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Edited by: Andrei Khrennikov, Linnaeus University, Sweden

Reviewed by: Jerome Busemeyer, Indiana University, USA; Irina Basieva, General Physics Institute, Russia; Giuseppe Sergioli, University of Cagliari, Italy

*Correspondence: Sandro Sozzo

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The scientific community is becoming more and more interested in the research that applies the mathematical formalism of quantum theory to model human decision-making. In this paper, we expose the theoretical foundations of the quantum approach to cognition that we developed in Brussels. These foundations rest on the results of two decade studies on the axiomatic and operational-realistic approaches to the foundations of quantum physics. The deep analogies between the foundations of physics and those of cognition lead us to investigate the validity of quantum theory as a general and unitary framework for cognitive phenomena, and the empirical success of the Hilbert space models derived by such investigation provides a strong theoretical confirmation of this validity. However, two situations in the cognitive realm, “question order effects” and “response replicability,” indicate that even the Hilbert space framework could be insufficient to reproduce the expected pattern. This does not mean that the mentioned operational-realistic approach would be incorrect, but simply that a larger class of measurements would be in force in human cognition, so that an extended quantum formalism may be needed to deal with all of them. As we will explain, the recently derived “extended Bloch representation” of quantum theory (and the associated “general tension-reduction” model) precisely provides such extended formalism, while remaining within the same unitary interpretative framework.

A fundamental problem in cognition concerns the identification of the principles guiding human decision-making. Identifying the mechanisms of decision-making would indeed have manifold implications, from psychology to economics, finance, politics, philosophy, and computer science. In this regard, the predominant theoretical paradigm rests on a classical conception of logic and probability theory. According to this paradigm, people take decisions by following the rules of Boole's logic, while the probabilistic aspects of these decisions can be formalized by Kolmogorov's probability theory [

In the last decade, an alternative scientific paradigm has caught on which applies a different modeling scheme. The research that uses the mathematical formalism of quantum theory to model situations and processes in cognitive science is becoming more and more accepted in the scientific community, having attracted the interest of renowned scientists, funding institutions, media, and popular science. And, quantum models of cognition showed to be more effective than traditional modeling schemes to describe situations like the “Guppy effect,” the “combination problem,” the “prisoner's dilemma,” the “conjunction and disjunction fallacies,” “similarity judgments,” the “disjunction effect,” “violations of the Sure-Thing principle,” “Allais,” “Ellsberg,” and “Machina paradoxes” (see, e.g., [

There is a general acceptance that the use of the term “quantum” is not directly related to physics, neither this research in “quantum cognition” aims to unveil the microscopic processes occurring in the human brain. The term “quantum” rather refers to the mathematical structures that are applied to cognitive domains. The scientific community engaged in this research does not instead have a shared opinion on how and why these quantum mathematical structures should be employed in human cognition. Different hypotheses have been put forward in this respect. Our research team in Brussels has been working in this domain since early nineties, providing pioneering and substantial contributions to its growth, and we think it is important to expose the epistemological foundations of the quantum theoretical approach to cognition we developed in these years. This is the main aim of the present paper.

Our approach was inspired by a two decade research on the mathematical and conceptual foundations of quantum physics, quantum probability, and the fundamental differences between classical and quantum structures [

Let us shortly explain the “operational-realistic” connotation characterizing our approach, because doing so we can easily point out its specific strength, and the reason why it introduces an essentially new element to the domain of psychology. “Operational” stands for the fact that all fundamental elements in the formalism are directly linked to the measurement settings and operations that are performed in the laboratory of experimentation. “Realistic” means that we introduce in an operational way the notion of “state of an entity,” considering such a “state” as representing an aspect of the reality of the considered entity at a specific moment or during a specific time-span. Historically, the notion of “state of a physical entity” was the “easy” part of the physical theories that were the predecessors of quantum theory, and it was the birth of quantum theory that forced physicists to take also seriously the role of measurement and hence the value of an operational approach. The reason is that “the reality of a physical entity” was considered to be a simple and straightforward notion in classical physics and hence the “different modes of reality of a same physical entity” were described by its “different states.” That measurements would intrinsically play a role, also in the description of the reality of a physical entity, only became clear in quantum physics for the case of micro-physical entities.

