digital biology’ that can be found
elsewhere.
8,33
I prefer to emphasize the stumbling block
that prevented Benveniste to achieve the intended
‘Unconventional’ experiments with optimized design
F Beauvais
56
Homeopathy
purpose of his work, namely, to demonstrate the role of
water as a support for biological activity.
For 20 years, approximately from 1984 to 2004, Benve-
niste’s team accumulated data from different biological
systems (mainly basophil degranulation, isolated rodent
heart and plasma coagulation) that were apparently in favor
of biological effects related to highly diluted compounds
and digital biology. However, one could wonder, if these
results were so obvious, why Benveniste did not succeed
to convince his peers and why these experiments were
not easily reproduced by other teams?
Indeed, although these results were consistent in open-
label and even in in-house blind experiments, the apparent
relationship between samples and biological changes van-
ished for unknown reasons during experiments that were
designed as proof of concept. In this latter case, a supervi-
sor coded the samples and kept the code secret until the end
of the measurements; he did not participate in the measure-
ments and was not informed before the end of testing.
When sample testing had been completed, he received
the list of results under code and he could then establish
the rate of success by comparing the two lists. Because
the results of experiments with an external supervisor
were always not better than chance, Benveniste could not
cross this hurdle. I described recently the details and the
analysis of a series of experiments including both in-
house and ‘external’ blinding.
34
Of interest, this stumbling block occurred with different
biological systems, different active molecules, different ex-
perimenters and different devices to ‘imprint’ water (high
dilutions, ‘transmission’ experiments, digital biology ex-
periments). The fact that a simple modification of the blind
design could have such consequences in these different
experimental models over an extended period of time is
in my opinion the scientific fact of this story.
35
Therefore,
understanding the nature of this obstacle could also cast
some light on other ‘unconventional’ experiments. In
2001, a team of experts mandated by the Defense
Advanced Research Projects Agency (DARPA) examined
a robot analyzer designed by Benveniste’s team. This ma-
chine automatically performed digital biology experiments
based on plasma coagulation, a quite simple biological
model.
8
The experts reported that they observed results
in favor of digital biology, but they concluded on the
absence of reproducible effects because they were unable
to replicate these experiments independently of Benve-
niste’s team. In the article reporting their observations,
they suggested that unknown experimenter factors could
be an explanation for this discrepancy.
36
Failures of proof-of-concept experiments with external
supervision were not interpreted by Benveniste as a ‘falsi-
fication’ ein the sense of K. Popper eof the possibility of
a ‘memory of water’. Mismatches of outcomes were
considered as the consequence of uncontrolled factors
such as electromagnetic waves in the environment, pollu-
tion of water, contaminations, human errors, unknown in-
terferences with experimenter, etc. The possibility that
the initial hypothesis ewater as a support of biological ac-
tivity ewas erroneous was not really considered. One can
understand this attitude to hang on to the ‘memory of wa-
ter’ interpretation. Indeed, a biological ‘signal’ (i.e. a
change of a biological parameter) repeatedly emerged
from background noise, although not always at the good
place, and there was no explanation for its presence in
the current state of knowledge.
37
However, I think that a
global view on all results eincluding unexpected findings
eis necessary. Indeed, the fact that in-house blind samples
eprepared in the same conditions as samples with external
supervision and submitted to the same supposed ‘distur-
bances’ ebehaved as ‘expected’ was inconsistent.
34
One must underscore that such a difference according to
blind design was not specific to Benveniste’s experiments.
Simply, mismatches were more obvious with protocols de-
signed to minimize experimental loopholes and with the
desire of Benveniste to convince other scientists with flaw-
less results. As an example, a trial performed independently
of Benveniste, namely the multicenter trial of Belon et al.
with highly diluted histamine on basophil degranulation, ex-
hibited also issues about blinding. Overall, the results ob-
tained with four laboratories in centralized conditions were
statistically significant, a result that was in favor of an effect
of high dilutions.
38
Nevertheless, a detailed analysis indi-
cates that the results were different according to the labora-
tories, sometimes at the opposite, and that one team did not
achieve a significant difference between controls and ‘active’
samples. Moreover, the regular pseudo-sinusoidal inhibitory
dose-responses that were previously reported with highly
diluted histamine were no longer observed.
39
Itwasasif
blinding scrambled the outcomes, a phenomenon that should
not be observed if only local mechanisms were prevailing.
Definition of an elementary
unconventional experiment
In experimental biology and medicine, the purpose of
most experiments is to explore a possible relationship be-
tween a supposed cause and a biological (or clinical) effect.
For the description of ‘unconventional’ experiments (e.g.
homeopathy clinical trial, ‘memory of water’ experi-
ments), we make no assumptions on physical differences
among the experimental ‘causes’ (e.g. high dilutions, ho-
meopathic granules). In other words, we assume that all
samples that are evaluated in an experiment are physically
comparable and interchangeable. Test samples differ only
by the label that is attributed in accordance with a defined
protocol, generally after a procedure (such as serial dilu-
tions or ‘impregnation of activity’ by various means). After
such a process, two categories of samples (or medications)
are defined: those with ‘inactive’ (or placebo) label and
those with ‘active’ (or ‘verum’) label. Note that ‘inactive’
versus ‘active’ naming does not prejudge the outcomes of
the experiment; it simply reflects the results ‘expected’
by the experimenter or the physician.
