The unreasonable success of quantum probability I: Quantum measurements as uniform fluctuations

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.