In psychology, things historically evolved in a different way. Here, one is in fact confronted with what we call “conceptual entities,” such as “concepts” or “conceptual combinations,” and more generally with any cognitive situation which is presented to the different participants in a psychology experiment. Due to their nature, conceptual entities and cognitive situations are “much less real than physical entities,” which makes the notion of “state of a conceptual entity” a highly non-obvious one in psychology. And, as far as we know, the notion of state is never explicitly introduced in psychology, although it appears implicitly within the reasoning that is made about experiments, their setups and results. Possibly, the notion of “preparation of the experiment” will be used for what we call “the state of the considered conceptual entity” in our approach. Often, however, the notion of state is also associated with the “belief system” of the participant in the experiment. In our approach we keep both notions of “state” and “measurement” on equal footing, whether our description concerns a physical entity or a conceptual entity. In this way, we can make optimal use of the characteristic methodological strengths of each one of the notions. It is in doing so that we observed that there is an impressive analogy between the operational-realistic description of a physical entity and the operational-realistic description of a conceptual entity, in particular for what concerns the measurement process and the effects of context on the state of the entity. As a matter of fact, one can give a SCoP description of a conceptual entity and its dynamics [

We put forward an alternative solution for these effects within a “hidden measurement formalism” elaborated by ourselves (see, e.g., [

For the sake of completeness, we summarize the content of this paper in the following.

In Section 2, we present the epistemological foundations of the quantum theoretical approach to human cognition we developed in Brussels. We operationally describe a conceptual entity in terms of concrete experiments that are performed in psychological laboratories. Specifically, the conceptual entity is the reality of the situation which every participant in an experiment is confronted with, and the different states of this conceptual entity are the different modes of reality of this experimental situation. There are contexts influencing the reality of this experimental situation, and the relevant ones of these contexts are elements of the SCoP structure, the theory of our approach, and their influence on the experimental situation is described as a change of state of the conceptual entity under consideration. There are also properties of this experimental situation, the relevant ones being elements of the SCoP structure, and they can be actual or potential, their “amount of actuality” (i.e., their “degree of availability in being actualized”) being described by a probability measure. The operational analogies between physical and conceptual entities suggest to represent the latter by means of the mathematical formalism of quantum theory in Hilbert space. Hence, we assume, in our research, the validity of quantum theory as a scientific paradigm for human cognition. On the basis of this assumption, we provide a unified presentation in Section 3 of the results obtained within a quantum theoretical modeling in knowledge representation, decision theory under uncertainty and behavioral economics. We emphasize that our research allowed us to identify new unexpected deviations from classical structures [

Many quantum physicists agree that the phenomenology of microscopic particles is intriguing, but what is equally curious is the quantum mathematics that captures the mysterious quantum phenomena. Since the early days of quantum theory, indeed, scholars have been amazed by the success of the mathematical formalism of quantum theory, as it was not clear at all how it had come about. This has inspired a long-standing research on the foundations of the Hilbert space formalism of quantum theory from physically justified axioms, resting on well defined empirical notions, more directly connected with the operations that are usually performed in a laboratory. Such an operational justification would make the formalism of quantum theory more firmly founded.

One of the well-known approaches to the foundations of quantum physics and quantum probability is the “Geneva-Brussels approach”, initiated by Jauch [

There are still difficulties connected with the interpretation of some of these axioms and their physical justification, in particular for what concerns compound physical entities [

Let us first consider the empirical phenomenology of cognitive psychology. Like in physics, where laboratories define precise spatio-temporal domains, we can introduce “psychological laboratories” where cognitive experiments are performed. These experiments are performed on situations that are specifically “prepared” for the experiments, including experimental devices, and, for example, structured questionnaires, human participants that interact with the questionnaires in written answers, or each other, e.g., an interviewer and an interviewed. Whenever empirical data are collected from the responses of several participants, a statistics of the obtained outcomes arises. Starting from these empirical facts, we identify in our approach entities, states, contexts, measurements, outcomes, and probabilities of outcomes, as follows.

The complex of experimental procedures conceived by the experimenter, the experimental design and setting and the cognitive effect that one wants to analyze, define a conceptual entity _{A} of the conceptual entity _{Fruits}, _{Vegetables}, and _{Fruits and Vegetables} of the conceptual entities

A context _{A} of the conceptual entity _{Fruits} of the conceptual entity _{Apple}, i.e., the state describing the situation “the fruit is an apple,” as a consequence of the contextual interaction with the participant.