For simplicity, we will use only the vocabulary related to
biology experiments; of course the logic is exactly the
same for clinical trials. A ‘successful’ experiment is thus
defined as the association of the ‘inactive’ label (IN) with
‘Unconventional’ experiments with optimized design
F Beauvais
57
Homeopathy
the resting state (i.e. a change not different from back-
ground noise noted ‘Y’) of the biological system or the as-
sociation of the ‘active’ label (AC) with a biological change
(i.e. a change above background noise noted ‘[’). ‘Failure’
is defined as AC associated with ‘Y’orIN associated with
‘[’(Figure 1). Because all samples are considered physi-
cally identical, experiments with a series of samples are
repetitions of assessments of the state of the experimental
model associated with either the label AC or the label IN.
The aim of the experiments is to establish whether the state
‘[’ is more frequently associated with the label AC than
with the label IN.
Description of experimental outcomes
with the relational interpretation
Some of the concepts of quantum physics (superposition
of states, entanglement, etc) are beyond understanding
through our daily concepts. To give a view of quantum
physics more compatible with our classical view of the
world, different interpretations have been proposed (Co-
penhagen’s interpretation, Everett’s relative states, etc).
Despite their differences, all these interpretations are
compatible with the mathematics of quantum physics.
Rovelli’s relational interpretation is one of these inter-
pretations.
40,41
This interpretation has the advantage to
apply not only to microscopic systems such as particles,
but also to macroscopic systems such as measurement
devices or human observers. In Rovelli’s interpretation, a
physical system can be said to possess a certain property
only relative to another system (called an ‘observer’). It
means that this property is not absolute, but that it
belongs in common to the object and to the observer. In
other words, any observation of a physical event must be
expressed relatively to an observer. An unavoidable
consequence that is at the heart of the relational
interpretation is that different observers can give different
reports of the same outcome (there is no meta-observer of
the reality). Nevertheless, all observers agree when they
interact (an interaction is equivalent to a measurement).
Consider, for example, the situation depicted in Figure 2.
In this picture, an observer O is measuring a quantum sys-
tem S (i.e. any microscopic or macroscopic system) that can
have two outcomes after measurement: ‘1’ or ‘2’. For O,
this system is in a defined ‘state’ after measurement (either
‘1’ or ‘2’). The external observer P has a full knowledge of
the initial conditions, but he does not interact with S and O
during their evolution. For P, the system OeSisinanunde-
fined ‘state’ after O has measured S: O
1
having observed ‘1’
or O
2
having observed ‘2’. More exactly, P knows that Ois
in a defined state, but he does not know what state.
The two different accounts of O (defined outcome) and P
(undefined outcome) are both correct. Only after interac-
tion the ‘state’ of O becomes defined for P. It must be un-
derscored that the interaction of P with O does not force P
to observe what O observed before interaction (there is no
‘hidden variable’). This does not make sense in the context
of the relational interpretation to speculate about what each
observer has really observed. Indeed, we can suppose an
observer Q who does not interact with S, O or P; for this
observer, the system PeOeS is in an undefined ‘state’
even after interaction of P with OeS. The properties of ob-
jects are relational and this interpretation deals only with
the consistency of reports of different observers, not with
elusive absolute properties of objects (there is no absolute
‘state’ of an object). In other words, for a non-participating
observer, a form (but not a content) can be assigned to the
information available for concrete observers.
Figure 1 The different possible associations of labels and states
of the experimental system in the modeling. The two labels are
‘inactive’ and ‘active’ and there are two possible states for the
experimental system: (1) ‘resting’ state or background (‘Y’) i.e.
no change of the biological parameter and (2) ‘activated’ state
or biological change above background (‘[’). Success is defined
as the association of ‘inactive’ label with no change or ‘active’ label
with biological change.
Figure 2 Internal and external observers in the relational interpre-
tation. The internal observer O measures the system S and the
external participant P assesses the evolution of the system
formed by S and O. The external observer P has full knowledge
of the initial conditions, but he does not interact with S and O. Ac-
cording to the relational interpretation, two observers can make
different accounts of an outcome; both accounts are nevertheless
correct and when observers interact they agree on their observa-
tions (interaction is also a measurement). In the modeling, P de-
scribes a team of interacting observers (named O and O0)
committed in the study of a relationship between labels and a bio-
logical system S. The evolution of O, O0and S is described from
the point of view of P (GNU Free Documentation License).
‘Unconventional’ experiments with optimized design
F Beauvais
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Homeopathy
Application of the relational
interpretation to unconventional
experiments
Description of the experimental system and observers
For the present modeling, we describe an experiment
from the point of view of P as defined above and in
Figure 1. We consider that P describes the evolution of a
team of observers who are committed in an ‘unconven-
tional’ experiment and who interact with each other. We
postulate that P has full information on the states of the
team of observers and the system S at the beginning of
the experiment and does not interact with them.
