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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

58

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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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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62

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’.

‘Unconventional’ experiments with optimized design

F Beauvais

64

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.

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