We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation as it appears in different domains of science–from cognitive science to physics–and use it to explain the emergence of quantum probabilities (the Born rule) as uniform fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states both of the system and the measuring apparatus, in a way that the measurement probabilities can be derived as the Lebesgue measure of suitably defined convex subregions of the simplex under consideration. Although the model we propose, which we call the uniform tension-reduction (UTR) model, is an abstract construct, it admits physical realizations. In this article we consider a very simple and evocative one, using a material point particle which is acted upon by special elastic membranes, which by breaking and collapsing are able to "release the tension" and produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of a measurement process, be it from a measurement associated with a situation in cognitive science or in physics, or in any other domain. We also show that the UTR-model can be further generalized into a model describing conditions of lack of knowledge generated by non-uniform fluctuations, which we call the general tension-reduction (GTR) model. In this more general framework, which is more suitable to describe typical experiments in cognitive science, we define and motivate a notion of universal measurement, describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of quantum probability in the description of different domains of reality. And more specifically, also providing a possible explanation for the success of quantum cognition, a research field in cognitive science employing the quantum formalism as a modeling tool. This is the first part of a two-part article. In the second part (Aerts and Sassoli de Bianchi, 2014a), the proof of the equivalence between universal measurements and uniform measurements, and its significance for quantum theory as a first order approximation, is given and further analyzed.
We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation as it appears in different domains of science–from cognitive science to physics–and use it to explain the emergence of quantum probabilities (the Born rule) as uniform fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states both of the system and the measuring apparatus, in a way that the measurement probabilities can be derived as the Lebesgue measure of suitably defined convex subregions of the simplex under consideration. Although the model we propose, which we call the uniform tension-reduction (UTR) model, is an abstract construct, it admits physical realizations. In this article we consider a very simple and evocative one, using a material point particle which is acted upon by special elastic membranes, which by breaking and collapsing are able to "release the tension" and produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of a measurement process, be it from a measurement associated with a situation in cognitive science or in physics, or in any other domain. We also show that the UTR-model can be further generalized into a model describing conditions of lack of knowledge generated by non-uniform fluctuations, which we call the general tension-reduction (GTR) model. In this more general framework, which is more suitable to describe typical experiments in cognitive science, we define and motivate a notion of universal measurement, describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of quantum probability in the description of different domains of reality. And more specifically, also providing a possible explanation for the success of quantum cognition, a research field in cognitive science employing the quantum formalism as a modeling tool. This is the first part of a two-part article. In the second part (Aerts and Sassoli de Bianchi, 2014a), the proof of the equivalence between universal measurements and uniform measurements, and its significance for quantum theory as a first order approximation, is given and further analyzed.
We introduce a model which allows to represent the probabilities associated with an arbitrary measurement situation as it appears in different domains of science–from cognitive science to physics–and use it to explain the emergence of quantum probabilities (the Born rule) as uniform fluctuations on this measurement situation. The model exploits the geometry of simplexes to represent the states both of the system and the measuring apparatus, in a way that the measurement probabilities can be derived as the Lebesgue measure of suitably defined convex subregions of the simplex under consideration. Although the model we propose, which we call the uniform tension-reduction (UTR) model, is an abstract construct, it admits physical realizations. In this article we consider a very simple and evocative one, using a material point particle which is acted upon by special elastic membranes, which by breaking and collapsing are able to “release the tension” and produce the different possible outcomes. This easy to visualize mechanical realization allows one to gain considerable insight into the possible hidden structure of a measurement process, be it from a measurement associated with a situation in cognitive science or in physics, or in any other domain. We also show that the UTR-model can be further generalized into a model describing conditions of lack of knowledge generated by non-uniform fluctuations, which we call the general tension-reduction (GTR) model. In this more general framework, which is more suitable to describe typical experiments in cognitive science, we define and motivate a notion of universal measurement, describing the most general possible condition of lack of knowledge in a measurement, emphasizing that the uniform fluctuations characterizing quantum measurements can also be understood as an average over all possible forms of non-uniform fluctuations which can be actualized in a measurement context. This means that the Born rule of quantum mechanics can be understood as a first order approximation of a more general non-uniform theory, thus explaining part of the great success of quantum probability in the description of different domains of reality. And more specifically, also providing a possible explanation for the success of quantum cognition, a research field in cognitive science employing the quantum formalism as a modeling tool. This is the first part of a two-part article. In the second part (Aerts and Sassoli de Bianchi, 2014a), the proof of the equivalence between universal measurements and uniform measurements, and its significance for quantum theory as a first order approximation, is given and further analyzed.