For simplicity, we suppose that this team is composed of
only two observers named O and O0who observe the exper-
imental system S. ‘Observation’ means expectation (which
requires an a priori framework on what is measured) and
then feedback (recording of the outcome). We suppose an
experiment where, for a given configuration of the experi-
mental device, the probability to observe ‘success’ as
defined above is p(the probability of ‘failure’ is equal to
qwith p+q= 1). Thus, before they interact, the probability
of success is pfor O and is also pfor O’.
According to the relational interpretation, each outcome
must be expressed relatively to a given observer. In other
words, a system has one ‘state’ relative to a given observer
and it has another ‘state’ relative to a second observer.
Therefore, from the point of view of P, for two observers
O and O0who have not yet interacted, the outcomes asso-
ciated relatively to O and O0are independent. To take
into account this independence, we have to remember
that the probability of two independent events Aand B
have well-known mathematical properties:
Prob ðAXBÞ¼Prob ðAÞProb ðBÞ(1)
Calculation of the probability of ‘success’ for a ‘team of
interacting observers’
Starting from Eq. (1), we continue to describe the exper-
imental situation from the point of view of P after the two
observers interact. As depicted in Figure 3, the joint prob-
ability of ‘success’ is pp(outcomes associated relatively
to O and O0are independent) divided by the sum of the
probabilities of all events (‘failure’ and ‘success’) allowed
by the intersubjective agreement that requires that all ob-
servers agree on the outcome:
a
Prob ðsuccessÞ¼ p2
p2þq2(2)
Eq. (2) can be written with only pas a variable by
dividing both the numerator and the denominator by p
2
and by taking into account that p+q=1:
Prob ðsuccessÞ¼ 1
1þ1
p12(3)
We can generalize Eq. (3) to Nobservers:
b
Prob ðsuccessÞ¼ 1
1þ1
p1N(4)
The importance of Eqs. (3) and (4) will appear in the next
section when probability fluctuations will be taken into
consideration.
Consequences of probability fluctuations
In the laboratory, obtaining the outcome of an exper-
iment, particularly with biological models, is not imme-
diate; it takes time during which small random
fluctuations occur. Indeed, fluctuations affect all macro-
scopic objects. At each elementary time, a tiny random
bias is inevitably introduced. Therefore, from the point
of view of P, Prob (success) must be updated after each
fluctuation.
We can calculate with Eq. (5) that Prob (success) is equal
to 1/2 in the absence of observers (i.e. with N= 0). There-
fore, we write out that the initial value of Prob (success)at
time t
0
before the first fluctuation is equal to p
0
= 1/2 for any
experiment.
An elementary random fluctuation of Prob (success) that
occurs during an elementary interval of time is noted
3
(with
3
positive or negative real random number such as
r
3
r<< 1).
c
The probability of success is obtained by
completing Eq. (5). After a first fluctuation
3
1
, a new prob-
ability is calculated which is based on p
0
= 1/2. One can
thus generalize the formula for any evaluation n+ 1 based
on previous evaluation nand fluctuation n+1.
d
The for-
mula of the mathematical sequence for calculating succes-
sive evaluations of Prob (success) taking into account
fluctuations is:
Probnþ1ðsuccessÞ¼pnþ1¼1
1þ1
pnþ
3
nþ11Nwith p0¼1=2
(5)
The computer calculations of the sequence from n=0to
n= 100 random elementary fluctuations with small
3
values
(about 10
15
) and with two observers (N= 2) show that the
a
The concomitant consideration of these two principles
(independence of the outcomes relative to O and O0and
intersubjective agreement) implies that the ‘shared reality’ of O
and O0does not pre-exist to their interaction from the point of
view of P. This is a characteristic of quantum measurements. In
the language of quantum mechanics, the ‘state’ of O would be
said ‘superposed’ before interaction (idem for O0); O and O0
would be said ‘entangled’ after interaction.
b
Note that for a number of observers N> 2, they interact anyway
by pairs; this equation will be useful for N=0.
c
This means that the probability to observe ‘[’ is not null, even if
this probability is very low.
d
We assume here that probability after fluctuation n+1is
dependent on probability after fluctuation n; this will be justified
in the section “Which experimental systems are appropriate for
‘unconventional’ experiments?”
‘Unconventional’ experiments with optimized design
F Beauvais
59
Homeopathy
initial situation (p
0
= 1/2) is, in fact, metastable (Figure 4).
Indeed, after several dozens of fluctuations, there is in all
cases (i.e. whatever the series of
3
terms) a dramatic tran-
sition and one of two mutually exclusive stable positions
is achieved:
Prob ðsuccessÞ¼1=2ðmetastable positionÞ
Y
Prob ðsuccessÞ¼1or 0ðtwo possible stable positionsÞ
(6)
Note that fluctuations are required for the transition of
probability toward 0 or 1: indeed, with
3
= 0, Prob (success)
remains equal to 1/2. Moreover, expressing the outcomes
relatively to each observer O and O0before the interaction
is also necessary to allow this transition.
