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ORIGINAL PAPER

‘‘Memory of Water’’ Without Water: The Logic

of Disputed Experiments

Francis Beauvais

Received: 18 February 2013 / Accepted: 16 July 2013 / Published online: 30 July 2013

ÓSpringer Science+Business Media Dordrecht 2013

Abstract The ‘‘memory of water’’ was a major international controversy that

remains unresolved. Taken seriously or not, this hypothesis leads to logical con-

tradictions in both cases. Indeed, if this hypothesis is held as wrong, then we have to

explain how a physiological signal emerged from the background and we have to

elucidate a bulk of coherent results. If this hypothesis is held as true, we must

explain why these experiments were difﬁcult to reproduce by other teams and why

some blind experiments were so disturbing for the expected outcomes. In this

article, a third way is proposed by modeling these experiments in a quantum-like

probabilistic model. It is interesting to note that this model does not need the

hypothesis of the ‘‘memory of water’’ and, nevertheless, all the features of Ben-

veniste’s experiments are taken into account (emergence of a signal from the

background, difﬁculties faced by other teams in terms of reproducibility, distur-

bances during blind experiments, and apparent ‘‘jumps of activity’’ between sam-

ples). In conclusion, it is proposed that the cognitive states of the experimenter

exhibited quantum-like properties during Benveniste’s experiments.

Keywords Memory of water Scientiﬁc controversy Quantum-like

probabilities Quantum cognition

Where would elementary principles such as the law

of mass action be if Benveniste is proved correct?

(Maddox 1988b)

F. Beauvais (&)

91, Grande Rue, 92310 Se

`vres, France

e-mail: beauvais@netcourrier.com

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Axiomathes (2014) 24:275–290

DOI 10.1007/s10516-013-9220-9

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1 Introduction

Scientiﬁc controversies often reveal the functioning of science and sometimes lead

to paradigm changes (Kuhn 1962). Thus, the ‘‘memory-of-water’’ controversy

exposed the role of leading scientiﬁc journals and the peer-review system in the

ﬁltering of ideas from emergent research ﬁelds (Schiff 1998). This speciﬁc topic has

been widely commented upon and details could be found elsewhere (Maddox

1988a; Maddox et al. 1988; Schiff 1998; Benveniste 2005; Beauvais 2007). For

many scientists, the affair with the journal Nature marked the end of the ‘‘memory-

of-water’’ controversy. Indeed, the hypothesis that a ﬂuid such as water could retain,

even if temporarily, information from large molecules after serial dilutions beyond

the limit of Avogadro was judged highly implausible (Teixeira 2007). Moreover,

other teams encountered difﬁculties to reproduce the effects with high dilutions with

either the same experimental protocol or other biological systems; these reasons led

to a disinterest in an idea that was considered at one time as a ‘‘new area for

biology’’ if it could be conﬁrmed (Benveniste 2005).

After 1988, Benveniste and his team continued to explore the new research

domain that they thought to have discovered. After the disputed basophils, two

biological models with promising results were successively developed: the isolated

rodent heart (Langendorff model) and the coagulation model. In parallel, after high

dilutions, Benveniste proposed other methods to ‘‘imprint’’ biological information

into water. Thus, in 1992, he reported that a speciﬁc electromagnetic radiation

emitted from a solution containing a biologically-active molecule could be

transmitted to water via an electronic ampliﬁer (Benveniste et al. 1992;Aı

¨ssa

et al. 1993; Benveniste et al. 1994;Aı

¨ssa et al. 1995). Finally, in 1996, he described

the storage of this ‘‘biological information’’ on a hard disk via the sound card of a

computer; the stored information could then be ‘‘played’’ to water to transmit this

speciﬁc ‘‘information’’ (Benveniste et al. 1996; Benveniste et al. 1997; Benveniste

et al. 1998).

Close examination of the whole ‘‘memory-of-water’’ saga supports the idea that

the controversy has not been closed satisfactorily (Beauvais 2007). Indeed,

substantiated arguments were made from both sides. On one side, the a priori

impossibility for writing ‘‘bits’’ in water was far from absurd and the proponents of

‘‘memory of water,’’ despite promising results, have not been able to offer

convincing proofs to the contrary. On the other side, the effects that were related to

the ‘‘memory of water’’ were reported by a laboratory with an excellent reputation.

Benveniste himself was a reputed senior director of INSERM, the French medical

research organization, and he was a member of the scientiﬁc establishment. He was

one of the discoverers of the platelet-activating factor, a new inﬂammatory molecule

discovered in the 1970s, and he had everything to lose with such extraordinary

claims. This research extended on for approximately 20 years and involved

successive experimenters who were experts in the management of different

biological models. Therefore, suggesting trivial explanations, such as artifact, fake,

or incompetence cannot explain the whole story.

