Superposition of quantum states. In quantum logic, a quantum system is a “superposed” state of all possible pure states. The states are represented in a Hilbert space ( i.e. , a vector space). In this formalism, the cognitive state A is symbolized by Ψ A . The probability associated with the pure 

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... For these blind experiments with the participation of observers that we call “type-1 controllers” (Figure 1), something like that was obtained: ↓↑↑↓↓↑↓↑ , i.e. a signal was observed at the expected places not better than random. Nevertheless, the important point is that we should expect: ↓↓↓↓↓↓↓↓ , i.e. only background noise and no signal. The fact that a signal ( ↑ ) was observed – whatever its place – was nevertheless a fascinating result (Beauvais, 2012). Initially, the apparent mismatches between “inactive” and “active” samples in these experiments were interpreted by Benveniste as errors during handling of samples. Then, it appeared that such trivial explanation was false and “jumps” of activity from one tube to another were thought to be due to electromagnetic cross- contamination of samples. Note that a series of samples prepared and treated in the same conditions but not blinded by a type-1 controller gave “expected” results. Therefore, the explanation by “electromagnetic contamination” was limited. It was as if the conditions of blinding (type-1 vs. type-2 controllers) influenced the outcomes (Beauvais, 2008; Beauvais, 2013c). In a next section, we will precisely define the role of the different observers who participated to the blinding of the experiments. The unexplained emergence of a “signal” from background noise (in other words, a biological parameter significantly changed) was the main reason why Benveniste’s considered that these experiments merited to be continued. Contamination whatsoever (from water or from the electromagnetic environment) were then called to the rescue to explain a posteriori these mismatches. As a consequence, after each failure of a “public demonstration”, Benveniste’s team engaged in a headlong rush to improve electronic devices and biological models. The quest of the crucial experiment became the main objective. These efforts culminated in 2001 when a team commissioned by the United States Defense Advanced Research Projects Agency (DARPA) investigated a robot analyzer designed by Benveniste’s team to automatically perform “digital biology” experiments with the coagulation model. In an article published in 2006, the team of investigators reported that some effects supporting “digital biology” had been observed when Benveniste’s team was present, but they concluded that they had been unable to reproduce these results after the departure of the team (Jonas et al., 2006). According to the authors of the article, experimenter’s effects could explain these results of Benveniste’s experiments, but they concluded that a theoretical framework was necessary before continuing this research. Therefore we face a dilemma: we can consider that “memory of water” (and its avatars) exists or does not exist; but, in both cases, we are uncomfortable. We can deem that “memory of water” exists because biological signal emerged from background noise and because there were numerous coherent results, particularly in some blind experiments. On the contrary, we can judge that “memory of water” does not exist because this hypothesis is not compatible with our knowledge of physics of water, because reproduction of these experiments by other teams was generally not convincing or because blind experiments during “public demonstrations” were not better than random. To solve the dilemma described in the previous section, we proposed in a series of articles to describe these experiments using notions from quantum logic (Beauvais, 2012; 2013a; Beauvais, 2013c; b). In the model that we described, there is no need of postulating “memory of water”. Taking into account the experimental context allows describing “success” and “failure” as two facets of the same phenomenon. The paradoxes described in the above section are therefore dissolved in this model. The use of quantum logic is reasonable since classical probabilities are only a special case of quantum probabilities and, at worst, calculations would show that classical probabilities are sufficient to describe these experiments. In quantum logic, the term “observable” refers to a physical variable. To each observable corresponds a set of possible “pure states”. Before any measurement/observation, the quantum system is in a “superposed” state of all possible pure states. The states are represented in a Hilbert space ( i.e. , a vector space with a finite or infinite number of dimensions and with a complex or real inner product). Note that the superposed state is not a simple mixture of the pure states; indeed, the unique pure state selected randomly by the measurement is undetermined before the measurement (there are no hidden variables that determine the outcome). We define the “cognitive state” A of an experimenter/observer as the information on all possible measures/observations (with their respective probabilities) performed by this experimenter/observer on this system. Mathematically, A is represented by a vector  A in a Hilbert vector space. In Figure 2, the state vector of the cognitive state A is the superposition of the two pure states A -1 and A +1, which are the tw  A  a A  1  b A  1 o possible outcomes of a measurement: with | a | 2 + | b | 2 = 1. The norm a of the vector obtained after projection of  A on the axis of the pure state A +1 , is assimilated to a probability amplitude; therefore, the probability for the cognitive state to be associated with A +1 is | a | 2 . In our model of Benveniste’s experiments, the first set of observables is the cognitive state A associated with expected results, in other words the “labels” of the samples: i.e. A associated with sample designated as “inactive” ( A IN ) or A associated with samples designated as active ( A AC ); all experimental samples are physically equivalent, only their “labels” differ. The numbers of inactive and active samples are defined for a series of experiments. For example, if the numbers of inactive and active samples are equal, then P ( A IN ) = P ( A AC ) = 0.5. The second set of observables is the cognitive state A associated with the concordance of pairs: the experimenter observes the outcome with the biological system (background noise “ ↓ ” or signal “ ↑ ”) and compares with the expected result ( i.e. , the label of the sample: “inactive” or “active”). The pure states of the second set of observables are A CP if pairs are concordant and A DP if pairs are discordant. Pairs are concordant if “expected” results fit observed results: i.e. , A ↓ associated with A IN and A ↑ associated with A AC . Otherwise, pairs are discordant. Note that the probability associated with A ↑ must be different of zero even though this probability can be low (signal must be present in background noise). The use of quantum logic outside quantum physics has been already described; many authors have proposed to describe cognitive mechanisms and information processing in human brain by using notions and tools initially developed for quantum mechanics. These new quantum-like tools allowed addressing problems in human memory, decision making, personality psychology, etc, which were paradoxical in a classical frame (see for example the special issue of Journal of Mathematical Psychology in 2009) (Bruza et al., 2009; Busemeyer et al., 2012) In our previous articles (Beauvais, 2013a; b ; c ), we showed that the paradoxes of Benveniste’s experiments disappeared if the possible “cognitive states” of the experimenter were described according to some principles from quantum physics (superposition and probability interferences). We obtained equations that correctly described the characteristics of Benveniste’s experiments, namely emergence of signal from background noise, disturbance of blind experiments (type 1 vs. type 2) and difficulties for other teams to reproduce the experiments. The cognitive state of A was described as: ( Eq. 1 ) where p and q are the rates of “inactive” and “active” labels, respectively; θ is the angle of the rotation matrix for change of basis (from A IN / A AC basis to A CP / A DP basis). According to Eq. 1, the probability to observe concordant pairs is: This probability is maximal (equal to 1) if sin   q . Why pairs were concordant remained however unexplained. Technically speaking, the couple of observables A IN / A AC and A CP / A DP behaved as noncommuting observables. In this article, we address this issue by considering what happens if several experimenters – not only one – perform experiments. First, we consider a simple situation with “Benveniste’s experiments” performed in two separate places by two experimenters: Alice with cognitive state A and Bob with cognitive state B ( Figure 3A ). We suppose that Alice and Bob make experiments in their respective laboratories with two experimental devices. They observe the rates of concordant and discordant pairs for the same series of samples (labeled “inactive” or “active”). According to the quantum logic, we can describe the two cognitive states A and B of Alice and Bob. Eq. 1 described in previous section is simplified with a  p cos   q sin  and b  q cos   p sin  ...