In stable position #1, the observed results are similar to
the ‘expected’ results, whereas, in stable position #2, there
is a systematic inverse relationship to what is expected
(Figures 1 and 4).
Therefore, an important consequence of the modeling is
the emergence of a relationship between labels and biolog-
ical outcomes. Moreover, in both stable positions, the prob-
ability to observe ‘[’ increases from w0 to 1/2.
Nevertheless, there is no reason in the formalism itself to
choose between stable position #1 (systematic ‘success’)
and stable position #2 (systematic ‘failure’) that are
randomly obtained. We can go further nevertheless if we
note that biological systems are prepared in an asymmet-
rical state. Indeed, the resting state (background noise) is
always implicitly associated with the ‘inactive’ label.
Therefore, only the stable position #1 is a possible state
for the observers and the ‘expected’ results in this case fit
the observed results. The only possible evolution of Prob
(success) is thus:
Prob ðsuccessÞ¼1=2ðmetastable positionÞ
Y
Prob ðsuccessÞ¼1ðstable positionsÞ
(7)
Consequences of blind experiments in
the modeling
Blind experiments with local assessment of ‘success’
In the case of local (in-house) blind experiments, the
automatic device or the observer who keeps secret the
code of the samples until the end of the experiment are
also elements of the experiment and the rates of ‘success’
are locally assessed. Therefore, these experiments can be
Figure 3 Schematic description of a team of observers (O and O0) of ‘unconventional’ experiments according to the relational interpretation.
We suppose a probability equal to pfor the event ‘success’ and equal to qfor the event ‘failure’ (with p+q= 1). The situation is described from
the point of view of P (see Figure 2). The outcome of an experiment is indexed relatively to O and O0, but these observers nevertheless agree
on the outcome after they interact. The white areas correspond to unauthorized situations where the outcomes are not consistent among
observers after they interact (e.g. ‘success’ for the experimenter and ‘failure’ for another observer). The white areas are consequently
excluded for the calculation of joint probability. The probability that both agents observe ‘success’ is thus calculated by the ratio of the central
gray area (‘success’ for both observers) divided by the probability of outcomes (either ‘success’ or ‘failure’) consistent for both observers (all
gray areas).
‘Unconventional’ experiments with optimized design
F Beauvais
60
Homeopathy
described with the same modeling as open-label experi-
ments and Prob (success)=1.
Blind experiments with the assessment of ‘success’ by
an external supervisor
In Benveniste’s experiments, blind experiments with an
external supervisor were performed as described above. In
clinical trials, a centralized design is also the rule for blind-
ing in accordance with modern methodological standards.
The distant/external supervisor who holds the code of the
samples does not interact with the experimenters before
all measurements are done.
e
When he receives the out-
comes for all samples, the external supervisor separately
assesses the rate of ‘success’ for labels IN and AC thus al-
lowing calculations of Prob (successrIN) and Prob (suc-
cessrAC) where Prob (xry) is the conditional probability
of xgiven y.
Note that the ‘inactive’ and ‘active’ labels are ‘expected’
to be present in the series; therefore there is a transition
from the metastable position toward one of the two stable
positions, but at random for the two labels; as a conse-
quence, Prob (successrIN) = Prob (successrAC) = 1/2.
Prob (success) is calculated according to the law of total
probability:
Prob ðsuccessÞ¼Prob ðINÞProb ðsuccessjINÞ
þProb ðACÞProb ðsuccessjACÞ(8)
¼1=21=2þ1=21=2¼1=2 (9)
This result means that a biological change is observed
but at random places. Consequently, statistical tests do
not evidence a significant difference of the effects associ-
ated with IN and AC labels. We see with Eq. (9) that the
random ‘spreading’ of outcomes between IN and AC sam-
ples (or ‘jumps of activity’) can be simply described ac-
cording to logic and does not require calling upon
external physical disturbances to explain failures with an
external supervisor.
Which experimental systems are
appropriate for unconventional
experiments?
It could be argued that this modeling could apply to any
experimental situation such as bets on coin flipping. The
use of Eq. (5) rests, however, on some conditions that
must be clarified.
The transition of Prob (success) from 1/2 to 1 (calculated
with Eq. (5) and described in Figure 4)supposesthatthe
experimental system S is based on a phenomenon that pos-
sesses an internal structure submitted to small random fluc-
tuations (e.g. thermal fluctuations). Moreover, Eq. (5)
Figure 4 Estimation of the probability for ‘success’ taking into account probability fluctuations. This figure describes the evolution of the prob-
ability of ‘success’ for a team composed of two members who interact (the experimenter and one observer for example). In this figure, the
probability defined in Figure 2 is computed by taking into account tiny random fluctuations. The equation in the cartouche defines a math-
ematical sequence that allows estimating this probability of ‘success’ at defined times after successive fluctuations. Each successive term
p
n+1
of the mathematical sequence is calculated by using p
n
and the random probability fluctuation
3
n+1
. The starting value of Prob (success)
at time t
0
is p
0
= 1/2. The values of
3
n+1
at each calculation step (corresponding to the successive times t
n+1
) are randomly obtained in the
interval 0.5 to +0.5 10
15
. One observes in this computer simulation that the probability of ‘success’ is metastable and, after a few dozens
of calculation steps, one of the two stable positions is achieved: either Prob (‘success’) = 1 or Prob (‘success’) = 0. Fluctuations
3
n+1
with
higher values lead to a transition that occurs after a lower number of calculation steps. The figure depicts the results obtained after eight
computer simulations.