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2 Why Did Benveniste’s Experiments Fail to Convince?

The initial program of Benveniste’s team was to assess a causal relationship

between water samples, which were supposed to have been ‘‘informed’’ by different

processes and the corresponding biological outcomes. Although the initial program

was a failure, signiﬁcant correlations were observed in these experiments.

The hypothesis of the ‘‘memory of water’’ was supported by experiments that were,

at ﬁrst sight, similar to classical pharmacological experiments. However, odd results

were repeatedly observed during the ‘‘public demonstrations’’ that Benveniste

organized to convince other scientists about the importance of his research (Table 1).

The aim of these public demonstrations was to establish a deﬁnitive proof of concept

for ‘‘electronic transmission’’ and ‘‘digital biology’’ with other scientists as witnesses.

During these demonstrations, scientists who were interested in these experiments

participated in the production of the experimental samples by using the electronic

tools devised by Benveniste’s team for the ‘‘transmission of biological activity.’’ The

samples received a code number from the participants and the samples were assessed

in Benveniste’s laboratory. The protocols and results of these public blinded

demonstrations have been previously described in detail (Beauvais 2007).

An unexpected phenomenon that was an obstacle for the establishment of a

‘‘deﬁnitive’’ proof repeatedly occurred. Indeed, after the unblinding of the masked

experiments, an effect on the biological system was frequently found associated with

the ‘‘control’’ tubes, whereas some of the samples supposed to be ‘‘active’’ were

without effect. Benveniste generally interpreted these mismatches as ‘‘jumps of

activity’’ between the samples owing to the electromagnetic nature of the speciﬁc

Table 1 The antagonistic pro and con arguments of the ‘‘memory-of-water’’ controversy and their

peaceful coexistence in a quantum-like model

Classical view Quantum-like view

Yes, ‘‘memory of

water’’ exists

a

No, ‘‘memory of water’’

does not exist

Arguments Arguments Description of the experiments with a quantum-

like probabilistic model taking into account

the experimental context

‘‘Success’’ and ‘‘failure’’ of the experiments are

described as the two facets of the same

phenomenon

No need of ‘‘memory-of-water’’ hypothesis

Emergence of signal

from the background

Numerous coherent

results

Success with blind

experiments (type-2

observer

b

)

Not compatible with our

knowledge of physics of

water

Reproductions of

experiments by other

teams generally failed

Blind experiments (type-1

observer

b

) failed

Paradox No paradox

a

‘‘Memory of water’’ is the hypothesis that speciﬁc biological information could be ‘‘imprinted’’ in water

samples in the absence of the original biological molecule. Highly diluted solutions of biologically active

compounds or other methods (‘‘electronic transmission’’ and ‘‘digital biology’’) were used

b

Blind experiments with type-1 and type-2 observers: see text

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‘‘molecular signal.’’ The logical consequence of this interpretation was trying to

protect the ‘‘informed’’ water samples and their controls from external inﬂuences, such

as electromagnetic waves. Despite additional precautions and further improvements

of the devices, this weirdness nevertheless persisted and ‘‘jumps’’ could not be

prevented (Benveniste 2005; Beauvais 2007,2008,2012; Thomas 2007).

At this stage, one could conclude that the initial hypotheses were ‘‘falsiﬁed’’ and that

the concepts of ‘‘memory of water,’’ ‘‘electronic molecular transmission,’’ and ‘‘digital

biology’’ were illusions. Benveniste, however, clung to the idea that a variation of the

biological parameters was nevertheless observed during these experiments, a phenom-

enon that was not explained by current scientiﬁc knowledge. For example, the

experimental outcomes were correlated after two successive measurements on the same

biological system or after measurements on two experimental devices (Beauvais 2007).

Therefore, Benveniste’s team constructed an automatic robot analyzer to perform

coagulation experiments with minimal intervention of the experimenter, which was

suspected to interfere, by unknown reasons, with the device.

In 2001, the United States Defense Advanced Research Projects Agency

(DARPA) that was amazed by Benveniste’s theories decided to investigate the

automatic robot analyzer and assess if the digital signals recorded on a hard disk

could be the source of speciﬁc biological effects. In the article that summarized their

study, the experts reported that some effects supporting the concepts of ‘‘digital

biology’’ were observed. However, they did not admit that the concepts of ‘‘digital

biology’’ were valid, but that an unknown ‘‘experimenter effect’’ could explain the

results. The experts concluded that a theoretical framework was necessary before

trying to apprehend these phenomena (Jonas et al. 2006).