In the first part of this two-part article (Aerts & Sassoli de Bianchi, 2014), we have introduced and analyzed a multidimensional model, called the general tension-reduction (GTR) model, able to describe general quantum-like measurements with an arbitrary number of outcomes, and we have used it as a general theoretical framework to study the most general possible condition of lack of knowledge in a measurement, so defining what we have called a universal measurement. In this second part, we present the formal proof that universal measurements, which are averages over all possible forms of fluctuations, produce the same probabilities as measurements characterized by uniform fluctuations on the measurement situation. Since quantum probabilities can be shown to arise from the presence of such uniform fluctuations, we have proven that they can be interpreted as the probabilities of a first-order non-classical theory, describing situations in which the experimenter lacks complete knowledge about the nature of the interaction between the measuring apparatus and the entity under investigation. This same explanation can be applied–mutatis mutandis–to the case of cognitive measurements, made by human subjects on conceptual entities, or in decision processes, although it is not necessarily the case that the structure of the set of states would be in this case strictly Hilbertian. We also show that universal measurements correspond to maximally robust descriptions of indeterministic reproducible experiments, and since quantum measurements can also be shown to be maximally robust, this adds plausibility to their interpretation as universal measurements, and provides a further element of explanation for the great success of the quantum statistics in the description of a large class of phenomena.
With D. Aerts, we showed in 2014, within the extended Bloch representation of quantum mechanics, that the Born rule can be understood as resulting from a universal average over all the possible ways in which the measurement-interactions can be selected. The cellular construction used there was since questioned, in the context of Bertrand's paradox, by N. Shackel, who observed that the result depends on the discretization adopted, that the sequence of averages is constant, and that the sense in which the average is taken over "all" densities remains unclear. Taking these criticisms as a stimulus, we re-examine the notion. The key is to recognize that a universal measurement is a random way of selecting a measurement which does not privilege any portion of the structure describing the available interactions, a property we call indifference. We prove that every indifferent universal measurement, however constructed, produces exactly the Born probabilities: the universal average is unique, and the cellular construction of 2014 is one among infinitely many ways of obtaining it, exact at every finite resolution. Non-trivial indifferent universal measurements exist, but must involve deterministic measurements: Dirichlet processes are indifferent universal measurements reaching every way of selecting an interaction, whereas an indifferent universal measurement free of deterministic components necessarily selects the uniform density, so that the Born rule holds not only on average but with certainty. Deviations from the quantum statistics in sequences of measurements are governed by a single parameter, in principle accessible to experiment. The only geometric inputs are the Euclidean length of the membrane and the orthogonal landing of the point-particle on it; the latter is where the content of Gleason's theorem lies. Generalizations to N outcomes and to Bertrand's paradox are discussed.
We present a very general geometrico-dynamical description of physical or more abstract entities, called the 'general tension-reduction' (GTR) model, where not only states, but also measurement-interactions can be represented, and the associated outcome probabilities calculated. Underlying the model is the hypothesis that indeterminism manifests as a consequence of unavoidable fluctuations in the experimental context, in accordance with the 'hidden-measurements interpretation' of quantum mechanics. When the structure of the state space is Hilbertian, and measurements are of the 'universal' kind, i.e., are the result of an average over all possible ways of selecting an outcome, the GTR-model provides the same predictions of the Born rule, and therefore provides a natural completed version of quantum mechanics. However, when the structure of the state space is non-Hilbertian and/or not all possible ways of selecting an outcome are available to be actualized, the predictions of the model generally differ from the quantum ones, especially when sequential measurements are considered. Some paradigmatic examples will be discussed, taken from physics and human cognition. Particular attention will be given to some known psychological effects, like question order effects and response replicability, which we show are able to generate non-Hilbertian statistics. We also suggest a realistic interpretation of the GTR-model, when applied to human cognition and decision, which we think could become the generally adopted interpretative framework in quantum cognition research.
A generalized Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a general solution to the measurement problem, inasmuch it allows to derive the Born rule as an average over hidden-variables, describing not the state of the quantum entity, but its interaction with the measuring system. According to this modelization, a quantum measurement is to be understood, in general, as a tripartite process, formed by an initial deterministic decoherence-like process, a subsequent indeterministic collapse-like process, and a final deterministic purification-like process. We also show that quantum probabilities can be generally interpreted as the probabilities of a first-order non-classical theory, describing situations of maximal lack of knowledge regarding the process of actualization of potential interactions, during a measurement.
Quantum mechanics emerged as the result of a successful resolution of stringent empirical and profound conceptual conflicts within the development of atomic physics at the beginning of the last century. At first glance, it seems to be bizarre and even ridiculous to apply ideas of quantum physics in order to improve current psychological and linguistic or semantic ideas. However, a closer look shows that there are some parallels in the development of quantum physics and advanced theories of cognitive science. In psychology, geometric models of meaning have a long tradition. However, they suffer from many shortcomings which are illustrated by discussing several puzzles of bounded rationality. In the first part of this article, we argue that the present account of quantum cognition - taking quantum probabilities rather than classical probabilities - can give a more systematic description of these puzzles than the alternate and rather eclectic treatments in the traditional framework of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always and does not automatically provide a deeper understanding and a true explanation of these puzzles. In the second part of this article, we explain the foundational issue from the perspective of Piron, Foulis, Randall, and others, and we apply it to the foundation of quantum cognition. In this connection, we show that quantum probabilities are of (virtual) conceptual necessity if grounded in an abstract algebraic framework of orthomodular lattices. This framework is motivated by assuming partial Boolean algebras (describing particular perspectives) that are combined into a uniform system while considering certain capacity restrictions. It is at this point that one important aspect of the whole idea of bounded rationality directly enters the theoretical scenery of quantum cognition: resource limitation.