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.
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The paper revisits whether the Born rule can arise by averaging over possible measurement-interaction distributions in the extended Bloch representation. In this model, a state projects orthogonally onto a measurement simplex, and an interaction sampled from a distribution $\rho$ determines the outcome. The question is how to define an average over distributions without discretization-dependent ambiguity.
A random probability measure is called indifferent when its law is invariant under permutations of equal-size cells at every resolution. Exchangeability and normalization force its mean measure to be uniform, yielding Born probabilities within the assumed geometry. For cellwise-constant densities, permutation invariance gives this result exactly at any finite resolution. Dirichlet processes provide indifferent laws with full weak support over probability measures. A variance argument establishes rigidity: an indifferent law selecting only non-atomic measures must select the uniform measure almost surely. Otherwise, second-order fluctuations are determined by $\beta=\mathbb{E}\sum_i w_i^2$, the expected squared atomic weights: $\operatorname{Var}[F_\rho(x)]=\beta p(1-p)$, with $p=(1+x)/2$. For a Dirichlet process of concentration $\alpha$, $\beta=1/(1+\alpha)$.
If the selected measure persists across identically prepared trials, the joint success probability is $p^2+\beta p(1-p)$; independent reselection restores independent Born statistics. Higher-order statistics need not be fixed by $\beta$. The construction assumes the geometric reference measure and orthogonal projection rather than deriving them. Indifference is sufficient, not necessary, for Born averages; weak support does not imply exact selection of every distribution. No experimental bound on persistence or $\beta$ is obtained.
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and a first draft of the manuscript, were developed by the author drawing also on a dialogue with the artificial intelligence assistant Claude (Anthropic).
Error found: The abstract overstates the role of beta: indifferent measurement laws with the same beta can produce different three-run probabilities, so beta does not determine general sequence deviations.
The abstract's identification of orthogonal landing with Gleason's theorem exceeds the body's restricted connection for scalar-product deformations in dimensions N >= 3.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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.
The paper revisits whether the Born rule can arise by averaging over possible measurement-interaction distributions in the extended Bloch representation. In this model, a state projects orthogonally onto a measurement simplex, and an interaction sampled from a distribution ρ determines the outcome. The question is how to define an average over distributions without discretization-dependent ambiguity.
A random probability measure is called indifferent when its law is invariant under permutations of equal-size cells at every resolution. Exchangeability and normalization force its mean measure to be uniform, yielding Born probabilities within the assumed geometry. For cellwise-constant densities, permutation invariance gives this result exactly at any finite resolution. Dirichlet processes provide indifferent laws with full weak support over probability measures. A variance argument establishes rigidity: an indifferent law selecting only non-atomic measures must select the uniform measure almost surely. Otherwise, second-order fluctuations are determined by β=E∑iwi2, the expected squared atomic weights: Var[Fρ(x)]=βp(1−p), with p=(1+x)/2. For a Dirichlet process of concentration α, β=1/(1+α).
If the selected measure persists across identically prepared trials, the joint success probability is p2+βp(1−p); independent reselection restores independent Born statistics. Higher-order statistics need not be fixed by β. The construction assumes the geometric reference measure and orthogonal projection rather than deriving them. Indifference is sufficient, not necessary, for Born averages; weak support does not imply exact selection of every distribution. No experimental bound on persistence or β is obtained.
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.
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.
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.
Abstract This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or ’t Hooft’s Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness (or algorithmic randomness) that in various guises was investigated by Kolmogorov and others and treat Bell’s (Physics 1:195–200, 1964) Theorem and the Free Will Theorem with their implications for randomness.