In a previous article, we analyzed a large set of experiments obtained by

Benveniste’s team in the 1990s with the Langendorff model including ‘‘public

demonstrations’’ (Beauvais 2012). Comparing the results obtained in different

blinding conditions, we concluded that the results of these experiments were related to

experimenter-dependent correlations. Although these results did not support the initial

‘‘memory of water’’ hypothesis, the signal that emerged from the background noise

remained puzzling. We proposed a model in which the emergence of a signal (i.e., a

change of biological parameter) from the background noise could be described by the

entanglement of the experimenter with the observed system. However, entanglement

is a notion that is borrowed to quantum physics and decoherence of any macroscopic

system was an obstacle to the general acceptance of such an interpretation. In a second

article, we showed that Benveniste’s experiments and quantum interference exper-

iments of single particles had the same logical structure. This parallel allowed

elaborating a more complete formalism of Benveniste’s experiments and we proposed

to see Benveniste’s experiments as the result of quantum-like probability interferences

of cognitive states (Beauvais 2013).

The purpose of this article is to present an original framework based on a quantum-

like description of Benveniste’s experiments. Biological systems will not be detailed

and will be considered as black boxes with inputs (sample labels) and outputs

(biological outcomes); only the logical aspects and the underlying mathematical

structures of these experiments will be taken into account. Some of the ideas presented

here have been previously published, but the present article offers a synthesis and takes

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a closer look at speciﬁc issues raised by the quantum-like formalism that were not

addressed before (Beauvais 2012,2013).

3 The Logic of Benveniste’s Experiments and Single-Particle Interference

Experiments

In Benveniste’s experiments and single-particle interference experiments, a

quantum object (photon) or quantum-like object (cognitive state of the experi-

menter) interact with macroscopic devices for measurement/observation. Therefore,

we drew a parallel between Benveniste’s experiments and single-particle interfer-

ence experiments in a Mach–Zehnder interferometer (Table 2; Fig. 1).

A Mach–Zehnder interferometer has the advantage of possessing only two

detectors and not a screen as the two-slit Young’s experiment. As seen in Fig. 1

(upper drawing), 50 % of light emitted from a source is transmitted by a beam

splitter (BS1) in path T and 50 % is reﬂected in path R (Scarani and Suarez 1998).

In BS2, the two beams are recombined and 50 % of light is transmitted to detector

D1 and 50 % to detector D2. If light is considered a wave, it can be demonstrated

that waves from the two paths are constructive when they arrive in D1 and are

destructive in D2. This is indeed what experiments show: only detector D1 clicks

after light detection. This result is in favor of the wave nature of light. Indeed, if

light is considered as a collection of particles, then they should be recorded

randomly into D1 or D2 (with a probability of 1/2 either D1 or D2). However, if

light intensity is decreased in order that particles are emitted one by one, the

interference pattern persists (only detector D1 clicks). Each particle behaves as if it

Table 2 Parallel between the single-particle interference experiment with the Mach–Zehnder interfer-

ometer and Benveniste’s experiments

Interferometer experiment Benveniste’s experiments

a

First ‘‘path’’ Path T A

IN

Second ‘‘path’’ Path R A

AC

k2

1Prob (path T) Prob (A

IN

)

k2

2Prob (path R) Prob (A

AC

)

Superposition (quantum probabilities) Path T and path R A

IN

and A

AC

Outcome 1 100 % detector D1 100 % ‘‘concordant’’ pairs

b

Outcome 2 0 % detector D2 0 % ‘‘discordant’’ pairs

c

No superposition (classical probabilities) Path T or path R A

IN

or A

AC

Outcome 1 50 % detector D1 50 % ‘‘concordant’’ pairs

b

Outcome 2 50 % detector D2 50 % ‘‘discordant’’ pairs

c

Acognitive state of the experimenter, IN ‘‘inactive’’ labels, AC ‘‘active’’ labels, ;background, :signal,

Ttransmission, Rreﬂection

a

For an experiment with optimal correlations between labels and biological outcomes (and with k2

1¼k2

2)

b

A

IN

with A

;

or A

AC

with A

:

c

A

IN

with A

:

or A

AC

with A

;

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would interfere with itself. This counterintuitive (i.e., nonclassical) behavior

disappears if the information on the initial path (T or R) is obtained by

measurement: then D1 or D2 click randomly with a probability of 1/2 for each

detector (classical probabilities apply in this case) (Fig. 1; lower drawing).