The change of the state of a conceptual entity due to a context may be either “deterministic,” hence in principle predictable under the assumption that the state before the context acts is known, or “intrinsically probabilistic,” in the sense that only the probability μ(_{A}) that the state _{A} of _{Apple}, _{Fruits}), where the context

Like in physics, an important role is played by experiments with only two outcomes, the so-called “yes-no experiments.” Suppose that in an opinion poll a participant is asked to answer the question: “Is Gore honest and trustworthy?” Only two answers are possible: “yes” and “no.” Suppose that, for a given participant, the answer is “yes.” Then, the state _{Honesty} of the conceptual entity _{Gy}, which is the state describing the situation “Gore is honest.” Hence, we can distinguish a class of yes-no measurements on conceptual entities, as we do in physics.

The third step is the mathematical representation. We have seen that the Hilbert space formalism of quantum theory is general enough to capture an operational description of any entity in the micro-physical domain. Then, the strong analogies between the realistic and operational descriptions of physical and conceptual entities, in particular for what concerns the measurement process, suggest us to apply the same Hilbert space formalism when representing cognitive situations. Hence, each conceptual entity _{A} of

The Born rule obviously applies to measurement with more than two outcomes too. For example, a typicality measurement involving a list of _{1}, …, _{n} with respect to a concept _{1}, …, _{n}}, where _{k}_{l} = δ_{kl}_{k}, such that the typicality μ_{k}(_{k} with respect to the concept _{k}(_{k}|

An interesting aspect concerns the final state of a conceptual entity _{1}, …, _{n} with respect to a concept _{k}, then the final state of the conceptual entity after the measurement is represented by the unit vector _{k}(_{k}, _{A}), where _{A} of the conceptual entity _{k}, represented by the unit vectors |_{k}〉.

Thus, how can a Hilbert space model be actually constructed for a cognitive situation? To answer this question let us consider again a conceptual entity _{A}, a cognitive measurement on _{1}, _{2}, …, _{n}. A quantum theoretical model for this situation can be constructed as follows. Let us assume, for the sake of simplicity, that the measurement outcomes can be considered to be non-degenerate—this is a special situation which does not hold for a wide class of cognitive measurements, see Section 3. Then, we associate _{1}〉, |_{2}〉, …, |_{n}〉} in ^{n}, the orthonormal base of ^{n}). Next, we represent the cognitive measurement described by _{1}, _{2}, …, _{n}}, where _{k} = |_{k}〉〈_{k}|, _{A} gives the outcome _{k} is given by

What about the interpretation of the Hilbert space formalism above? Two major points should now be reminded, namely:

the states of conceptual entities describe the “modes of being” of these conceptual entities;

in a cognitive experiment, a participant acts as a (measurement) context for the conceptual entity, changing its state.

This means that, as we mentioned already, the state _{A} of the conceptual entity _{k} of the experiment by the base vectors |_{k}〉, and the action of a participant (or the overall action of the ensemble of participants) as the state transformation |_{k}〉 induced by the orthogonal projection operator _{k} = |_{k}〉〈_{k}|, if the outcome _{k} is obtained, so that the probability of occurrence of _{k} can also be written as μ_{k}(_{k}〉, _{1}, _{2}, …, _{n}}.

It follows from (i) and (ii) that a state, hence a unit vector in the Hilbert space representation of states, does not describe the subjective beliefs of a person, or collection of persons, about a conceptual entity. Such subjective beliefs are rather incorporated in the cognitive interaction between the cognitive situation and the human participants deciding on that cognitive situation. In this respect, our operational quantum approach to human cognition is also a realistic one, and thus it departs from other approaches that apply the mathematical formalism of quantum theory to model cognitive processes [

We stress a third point that is important, in our opinion. For most situations, we interpret the effect of the cognitive context on a conceptual entity in a decision-making process as an “actualization of pure potentiality.” Like in quantum physics, the (measurement) context does not reveal pre-existing properties of the entity but, rather, it makes actual properties that were only potential in the initial state of the entity (unless the initial state is already an eigenstate of the measurement in question, like in physics) [

It follows from the previous discussion that our research investigates the validity of quantum theory as a general, unitary and coherent theory for human cognition. Our quantum theoretical models, elaborated for specific cognitive situations and data, derive from quantum theory as a consequence of the assumptions about this general validity. As such, these models are subject to the technical and epistemological constraints of quantum theory. In other terms, our quantum modeling rests on a “theory based approach,” and should be distinguished from an “

We present in Section 3 the results obtained in our quantum theoretical approach in the light of the epistemological perspective of this section.