e
The remote supervisor should not be confused with the
uninvolved observer P who describes the experiment. Indeed, P
has no interaction with the system and the team members and,
from his point of view, labels and corresponding outcomes
remain undefined.
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F Beauvais
61
Homeopathy
assumes that each p
n+1
value is strongly dependent on p
n
value; in other words, the probabilities p
n+1
are correlated
with the probabilities p
n
. This characteristic is known as tem-
poral autocorrelation. According to these considerations,
different types of experimental systems can be described:
Experimental systems based on a phenomenon not sub-
mitted to internal fluctuations such as radioactive decay
(Schr€
odinger’s cat) or systems with sufficient mechanical
inertia to be not influenced (‘rigid’ systems; e.g. coin flip-
ping, dice rolling). In Eq. (5),
3
is equal to zero and there
is no transition.
Experimental systems submitted to internal fluctuations,
but with successive states that are not autocorrelated due
to strong restoring forces (‘elastic’ systems). An example
of such system is a beam splitter that randomly transmits
or reflects photons. In Eq. (5),p
n
is replaced with 1/2 and
there is no transition (only fluctuations of about 1/2 are
observed).
Experimental systems with internal fluctuations but with
successive states that are not autocorrelated due to large
random fluctuations. Examples of such systems are de-
vices based on electronic noise. For these systems, there
is no correlation between p
n
and p
n+1
and no transition to-
wards 0 or 1.
Experimental systems based on a random phenomenon
with successive autocorrelated states. Examples of such
systems are structures submitted to Brownian motion or
biological systems. Indeed, temporal autocorrelation is
characteristic of phenomena with slow random fluctua-
tions.
Only the last type of experimental systems appears to be
suitable for evidencing quantum-like correlations between
‘labels’ and experimental outcomes. The appropriateness
of biological models for the appearance of quantum-like
correlations could explain why the question of ‘unconven-
tional’ experiments arose in medical and biological exper-
imental contexts.
Role of the observers and their
commitment in the experiments
In this section we will deepen the role of the observers in
the outcomes of the experiments. We have seen that ex-
pressing the outcomes relatively to each observer O and
O0was a prerequisite for a transition of the relationship be-
tween labels and biological outcomes from 1/2 towards
0 or 1. The joint probability of two independent events A
and Bis equal to the product of the separate probabilities
of the events as reported in Eq. (1). We now generalize
this equation for two events whatever their degree of inde-
pendence:
Prob ðAXBÞ¼Prob ðAÞProb ðBÞþdðwith 0#d#1Þ
(10)
If d= 0, the two events are independent; the degree of in-
dependence decreases when dincreases (i.e. the correlation
between the two events increases). For our modeling, the
estimation of the joint probability for ‘success’ as
described in Eq. (2) can be easily modified (see Figure 5
and legend for details):
Prob ðsuccessÞ¼ p2þd
p2þq2þ2dðwith 0#d#1=4Þ(11)
Therefore, a transition of Prob (success) is progressively
allowed when the parameter dchanges from d=pq = 1/4
(outcomes expressed relatively to the observed system;
classical interpretation) to d= 0 (outcomes expressed rela-
tively to each observer; relational interpretation).
As seen in Figure 4, the calculation of Prob (success) re-
quires at each step a definition of the expected relationship
between labels and biological outcomes. Moreover, labels
are arbitrarily defined and the expected relationship is an
abstract idea (remember that no physical difference be-
tween samples is postulated in the modeling). The transi-
tion of the probability supposes observation (defined as
expectation followed by feedback). From the point of
view of P, no transition of Prob (success) towards the stable
position is possible in the absence of observation by the
team’s observers (N= 0 in Eq. (5)). The conclusion is the
same if the observers are physically present in the labora-
tory, but with attention not focused on this specific relation-
ship (they expect nothing about the system and do not
receive feedback). Therefore, the parameter dcan be
considered as an evaluation of the persistence of commit-
ment to observe the relationship between labels and biolog-
ical outcomes. When d= 0, the observers are fully
committed and for d=pq, there is no commitment at all
to observe the relationship. For intermediate values, the
persistence of commitment is more or less high.
Therefore, the modeling suggests a possible explanation
for the issues of reproducibility of ‘unconventional’ experi-
ments by other teams, as it was the case, for example, with
Benveniste’s experiments. Indeed, experimenters’ qualities,
such as attentiveness, commitment and persistence, appear
to be needed for the emergence of quantum-like correlations.