Fig. 1 Experiments of single-particle interference have the same logical structure as Benveniste’s

experiments. When a unique particle interferes with itself, interferences are constructive in the detector

D1 and destructive in the detector D2. Therefore, only detector D1 clicks (upper drawing). If one

evidences by measurement the path of the particle (R or T), then both detectors click with a probability of

0.5 for each (classical probability apply because information on the path must be taken into account)

(lower drawing). In Benveniste’s experiments, signiﬁcant correlations of the labels and outcomes (IN with

‘‘;’’ and AC with ‘‘:’’; concordant pairs) were observed in the open-label experiments (or after blinding

by a type-2 observer) (upper drawing). In case of blinding by a type-1 observer, correlations vanished and

the association between labels and outcomes were broken and were randomly distributed in concordant

and discordant pairs (lower drawing)

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Therefore, the logic of the two experimental situations (Benveniste’s experiments

and one-particle interference experiment) is comparable. In Benveniste’s experiments,

‘‘active’’ samples were expected to be associated with a change in the experimental

biological system (we name it a ‘‘signal’’) whereas ‘‘inactive’’ samples were expected to

be associated with background. According to the context of the experiment (detection or

not of the ‘‘initial path,’’ i.e., sample labels), either only concordant pairs (equivalent to

only detection in D1) or both concordant/discordant pairs (i.e., equivalent to random

detection by D1 and D2) were obtained (Fig. 2;Table 2).

4 The Quantum Probabilities in Brief

In the classical world, the probabilities P1 and P2 of two incompatible events E1

and E1 add (for example, head or tail after coin toss):

Prob

class

(E1 or E2) =P1 ?P2

This is not the case for quantum probabilities where probability amplitudes add;

probability is obtained by the squaring of the sum of probability amplitudes. If we

deﬁne the complex numbers aand b(probability amplitudes), such as P1 =a

2

and

P2 =b

2

, then:

Prob

quant

(E1 or E2) =(a?b)

2

=P1 ?P2 ?‘‘interference term.’’

Therefore, in quantum probabilities, the probability amplitudes of two events can

interfere constructively or destructively (as, for example, in the interference pattern

on the screen of the two-slit Young’s experiment).

Fig. 2 Roles of type-1 and type-2 observers. The role of the type-1 and type-2 observers was to check the

results of Benveniste’s experiments in the blind experiments. These observers replaced the initial label of

all the experimental samples by a code number. The type-2 observer was inside the laboratory where he/

she could interact with the experimenter and the experimental system. The type-1 observer was outside

the laboratory, and he did not interact with the experimenter or experimental device and had no

information on the on-going measurements. When all the samples had been tested, the results of the

biological effects were sent to the type-1 observer and the two observers could assess the rate of

concordant pairs by comparing the two lists: biological effects (background or signal) and corresponding

labels under code number (‘‘inactive’’ and ‘‘active’’ samples)

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In quantum logic, the term ‘‘observable’’ is used to designate a ‘‘physical

variable.’’ To each observable (for example, the outcome of Schro

¨dinger’s cat

experiment) corresponds a set of possible pure states obtained after measurement

(dead cat; alive cat). Before measurement, the quantum system is said to be in a

superposed state of all possible pure states. Vectors in a vector space called

Hilbert’s space represent the states. Thus, before measurement, the state of the

Schro

¨dinger’s cat in this vector space is:

Cat

ji

¼1

ﬃﬃﬃ

2

pdead

ji

þ1

ﬃﬃﬃ

2

palive

ji

In this equation, each pure state is associated to a probability amplitude 1

ﬃﬃ2

p;1

ﬃﬃ2

p

and the probability to obtain a pure state after measurement is calculated by

squaring the probability amplitude (1/2; 1/2). The quantum formalism involves

that, before measurement, the quantum object is in an undetermined state

(superposed state), which is not a mixture of the different possible pure states.

Moreover, there are no ‘‘hidden variables’’ that predetermine the future outcome

after measurement.

5 The Quantum-Like Formalism of Benveniste’s Experiments

5.1 Open-Label Experiments

In open-label experiments, the experiments are performed without blinding; the

experimenter assesses the rate of concordant pairs by associating the changes of a

biological parameter with the ‘‘labels’’ of the samples to be assessed. Samples are

said to be ‘‘active’’ (AC) if a change of biological parameter (‘‘signal’’ or ‘‘:’’ ) i s

expected and ‘‘inactive’’ if a change of the biological parameter that is not different

from the background (‘‘;’’) is expected.

The cognitive state Ais described in a superposed state for the ﬁrst observable:

wA

ji

¼k1AIN

ji

þk2AAC

ji

for each sample ð1Þ

In Eq. 1, that describes the cognitive state Awith regard to the label of a given

sample, A

AC

is the cognitive state Aassociated with the ‘‘active’’ label (and A

IN

is

associated with the ‘‘inactive’’ sample). This equation means that the probabilities

for Ato be associated with an ‘‘inactive’’ or ‘‘active’’ label for this sample are k2

1and

k2

2, respectively.

The second observable is the concordance of pairs with A

CP

for concordant pairs

and A

DP

for discordant pairs. The observable is said to be concordant if A

IN

is

associated with A

;

(Ais associated with background, i.e., no change of biological

parameter) or if A

AC

is observed with A

:

(Ais associated with the signal, i.e., change

of biological parameter). Otherwise, the observable is said to be discordant (A

IN

is

associated with A

:

and A

AC

is associated with A

;

).