The quantum approach to cognition described in Section 2 produced concrete models in Hilbert space, which faithfully matched different sets of experimental data collected to reveal “decision-making errors” and “probability judgment errors.” This allowed us to identify genuine quantum structures in the cognitive realm. We present a reconstruction of the attained results in the following.

The first set of results concerns knowledge representation and conceptual categorization and combination. James Hampton collected data on how people rate membership of items with respect to pairs of concepts and their combinations, conjunction [

In the data collected on the membership estimations of items with respect to pairs (

It is interesting to note that the size of deviation of classical probabilistic rules due to overextension and underextension generally depends on the item

Different concepts entangle when they combine, where “entanglement” is meant in the standard quantum sense. We proved this feature of concepts in two experiments. In the first experiment, we asked the participants to choose the best example for the conceptual combination ^{4} as the (tensor) product of a measurement performed on the concept

Since concepts exhibit genuine quantum features when they combine pairwise, it is reasonable to expect that these features should be reflected in the statistical behavior of the combination of several identical concepts. Indeed, we detected quantum-type indistinguishability in an experiment on the combination of identical concepts, such as the combination

The second set of results concern “decision-making errors under uncertainty.” In the “disjunction effect” people prefer action ^{3} allowed us to reproduce the data in both experiments on the disjunction effect [

Ellsberg's thought experiments, much before the disjunction effect, revealed that the Sure-Thing principle is violated in concrete decision-making under uncertainty, as people generally prefer known over unknown probabilities, instead of maximizing their expected utilities. In the famous “Ellsberg three-color example,” an urn contains 30 red balls and 60 balls that are either yellow or black, in unknown proportion. One ball will be drawn at random from the urn. The participant is firstly asked to choose between betting on “red” and betting on “black.” Then, the same participant is asked to choose between betting on “red or yellow” and betting on “black or yellow.” In each case, the “right” choice will be awarded with $100. As the events “betting on red” and “betting on black or yellow” are associated with known probabilities, while their counterparts are not, the participants will prefer betting on the former than betting on the latter, thus revealing what Ellsberg called “ambiguity aversion,” and violating the Sure-Thing principle [

The results above provide a strong confirmation of the quantum theoretical approach presented in Section 2, and we expect that further evidence will be given in this direction in the years to come. In the next section we instead intend to analyze some situations where deviations from Hilbert space modeling of human cognition apparently occur. We will see in Section 5 that these deviations are however compatible with the general operational-realistic framework portrayed in Section 2.

As mentioned in Section 2, if suitable axioms are imposed on the SCoP formalism, the Hilbert space structure of quantum theory can be shown to emerge uniquely, up to isomorphisms [

Let us first remark that the mere situation of having to deal with a set of data for which we do not yet have a faithful Hilbert space model should not make one necessarily search for an alternative more general quantum-like mathematical structure as a modeling framework. Indeed, it is very well possible that the adequate Hilbert space model has not yet been found. Recently, however, a specific situation was identified and analyzed indicating that the standard quantum formalism in Hilbert space would not be able to be used to model it [

For this we come back to the yes-no experiment of Section 2, where participants are asked: “Is Gore honest and trustworthy?” This experiment gives rise to a two-outcome measurement performed on the conceptual entity _{H}, represented by the unit vector |_{Gy}(_{G}|_{Cy}(_{C}|

Starting from these two measurements, it is possible to conceive sequential measurements, corresponding to situations where the respondents are subject to the Gore and Clinton questions in a succession, one after the other, in different orders. Statistical data about “Clinton/Gore” sequential measurements were reported in a seminal article on question order effects [

Quantum theory is equipped with a very natural tool to model question order effects: “incompatible measurements,” as expressed by the fact that two self-adjoint operators, and the associated spectral families, in general do not commute. More precisely, the Hilbert space expression for the probability that, say, we obtain the answer _{GnCy}(_{CyGn}(_{G}, _{C}] ≠ 0, i.e., if the spectral families associated with the two measurements do not commute. In the following we will analyze whether non-compatibility within a standard quantum approach can cope in a satisfying way with these question order effects, and show that a simple “yes” to this question is not possible. Indeed, a deep problem already comes to the surface in relation to the phenomenon of “response replicability.”