By the way, this modeling suggests a possible link be-
tween psychological and physical parameters. Note that
this link does not allow a causal relationship between
mental states and physical states. We will see in the next
section that only quantum-like correlations are allowed.
Emergence of a quantum-like
relationship from classical probability
In this section, we will see that, although we did not
formally use quantum mathematical tools in the modeling,
quantum-like logic was nevertheless at work unbeknown to
us. We start the demonstration by squaring Prob
(IN) + Prob (AC)=1:
½Prob ðINÞþProb ðACÞ2¼½Prob ðINÞ2þ½Prob ðACÞ2
þ2Prob ðINÞProb ðACÞ¼ 1
(12)
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Homeopathy
Let Prob (IN)=a
2
(or a$a) and Prob (AC)=b
2
(or b$b);
this situation corresponds to the stable position #1 (note
that for position #2, b
2
must be taken equal to bb):
½a$aþb$b2¼ða$aÞ2þðb$bÞ2þ2ða$bÞ2¼1 (13)
½a$aþb$b2þ½b$aa$b2¼ða$aÞ2þðb$bÞ2þðb$aÞ2
þða$bÞ2¼1 (14)
1þ0¼1=2þ1=2¼1 (15)
With the help of Figure 6, we easily recognize in the left
arm of Eq. (14) the sum of Prob (success) plus Prob (fail-
ure) without an external supervisor and in the right arm
the sum of Prob (success) plus Prob (failure) with an
external supervisor. We also identify aand bas probability
amplitudes (their squaring gives the corresponding proba-
bilities).
In Figure 6, the probability of ‘success’ in the absence
of external supervisor is calculated by doing the sum of
the probability amplitudes of the two paths that lead to
‘success’ and then by squaring it. With an external super-
visor, the probability of ‘success’ is obtained by squaring
the probability amplitude of each path that leads to ‘suc-
cess’ and then by making the sum of the probabilities of
the two paths. This logic is thus reminiscent of single-
photon interferences such as in Young’s double-slit exper-
iment.
Concordance of the different points of
view
The modeling has been built from the point of view of P.
From the point of view of O, if he observes ‘success’ or
‘failure’, then he is sure that O0will tell him that he ob-
serves the same event. Therefore the ‘joint’ probability of
O and O0is pas stated by classical probability, a result
that is different from the point of view of P according to
the relational interpretation (Eq. (2)). The points of view
of P and OeO0are concordant when:
p¼p2
p2þq2and q¼q2
p2þq2(16)
We can easily calculate that these two equations are
equivalent to (2p1)(p1) = 0 and
(2q1)(q1) = 0, respectively. Therefore, there are
only three possible values for p, namely 1/2, 1 or 0, which
are the probabilities of initial position, stable position #1
and stable position #2, respectively. Only P who is not
involved in the experiment is able to describe the
quantum-like ‘interferences’ (cross-terms with probability
amplitudes band ebin Figure 6).
The discrepancy between the points of view of O and P is
in line with the demonstration of Breuer, which showed
that a complete self-measurement is impossible. Thus, a
measurement apparatus (or an observer) cannot distinguish
all the states of a system in which he is contained,
Figure 5 General case for the calculation of the probability for ‘success’. This figure is a generalization of Figure 2 with variations the param-
eter d. The values of the two areas with unauthorized configurations (‘success’ for one observer and ‘failure’ for the other one) are easily
calculated: p(p
2
+d)=p(1 p)d=pq d. When d= 0, quantum-like probabilities emerge; when d=pq, the joint probability of
‘success’ is equal to pas in classical probability.
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Homeopathy
irrespective of whether this system is classical or quantum
mechanical.
42,43
All correlations between an apparatus (O)
and the observed system (S) are only measurable by a
second external apparatus (P) that observes both the
system (S) and the first apparatus (O).
Experimental arguments in favor of the
present modeling
Initially designed after a reflection on Benveniste’s ex-
periments, the present modeling describes all their charac-
teristics: emergence of a ‘signal’ (biological change),
concordance between labels and biological outcomes and
erratic ‘jumps’ of the biological ‘signal’ in blind experi-
ments with an external distant supervisor. The random
‘jumps’ or spreading of the ‘biological activities’ among
samples is thus described without ad hoc explanations
such as cross-contaminations or electromagnetic perturba-
tions. Although the hypothesis of ‘memory of water’ or any
other local explanation cannot be formally discarded, no
hypothesis on the physical differences between test sam-
ples was introduced (only labels of samples are different).
In a letter published in 2008, I draw the attention on the
importance of blind design in Benveniste’s experiments.
35
In an article in 2013, I made a parallel between homeopa-
thy clinical trials and single-photon interference in a
MacheZehnder apparatus, a device whose principle is
similar to Young’s two-slit experiment.
25
On the basis of
this modeling resting on quantum-like logic, I predicted
that higher successes should be achieved in blind homeop-
athy clinical trials with local blind design.