We introduce the possibility for the observables to be noncommuting.

Technically speaking, this means that two bases to describe any vector in the

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vector subspace where Ais described exist. When the two bases are confounded, the

observables commute (classical probabilities are therefore only a special case of

quantum probabilities).

The four vectors AIN

ji

,AAC

ji

,ACP

ji

and ADP

ji

are four unitary vectors; the two

pairs AIN

ji

=AAC

ji

and ACP

ji

=ADP

ji

form two bases of the vector subspace. We can

express one basis as a function of the other basis with four coefﬁcients named l

11

,

l

12

,l

21

, and l

21

:

AIN

ji¼l11 ACP

jiþl12 ADP

ji ð2Þ

AAC

ji

¼l21 ACP

ji

þl22 ADP

ji ð3Þ

Therefore, wA

ji

can be expressed as a superposed state of ACP

ji

and ADP

ji

:

wA

ji

¼ðk1l11 þk2l21 ÞACP

ji

þðk1l12 þk2l22ÞADP

ji ð4Þ

The quantum probability (Prob

quant

)ofA

CP

is the square of the probability

amplitude of this state:

ProbquantðACPÞ¼ k1l11 þk2l21

jj

2ð5Þ

Similarly, ProbquantðADPÞis calculated:

ProbquantðADPÞ¼ k1l12 þk2l22

jj

2ð6Þ

Since l2

11 þl2

12 ¼1;l2

21 þl2

22 ¼1;and Prob

quant

(A

CP

)?Prob

quant

(A

DP

)=1,

it means that the matrix for change of basis is a rotation matrix. Two rotations

matrixes with opposite directions are solutions. We choose the matrix that allows

the correct association of A

IN

with A

;

and A

AC

with A

:

:

l11 l12

l21 l22

¼l11 l21

l21 l11

¼cos hsin h

sin hcos h

Therefore, we can replace the probability amplitudes in the equations calculated

above:

AIN

ji

¼cos hACP

ji

sin hADP

ji ð7Þ

AAC

ji

¼sin hACP

ji

þcos hADP

ji ð8Þ

wA

ji

¼ðk1cos hþk2sin hÞACP

ji

þðk2cos hk1sin hÞADP

ji ð9Þ

ProbquantðACPÞ¼ k1cos hþk2sin h

jj

2ð10Þ

ProbquantðADPÞ¼ k2cos hk1sin h

jj

2ð11Þ

We can easily see that the rate of concordant pairs is maximal for k

1

=sin h(and

consequently k

2

=cos h):

ProbquantðACPÞ¼ k1cos hþk2sin h

jj

2¼k2

1þk2

2

2¼1ð12Þ

ProbquantðADPÞ¼ k2cos hk1sin h

jj

2¼k2k1k1k2

jj

¼0ð13Þ

In this case, all pairs (samples labels and biological outcomes) associated with

the cognitive state Aare concordant.

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5.2 Angle hand Emergence of Signal from the Background

The quantum-like formalism allows describing the results of Benveniste’s

experiments without the notion of the ‘‘memory of water’’ or its avatars, such as

‘‘digital biology.’’ In the present model, changing the value of the angle hallows the

passage from the logic of classic physics to quantum logic. The logic of classic

physics appears as a particular case (h=0) of a generalized probability theory

(with any hvalue). If his equal to zero, then the observables commute:

AIN

ji

¼1ACP

ji

0ADP

ji

¼ACP

ji ð2Þ

AAC

ji

¼0ACP

ji

þ1ADP

ji

¼ADP

ji ð3Þ

Therefore, with h=0, A

IN

is always associated with A

CP

and A

AC

is always

associated with A

DP

. In other words, concordant pair for the IN label is the back-

ground, and the discordant pair for the AC label is also the background: only the

background is associated with the cognitive state Aif h=0.

These results mean that h=0 is necessary not only for the concordant pairs, but

also for the emergence of the signal. The state A

:

must exist in the background of all

the possible states of A, even with a low probability. The superposition of the states

and the noncommuting observables allow the emergence of A

:

.

Some questions however remain. Thus, the origin of the noncommuting

observables remains unknown. Moreover, we chose one direction for the rotation

matrix to associate the ‘‘inactive’’ label with the background on one hand and the

‘‘active’’ label and the background on the other hand. However, the other rotation

matrix with the angle hin the opposite direction was also allowed by the formalism

(‘‘inactive’’ label with signal and ‘‘active’’ signal with background). How

asymmetry could be introduced in this formalism remains undeﬁned. These

questions will be explored in a future article.