Consider again the Gore/Clinton measurements: if a respondent says “yes” to the Gore question, then is asked the Clinton question, then again is asked the Gore question, the answer given to the latter is expected to be “yes,” independently of the answer given to the intermediary Clinton question. This conjectured phenomenon, still necessitating a clear experimental confirmation, is called “response replicability^{1}

Although refined experiments would be needed to reveal the possible reasons for response replicability, it is worth to put forward some intuitive ideas, as we have been developing a quantum-like but more general than Hilbert space formalism within our Brussels approach to quantum cognition [

We presented in Section 4 two paradigmatic situations in human cognition that cannot be modeled together using the standard quantum formalism. We want now to explain how the latter can be naturally extended to also deal with these situations, still remaining in the ambit of a unitary and coherent framework for cognitive processes.

For this, we introduce a formalism where the probabilistic response with respect to a specific experimental situation, i.e., a state of the conceptual entity under consideration, can vary, and hence can be different than the one compatible with the Born rule of standard quantum theory. This formalism, called the “extended Bloch representation” of quantum mechanics [

We do not enter here into the details of this remarkable process, and refer the reader to the detailed descriptions in Aerts and Sassoli de Bianchi [

_{1}, a _{2}, and _{3}, with the abstract point particle attached to it, at point _{1}, _{2}, and _{3}_{2}, indicates the initial point of disintegration of the membrane, which by collapsing brings the point particle to point _{2}, corresponding to the final outcome of the measurement.

Thus, when the membrane is uniform, the “way of choosing” an outcome is precisely the “Born way.” However, a uniform membrane is a very special situation, and it is natural to also consider membranes whose points do not all have the same probability of disintegrating, i.e., membranes whose disintegrative processes are described by non-uniform probability densities ρ, which we simply call ρ-membranes. Non-uniform ρ-membranes can produce outcome probabilities different from the standard quantum ones and give rise to probability models different from the Hilbertian one (even though the state space is a generalized Bloch sphere derived from the Hilbert space geometry^{2}

We thus see that it is possible to naturally complete the quantum formalism to obtain a finer grained description of psychological experiments in which the probabilistic response of a measurement with respect to a state can be different to the one described by the Born rule. Additionally, our generalized quantum-like theory also explains why, despite the fact that individual measurements are possibly associated with different non-Born probabilities, the Born rule nevertheless appears to be a very good approximation to describe numerous experimental situations. This is related to the notion of “universal measurement,” firstly introduced by one of us in Aerts [

In Aerts and Sassoli de Bianchi [

First, we note that in case one chooses a two-dimensional Hilbert space, additional equalities can be written which are strongly violated by the data this time. As an example, consider the quantity [_{G} = |_{C} = |

Second, let us repeat our intuitive reasoning as to why measurements in the situation of response replicability carry non-Bornian probabilities. Due to the local contexts of the collection of sequential measurements, Gore, Clinton, and then Gore again, the third measurement internally changes into a non-Bornian one, and more specifically a deterministic one for the considered state, since response replicability means that for all subsequent Gore measurements the same outcome is assured. It might well be the case, although an intuitive argument would be more complex to give in this case, that also for the situation of question order effects, precisely because they only appear if a same human mind is sequentially interrogated, non-Bornian probabilities would be required. An even stronger hypothesis, which we plan to investigate in the future, is that most individual human minds, and perhaps even all, would carry in general non-Bornian probabilities, so that the success of Hilbert space quantum theory and Bornian probabilities would be mainly an effect of averaging over a sufficiently large set of different human minds, which effectively is what happens in a standard psychological experiment. If this last hypothesis is true, the violation of the Born rule for question order effects and response replicability would be quite natural, since the same human mind is needed to provoke these effects. Indeed, our analysis in Aerts and Sassoli de Bianchi [

If the above considerations provide an interesting piece of explanation as to why the Born rule is generally successful also beyond the micro-physical domain, at the same time it also contains a plausible reason of why it will possibly be not successful in all experimental situations, i.e., when the average is either not large enough, or when the experiment is so conceived that it doesn't apply as such. This could be the typical situation of question order effects and response replicability, since in this case we do not consider an average over single measurements, but over sequential (conditional) measurements. And this could be an explanation of why Hilbertian symmetries like those described above can be easily violated and that it will not be possible, by means of the Born rule, to always obtain an exact fit of the data [

Additionally, as we said, it allowed us to precisely fit the data by using the extended Bloch representation, and more specifically simple one-dimensional locally uniform membranes inscribed in a 3-dimensional Bloch sphere that can disintegrate (i.e., break) only inside a connected internal region [