25
Indeed, as we
have seen, a local or in-house blind design is equivalent to
an open-label design according to the modeling. In
contrast, the spreading of outcomes between placebo and
verum was predicted for centralized blind design. In other
words, no statistical difference between treatment groups
could be evidenced in this latter situation.
An editorial of Homeopathy encouraged scientists to
test the hypothesis of an improvement of the difference
of outcomes between treatment groups with local blind
design.
44
Thieves et al. have taken up the challenge and
they recently reported results comparing local and.
centralized blind designs for a homeopathic compound.
45
Before designing a clinical trial, these authors studied the
effect of homeopathic sulfur on wheat germination. The
initial hypothesis was confirmed: there was a statistical
difference for local versus centralized blind designs
(p= 0.003 for the interaction test). These results are there-
fore a strong argument in favor of the quantum-like logic
of ‘unconventional’ experiments.
f
Indeed, hypotheses
such as ‘memory of water’, modifications of water struc-
ture or contaminations with active compounds cannot
explain this difference between the two blind designs.
Moreover, beyond ‘unconventional’ experiments, these
results are also unexplainable and counterintuitive in a
classical framework. From a historical point of view, it
is also pleasing to note that these experiments reproduced
in a different model the stumbling block that prevented
Benveniste to convince his peers as explained at the begin-
ning of the article.
Figure 6 Probability of ‘success’ without or with an external supervisor. The probabilities of ‘success’ are different without or with an external
supervisor. Indeed, quantum-like probability is calculated as the square of the sum of the probability amplitudes of the different possible
‘paths’. With an external supervisor, classical probabilities apply and they are calculated as the sum of squares of the probability amplitudes
of the ‘paths’.
f
Note that Rovelli’s interpretation preserves the principle of
locality; therefore, quantum correlations cannot be considered in
this framework as ‘non local’.
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Homeopathy
Which design for clinical trials?
For scientists or physicians seeking toreproduce the study
of Thieves et al. either in in vitro studies or in clinical trials,
it is important to underscore that experiments comparing
central and local blind designs are very demanding because
they require performing a double trial. These ‘meta-experi-
ments’ should be performed only if the purpose is to test the
quantum-like nature of a relationship. If the main objective
is to improve the difference of outcomes between placebo
and homeopathy medicine in blind randomized trials, then
a local blind design is sufficient and only slight adjustments
of a classical blind RCT are required.
We can hope that both the present simplified theoretical
description and the positive results of Thieves et al. will
encourage other authors to design new experiments to
confirm these promising results. Moreover, it is not
excluded that such quantum-like phenomena could add to
the classical local causal relationship in ‘conventional’
clinical trials.
Conclusion
A simple modeling of ‘unconventional’ experiments
based on classical probability is now available and its pre-
dictions can be tested. The underlying concepts are suffi-
ciently intuitive to be spread into the homeopathy
community and beyond. It is hoped that this modeling
will encourage new studies with optimized designs for
in vitro experiments and clinical trials.
Conflict of interest statement
No conflict of interest.
References
1Ball P. H
2
O: a biography of water. UK: Hachette, 2015.
2Teixeira J. Can water possibly have a memory? A sceptical view.
Homeopathy 2007; 96: 158e162.
3Shang A, Huwiler-Muntener K, Nartey L, et al. Are the clinical ef-
fects of homoeopathy placebo effects? Comparative study of
placebo-controlled trials of homoeopathy and allopathy. Lancet
2005; 366: 726e732.
4Ernst E. A systematic review of systematic reviews of homeopathy.
Br J Clin Pharmacol 2002; 54: 577e582.
5Mathie RT, Lloyd SM, Legg LA, et al. Randomised placebo-
controlled trials of individualised homeopathic treatment: system-
atic review and meta-analysis. Syst Rev 2014; 3: 142.
6Hahn RG. Homeopathy: meta-analyses of pooled clinical data.
Forsch Komplementmed 2013; 20: 376e381.
7Hansson SO. Homoeopathy and consumers’ right to know. J Intern
Med 2013; 274: 493.
8 Beauvais F. Ghosts of molecules ethe case of the “memory of wa-
ter”. Collection Mille Mondes (ISBN: 978-1-326-45874-4); Avail-
able at: http://www.mille-mondes.fr; 2016.
9Demangeat JL. Gas nanobubbles and aqueous nanostructures: the
crucial role of dynamization. Homeopathy 2015; 104: 101e115.
10 Cartwright SJ. Solvatochromic dyes detect the presence of homeo-
pathic potencies. Homeopathy 2016; 105:55e65.
11 A€
ıssa J, Litime MH, Attias E, Allal A, Benveniste J. Transfer of mo-
lecular signals via electronic circuitry. Faseb J 1993; 7: A602.
12 Benveniste J. Ma v
erit
e sur la m
emoire de l’eau. Paris: Albin Mi-
chel, 2005.
13 Del Giudice E, Preparata G, Vitiello G. Water as a free electric
dipole laser. Phys Rev Lett 1988; 61: 1085e1088.
14 Temgire MK, Suresh AK, Kane SG, Bellare JR. Establishing the
interfacial nano-structure and elemental composition of homeo-
pathic medicines based on inorganic salts: a scientific approach. Ho-
meopathy 2016; 105: 160e172.