In the next sections, we discuss how the other characteristics of Benveniste’s

experiments (such as ‘‘jumps of activity’’) are also described by the quantum-like

formalism.

5.3 Deﬁnition of Type-1 and Type-2 Observers

As explained above, some observers checked the results of Benveniste by using a

blind procedure. After samples had received a code number, the experimenter did

not know which sample (inactive or active) was tested and the outcome of the

experiment could not be unconsciously inﬂuenced. Since it appeared that the

outcomes (rate of concordant pairs) varied according to the circumstances of the

blinding in Benveniste’s experiments, we will ﬁrst precisely describe the roles and

characteristics of the different observers.

The deﬁnitions of the observers are based on the ‘‘Wigner’s friend,’’ a thought

experiment proposed by Wigner (1983). In this thought experiment, Wigner

imagines that a quantum experiment with two possible outcomes is performed in his

laboratory by his friend; Wigner is outside the laboratory for the duration of the

experiment (Fig. 2). At the end of the experiment, from the point of view of Wigner

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who has no information on the experiment’s outcome, his friend and the complete

chain of measurements are in an undetermined state (superposed state). When

Wigner enters the laboratory, he learns the outcome of the experiment. Therefore,

from his point of view, the quantum wave ‘‘collapses’’ at this moment. However,

from the point of view of Wigner’s friend, the ‘‘collapse’’ occurred when he looked

at the measurement apparatus at the end of the experiment and he never felt himself

in a superposed state. On the contrary, he felt that one and only one of the two

possible outcomes occurred with certainty. Therefore, according to this thought

experiment, two valid but different descriptions of the reality coexist: there is a

‘‘collapse’’ of the quantum wave at different times according to the information that

the observers get on the quantum system.

This interpretation of a quantum measurement, however, is now generally

considered out-of-date. Wigner himself subsequently adopted the theory of

decoherence when this theory was proposed in the 1970s. Decoherence occurs

when a quantum system interacts with its environment in a way that is

thermodynamically irreversible. Consequently, the different elements of the wave

function in the quantum superposition cannot interfere and the interferences become

negligible. Therefore, quantum decoherence has the appearance of a wave collapse.

However, in contrast with the Wigner’s thought experiment, no conscious observer

is necessary in the decoherence theory.

It is important to note that we do not endorse Wigner’s interpretation for the

quantum measurement (in fact, we are agnostic on this issue). This well-known

thought experiment simply allows a precise and immediately understandable

deﬁnition of the different observers/participants in Benveniste’s experiments.

Indeed, the type-1 observer and type-2 observer are respectively at the same

positions as Wigner and Wigner’s friend in the thought experiment (Fig. 2).

5.4 Blinding by Type-1 or Type-2 Observers into Practice

In blind experiments, the type-1 and type-2 observers replaced the initial labels of

the samples to be tested by a code number. The type-1 and type-2 observers were at

their respective places as deﬁned before (Fig. 2). When Benveniste’s team

completed all the measurements with samples, the results were sent to the type-1

observer (generally by fax or e-mail). The type-1/type-2 observers compared the

two lists: biological effects (background or signal) and labels (‘‘inactive’’ and

‘‘active’’ samples); then, she/he could assess the rate of concordant pairs (i.e.,

‘‘inactive’’ with the background and ‘‘active’’ with the signal).

5.5 Quantum Formalism with Blinding by a Type-2 Observer

In case of blind experiments by a type-2 observer with cognitive state O, Eq. 1is

modiﬁed:

wO

ji

¼k1OIN

ji

þk2OAC

ji

wAO

ji

¼k1AIN

ji

OIN

ji

þk2AAC

ji

OAC

ji

ð1bisÞ

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Finally, we obtain:

wAO

ji

¼ðk1cos hþk2sin hÞACP

ji

OCP

ji

þðk2cos hk1sin hÞADP

ji

OCP

ji

ð4bisÞ

Therefore, this experimental situation is formally not different from open-label

experiments described above since the cognitive states of both the experimenter

(A) and type-2 observer (O) are on the same ‘‘branch’’ of the state vector (Eq. 1bis;

Fig. 2). The type-2 observer can be considered as an integral part of the experiment

as the biological system or any automatic device for blinding.