In this article we explained the essence of the operational-realistic approach to cognition developed in Brussels, which in turn originated from the foundational approach to quantum physics elaborated initially in Geneva and then in Brussels (in what has become known as the “Geneva-Brussels school”). Our emphasis was that this approach is sufficiently general, and fundamental, to provide a unitary framework that can be used to coherently describe, and realistically interpret, not only quantum theory, but also its natural extensions, like the extended Bloch model and the GTR-model. In this final section we offer some additional comments on our approach to cognition, taking into consideration the confusion that sometimes exists between “

In that respect, it is worth emphasizing that the principal focus of our “theory of human cognition” is not to model as precisely as possible the data gathered in psychological measurements. A faithful modeling of the data is of course an essential part of it, but our aim is actually more ambitious. In putting forward our methodology, consisting in looking at instances of decision-making as resulting from an interaction of a decision-maker with a conceptual entity, we look first of all for a theory truly describing “the reality of the cognitive realm to which a conceptual entity belongs,” and additionally also “how human minds can interact with the latter so that decision-making can occur.”

In this sense, each time we have put forward a model for some specific experimental data, it has always been our preoccupation to also make sure that (i) the model was extracted following the logic that governs our theory of human cognition, and (ii) that whatever other experiments would be performed by a human mind interacting with that same cognitive-conceptual entity under consideration, also the data of these hypothetical additional experiments could have been modeled exactly in the same way. Clearly, this requirement—that “all possible experiments and data” have to be modeled in an equivalent way—poses severe constraints to our approach, and it is not a priori evident that this would always be possible. However, we are convinced that the fundamental idea underlying our methodology, namely that of looking upon a decision as an interaction of a human mind with a conceptual entity in a specific state (with such state being independent of the human minds possibly interacting with it), equips the theory of exactly those degrees of freedom that are needed to model “all possible data from all possible experiments.”

As we already explained in the foregoing, in all this we have been guided by how physical theories deal with data coming from the physical domain. They indeed satisfy this criterion and are able to model all data from all possible experiments that can be executed on a given physical entity. What we have called “conceptual entity” is what in physics corresponds to the notion of “physical entity.” Now, in our approach we might be classified as adhering to an idealistic philosophy, i.e., believing that the conceptual entities “really exist,” and are not mere creations of our human culture. Our answer to this objection is the following: to profit of the strength of the approach it is not mandatory to take a philosophical stance in the above mentioned way, in the sense that we are not obliged to attribute more existence to what we call a conceptual entity than that attributed, for example, to “human culture” in its entirety. The importance of the approach lies in considering such a conceptual entity as independently existing from any interaction with a human mind, and describe the continuously existing interactions with human minds as processes of the “change of state of the conceptual entity,” and whenever applicable also as processes of the “change of context.” And again, let us emphasize that this “hidden-interaction” methodology is inspired by its relevance to physical theories. Our working hypothesis is that in this way it will be possible to advantageously model, and better understand, all of human cognition experimental situations.

Having said this, we observe that the interpretation of the quantum formalism that is commonly used in cognitive domains is a subjectivist one, very similar to that interpretation of quantum theory known as “quantum Bayesianism,” or “QBism” [

It is important to emphasize that the subjectivist view is also a consequence of the absence, in the usual quantum formalism, of a meaningful description of what goes on “behind the scenes” during a measurement. On the other hand, the hidden-measurement paradigm, as implemented in the extended Bloch representation [

To conclude, a final remark is in order. Quantum cognition is undoubtedly a fascinating field of investigation also for physicists, as it offers the opportunity to take a new look at certain aspects of the quantum formalism and use them to possibly make discoveries also in the physical domain. We already mentioned the example of “entangled measurements,” that were necessary to exactly model certain correlations. Entangled (non-separable) measurements are usually not considered in the physics of Bell inequalities, while they are widely explored in quantum cryptography, teleportation and information. However, it is very possible that this stronger form of entanglement will prove to be useful for the interpretation of certain non-locality tests and the explanation of “anomalies” that were identified in EPR-Bell experiments [

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The Handling Editor declared recent co-publications, though no other collaboration, with the reviewers (IB) and (JB) and states that the process nevertheless met the standards of a fair and objective review.

^{1}We stress here that such a conjecture does require that “all' psychological measurements should satisfy “response replicability.” It rather claims that the latter should hold for a non-empty class of these measurements.

^{2}More general state spaces can also be considered, in what has been called the “general tension-reduction” (GTR) model [