15 Milgrom LR. Gold standards, golden calves, and random reproduc-
ibility: why homeopaths at last have something to smile about. J Al-
tern Complement Med 2009; 15: 205e207.
16 Weatherley-Jones E, Thompson EA, Thomas KJ. The placebo-
controlled trial as a test of complementary and alternative medicine:
observations from research experience of individualised homeo-
pathic treatment. Homeopathy 2004; 93: 186e189.
17 Walach H. Magic of signs: a non-local interpretation of homeopa-
thy. Br Homeopath J 2000; 89: 127e140.
18 Atmanspacher H, R€
omer H, Walach H. Weak quantum theory:
complementarity and entanglement in physics and beyond. Found
Phys 2002; 32: 379e406.
19 Walach H. Generalized entanglement: a new theoretical model for
understanding the effects of complementary and alternative medi-
cine. J Altern Complement Med 2005; 11: 549e559.
20 Walach H, von Stillfried N. Generalised quantum theory ebasic
idea and general intuition: a background story and overview. Axio-
mathes 2011; 21: 185e209.
21 Milgrom LR. Patient-practitioner-remedy (PPR) entanglement. Part
1: a qualitative, non-local metaphor for homeopathy based on quan-
tum theory. Homeopathy 2002; 91: 239e248.
22 Milgrom LR. Journeys in the country of the blind: entanglement
theory and the effects of blinding on trials of homeopathy and ho-
meopathic provings. Evid Based Complement Alternat Med 2007;
4:7e16.
23 Hyland ME. Extended network generalized entanglement theory:
therapeutic mechanisms, empirical predictions, and investigations.
J Altern Complement Med 2003; 9: 919e936.
24 Weingartner O. What is the therapeutically active ingredient of ho-
meopathic potencies? Homeopathy 2003; 92: 145e151.
25 Beauvais F. A quantum-like model of homeopathy clinical trials:
importance of in situ randomization and unblinding. Homeopathy
2013; 102: 106e113.
26 Beauvais F. Description of Benveniste’s experiments using
quantum-like probabilities. J Sci Explor 2013; 27:43e71.
27 Thomas Y. From high dilutions to digital biology: the physical na-
ture of the biological signal. Homeopathy 2015; 104: 295e300.
28 Thomas Y. The history of the memory of water. Homeopathy 2007;
96: 151e157.
29 Chaplin MF. The memory of water: an overview. Homeopathy
2007; 96: 143e150.
30 Fisher P. The memory of water: a scientific heresy? Homeopathy
2007; 96: 141e142.
31 Poitevin B. The continuing mystery of the memory of water. Home-
opathy 2008; 97:39e41.
32 Poitevin B. Survey of immuno-allergological ultra high dilution
research. Homeopathy 2015; 104: 269e276.
33 Schiff M. The memory of water: homoeopathy and the battle of
ideas in the new science. London: Thorsons Publishers, 1998.
34 Beauvais F. Quantum-like interferences of experimenter’s mental
states: application to “paradoxical” results in physiology. Neuro-
Quantology 2013; 11: 197e208.
35 Beauvais F. Memory of water and blinding. Homeopathy 2008; 97:
41e42.
36 Jonas WB, Ives JA, Rollwagen F, et al. Can specific biological sig-
nals be digitized? FASEB J 2006; 20:23e28.
37 Beauvais F. Emergence of a signal from background noise in the
“memory of water” experiments: how to explain it? Explore (NY)
2012; 8: 185e196.
‘Unconventional’ experiments with optimized design
F Beauvais
65
Homeopathy
38 Belon P, Cumps J, Ennis M, et al. Inhibition of human basophil
degranulation by successive histamine dilutions: results of a Euro-
pean multi-centre trial. Inflamm Res 1999; 48(suppl. 1): S17eS18.
39 Sainte-Laudy J, Belon P. Inhibition of basophil activation by hista-
mine: a sensitive and reproducible model for the study of the biolog-
ical activity of high dilutions. Homeopathy 2009; 98: 186e197.
40 Rovelli C. Relational quantum mechanics. Int J Theor Phys 1996;
35: 1637e1678.
41 Smerlak M, Rovelli C. Relational EPR. Found Phys 2007; 37:
427e445.
42 Breuer T. The impossibility of accurate state self-measurements.
Philos Sci 1995; 62: 197e214.
43 Laudisa F, Rovelli C. In: Zalta EN (ed) Relational Quantum Me-
chanics. The Stanford Encyclopedia of Philosophy, Summer 2013
edn. Available at: http://plato.stanford.edu/archives/sum2013/
entries/qm-relational/.
44 Fisher P. Local, entangled or both? Homeopathy 2013; 102:85e86.
45 Thieves K, Gleiss A, Kratky KW, Frass M. First evidence of Beau-
vais’ hypothesis in a plant model. Homeopathy 2016; 105:
270e279.
‘Unconventional’ experiments with optimized design
F Beauvais
66
Homeopathy