5.6 Quantum Formalism with Blinding by a Type-1 Observer

When a blind experiment is performed by a type-1 observer, he/she assesses the rate

of concordant pairs by comparing labels and biological outcomes. This experimen-

tal situation is then formally comparable to a ‘‘which-path’’ measurement in the

Mach–Zehnder interferometer experiment and therefore, classical probabilities

apply. Indeed, the information gained by the type-1 observer on the label has to be

taken into account for the calculation of the probability for Ato be associated with

the concordant pairs:

ProbclassðACP Þ¼ProbðAIN ÞProbðACP jAIN Þþ ProbðAAC ÞProbðACP jAAC Þ

ð14Þ

with ProbðACPjAIN Þ¼cos2hand ProbðACP jAAC Þ¼sin2h, then:

ProbclassðACP Þ¼k2

1cos2hþk2

2sin2hð15Þ

Prob

class

(A

DP

) is calculated similarly:

ProbclassðADP Þ¼k2

2cos2hþk2

1sin2hð16Þ

The important point is that Prob

quant

(A

CP

)=Prob

class

(A

CP

) in the general case

(compare Eqs. 10 and 16). In the squaring of the sum of probability amplitudes,

there is an additional term 2 k

1

k

2

cos hsin h, which is typical of all the quantum

mechanical interference effects.

The calculations for the different classical and quantum probabilities are

summarized in Fig. 3. Quantum probabilities are calculated as the square of the sum

of probability amplitudes and classical probabilities (in case of measurement/

observation of the ﬁrst observable by the type-1 observer) are obtained as the sum of

the squares of all probability amplitudes.

5.7 Consequence of the Formalism: ‘‘Jumps’’ of Activity

As we have seen, the apparent ‘‘jumps of activity’’ between the samples was a

strange phenomenon that poisoned Benveniste’s experiments, particularly, during

public demonstrations (Table 3). The design of these experiments involved blinding

by a type-1 observer, and in our quantum-like probabilistic model, this phenomenon

is simply explained.

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If we suppose that the number of ‘‘inactive’’ samples (labels IN) and ‘‘active’’

samples (labels AC) are equal (i.e., k2

1¼k2

2¼0:5) and that the concordance is

optimal (i.e., cos h=k

1

and sin h=k

2

), we can calculate the respective

probabilities according to Eqs. 10,11,15, and 16.

For open-label experiments (or with blinding by a type-2 observer),

ProbquantðACPÞ¼ k1coshþk2sinh

jj

2¼1

ProbquantðADPÞ¼ k2coshk1sinh

jj

2¼0

For experiments with blinding by a type-1 observer,

ProbclassðACPjAIN Þ¼cos2h¼0:5

ProbclassðACPjAAC Þ¼sin2h¼0:5

These calculations indicate that in the open-label experiments (or with blinding by

a type-2 observer), A

IN

is always associated with A

;

and A

AC

is always associated

with A

:

. In contrast, after blinding with a type-1 observer, Prob

class

(A

CP

)=0.5 and

Prob

class

(A

DP

)=0.5.

Therefore, for a participant in these blind experiments with a type-1 observer, the

proportion of samples with the AC labels associated with the signal decreases from

Fig. 3 Design of a quantum-like experiment (application to Benveniste’s experiments). The quantum-

like object (cognitive state Aof the experimenter) is ‘‘measured’’ through two successive noncommuting

observables (h=0), which are mathematical operators. The ﬁrst observable (‘‘labels’’) splits the state of

the cognitive state Ainto two orthogonal (independent) states (‘‘inactive’’ and ‘‘active’’ labels). Each of

these two states is split by the second observable (‘‘concordance of pairs’’) into two new orthogonal

states, concordant pairs and discordant pairs. If the events inside the box are not measured or observed,

the system is in a superposition of states. If the events inside the box are measured, then, classical

probabilities apply because we have to take into account the information obtained on the path

(consequently, there is no superposition of the initial ‘‘path’’). The probabilities for the concordance of

pairs are different according to the quantum or classic probabilities. Indeed, quantum probabilities are

calculated as the square of sum of the probability amplitudes of paths. Classical probabilities are

calculated as the sum of squares of the probability amplitudes of paths. Interferences of the two initial

paths (in area with dashed line) are possible with the probability amplitudes (quantum probabilities) but

not with probabilities (classical probabilities)

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100 to 50 % and the proportion of samples with the IN labels associated with the

signal increases from 0 to 50 %. It is as if the ‘‘biological activity’’ (signal)

‘‘jumped’’ from some samples with the AC label to samples with the IN label

(Table 3).

This is a chief consequence of the quantum-like formalism that easily describes

this phenomenon without supposing ad hoc hypotheses involving uncontrolled

‘‘external’’ causes or artifacts.

5.8 ‘‘Success’’ and ‘‘Failure’’ in Benveniste’s Experiments

Quite different results are obtained in the Mach–Zehnder interferometer experiment

(or in the two-slit Young’s experiments) based on the decision to measure the initial

path or not. In one case (interference pattern), light behaves as a wave and in the

other case (no interference pattern), it behaves as a collection of particles. In

Benveniste’s results, the experimental context also appeared to play an important

role (blinding by a type-1 observer vs. a type-2 observer) (Beauvais 2007,2008,

2012,2013). According to the blinding conditions, different results were obtained

that were considered as ‘‘successes’’ or ‘‘failures’’ (Table 3). In the two-slit

experiment, observing interferences on the screen or not, according to the

experimental context, is not considered as a success or a failure: both results are

Table 3 Summary of the quantum-like probabilistic model describing Benveniste’s experiments in

different experimental contexts

Patterns of results

Expected results

a

;;;;:::: ;;;;:::: ;;;;::::

Observed results ;;;;:::: ;;::;:;: ;;;;;;;;

Description Signal present at expected

places

Signal present but at

random places

No signal

Conclusion

according to

classic logic

Success Failure (‘‘jumps of

activity’’ between

samples)

Failure

Conclusion

according to

quantum logic

h=0 with interferences of

quantum states

h=0 without

interferences of quantum

states

h=0

Probability of

concordant

pairs

a

1

ﬃﬃ2

pcos hþ1

ﬃﬃ2

psin h

2¼11

2cos2hþ1

2sin2h¼1

2

1

2

Corresponding

experimental

situations

Benveniste’s experiment

without blinding by a type-

1 observer

b

Benveniste’s experiment

with blinding by a type-1

observer

Results as predicted

by classical

probabilities

c

;Background, :signal

a

Experiments with k2

1¼k2

2¼0:5 (i.e., numbers of ‘‘inactive’’ and ‘‘active’’ labels are equal); we sup-

pose that sin h=k

2

for the ﬁrst two columns (quantum interferences are maximal in the ﬁrst column)

b

Open-label experiment or experiment blinded by a type-2 observer

c

Such results (i.e., no signal with all samples) were generally obtained by other scientiﬁc teams that tried

to reproduce Benveniste’s experiments

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necessary to describe the physics of light. In Benveniste’s experiments also, the

‘‘successes’’ and ‘‘failures’’ were the two faces of the same coin. Results of the

blinding with the type-1 observer played the same role as the measurement of the

path entered by the particle in the single-particle interference experiments.

6 Quantum-Like Formalism and Decoherence

Decoherence is an obstacle to the general acceptance of any quantum or quantum-like

model that deals with macroscopic phenomena. In our quantum-like model, it is

important to note that we borrow only some notions from the quantum logic, such as

Hilbert’s space, vector superposition, and noncommuting observables. However,

there is no term equivalent to the Planck constant and no Schro

¨dinger equation. The

cognitive state itself is an abstract ‘‘object,’’ which is involved in measurement/

observation processes involving nonphysical observables. Thus, labels take the

meaning that the experimenter decides (samples are considered as physically the same

in the formalism). Deﬁnition of a concordant pair is also arbitrary and assessing pair

concordance requires information processing for ‘‘interpretation.’’ Therefore, the

formalism deals not with the events themselves, but with the relationships between

these events. For all these reasons, the superposition of the different possible states of

the cognitive state is supposed to be not exposed to a decoherence process (except in

the case of a blind experiment with a type-1 observer).

Such an approach has never been proposed for these experiments but there are

comparable uses of notions from the quantum physics in other domains. Thus,

Walach proposed to use a ‘‘generalized’’ version of the quantum theory by

weakening some constraints of the original quantum formalism. Therefore, the

theory is applicable in more general contexts than in the original quantum physics

(Walach and von Stillfried 2011). In quantum cognition, which is an emerging

research ﬁeld, the cognitive mechanisms and information processing in the human

brain are modeled by using some notions from the formalism of quantum physics.

This approach allowed addressing problems, that were until now considered

paradoxical, and has been applied to human memory, decision making, personality

psychology, etc. (see, for example (Bruza et al. 2009) for the special issue of

Journal of Mathematical Psychology in 2009).

7 Conclusions

Our description of Benveniste’s experiments can be summarized with only two

equations, whose general form is a

2

?b

2

and |a?b|

2

. Only one parameter (the

angle h) is necessary for the passage from classical (h=0) to quantum-like

(h=0) probabilities.

We understand now why Benveniste’s experiments were the ideal ground for a

controversy. Indeed, as soon as one tried to ‘‘measure/observe’’ the initial ‘‘path’’

(namely, the cognitive state Aassociated with sample labels), correlations between

the effects and supposed causes vanished. Nevertheless, a signal persisted and that

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was the reason why Benveniste’s team pursued its technical quest for the best

experimental device and the crucial experiment. Moreover, other teams (for

which—in our interpretation—observables were commuting) could not conﬁrm

these experiments and their results were as expected according to the logic of

classical physics.

In conclusion, the use of a quantum-like probabilistic model allows describing all

the characteristics of Benveniste’s experiments and brings a new light on this major

controversy. We propose that the outcomes of these experiments were related to

quantum-like interferences of the cognitive states of the experimenter.

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