Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Linear representations of probabilistic transformations induced by context transitions

Andrei Yu. Khrennikov

Published Jan 1, 2001• 89 citations
OpenAlex

Abstract

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. Each class is characterized by a perturbation of the `classical probabilistic rule': (a) $\\cos\\theta$, (b) $\\cosh \\theta$, (c) both $\\cos\\theta$ and $\\cosh \\theta$. Trigonometric transformations correspond to context-transitions that induce statistical deviations of relatively small magnitudes (in classical physics - negligibly small); hyperbolic - relatively large magnitudes. We found that not only preparation procedures described by conventional quantum formalism can have trigonometric probabilistic behaviour. We propose generalizations of ${\\bf C}$-linear space probabilistic calculus to describe non quantum (trigonometric and hyperbolic) probabilistic transformations.

Related papers

Linear representations of probabilistic transformations induced by context transitions

Andrei Yu. Khrennikov

By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C -linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.

arXiv:quant-ph/0105059•Journal of Physics A Mathematical and General•Nov 21, 2001•73 citations

`Quantum probabilities' as context depending probabilities

Andrei Yu. Khrennikov

We study transformations of conventional (`classical') probabilities induced by context transitions. It is demonstrated that the transition from one complex of conditions to another induces a perturbation of the classical rule for the addition of probabilistic alternatives. We classify such perturbations. It is shown that there are two classes of perturbations: (a) trigonometric interference; (b) hyperbolic interference. In particular, the well known `quantum interference of probabilistic alternatives' can be obtained in classical (but contextual) probabilistic framework. Therefore we need not apply to wave arguments (or consider superposition principle) to get Quantum Statistics. In particular, interference could be a feature of experiments with purely corpuscular objects.

arXiv:quant-ph/0106073•Jun 13, 2001•12 citations

Classification of transformations of probabilities for preparation procedures: trigonometric and hyperbolic behaviours

Andrei Yu. Khrennikov

We provide frequency probabilistic analysis of perturbations of physical systems by preparation procedures. We obtained the classification of possible probabilistic transformations connecting input and output probabilities that can appear in physical experiments. We found that so called quantum probabilistic rule is just one of possible rules. Besides the well known trigonometric transformation (for example, for the polarization of light), there exist the hyperbolic transformation of probabilities. In fact, `hyperbolic polarization' have laready been observed in experiments with elementary particles. However, it was not interpreted in such a way. The situation is more complex with the hyperbolic interference of alternatives.

arXiv:quant-ph/0012141•Dec 24, 2000•2 citations

On the mystery of quantum probabilistic rule: trigonometric and hyperbolic probabilistic behaviours

Andrei Yu. Khrennikov

We demonstrate the quantum probabilistic rule (which differ from classical Bayes' formula by the cosinus factor) can be obtained on purely classical basis as a consequence of the perturbation effect of preparation procedures. In any case the interference can be easily simulated for macrosystems. Moreover, we obtain the classification of all possible transformations of probablities which can arise via perturbation effects. There are two main transformation classes: with trigonometric and hyperbolic ("interference") perturbations of classical Bayes' formula. Quantum probabilistic rule is just a particular case of trigonometric probablistic behaviour. Hyperbolic probablistic behaviour can be easily simulated. However, the question of its observation for "real physical phenomena" is open.

arXiv:quant-ph/0011104•Nov 25, 2000•0 citations

Quantum statistics via perturbation effects of preparation procedures

Andrei Yu. Khrennikov

We study the following problem: Is it possible to explain the quantum interference of probabilities in the purely corpuscular model for elementary particles? We demonstrate that (by taking into account perturbation effects of measurement and preparation procedures) we can obtain $\cosθ$-perturbation (interference term) in probabilistic rule connecting preparation procedures for purely corpuscular objects. On one hand, our investigation demonstrated that there is nothing special in so called `quantum probabilities': the right choice of statistical ensembles gives the possibility to escape all `pathologies'. On the other hand, we found that the standard trigonometric interference of alternatives (observed, in particular, in quantum mechanics) is not the unique possibility to extend (disturb) the conventional probabilistic rule for addition of alternatives. There exist two other probabilistic rules that connect three preparation procedures: hyperbolic and hyper-trigonometric interferences.

arXiv:quant-ph/0103065•Mar 13, 2001•11 citations

Representation of the contextual statistical model by hyperbolic amplitudes

Andrei Yu. Khrennikov

We continue the development of a so-called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric cos-interference, there exist contexts producing the hyperbolic cos-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born’s rule. Incompatible observables are represented by noncommutative operators. This paper can be considered as the first step towards hyperbolic quantum probability. We also discuss possibilities of experimental verification of hyperbolic quantum mechanics: in physics of elementary particles, string theory as well as in experiments with nonphysical systems, e.g., in psychology, cognitive sciences, and economy.

arXiv:quant-ph/0408188•Journal of Mathematical Physics•Jun 1, 2005•9 citations
By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. Each class is characterized by a perturbation of the `classical probabilistic rule': (a) costheta\\cos\\thetacostheta, (b) coshtheta\\cosh \\thetacoshtheta, (c) both costheta\\cos\\thetacostheta and coshtheta\\cosh \\thetacoshtheta. Trigonometric transformations correspond to context-transitions that induce statistical deviations of relatively small magnitudes (in classical physics - negligibly small); hyperbolic - relatively large magnitudes. We found that not only preparation procedures described by conventional quantum formalism can have trigonometric probabilistic behaviour. We propose generalizations of bfC{\\bf C}bfC-linear space probabilistic calculus to describe non quantum (trigonometric and hyperbolic) probabilistic transformations.
By using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure (context) to another. There are three classes of transformations corresponding to statistical deviations of different magnitudes: (a) trigonometric; (b) hyperbolic; (c) hyper-trigonometric. It is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour. We propose generalizations of C -linear space probabilistic calculus to describe non-quantum (trigonometric and hyperbolic) probabilistic transformations. We also analyse the superposition principle in this framework.
We study transformations of conventional (`classical') probabilities induced by context transitions. It is demonstrated that the transition from one complex of conditions to another induces a perturbation of the classical rule for the addition of probabilistic alternatives. We classify such perturbations. It is shown that there are two classes of perturbations: (a) trigonometric interference; (b) hyperbolic interference. In particular, the well known `quantum interference of probabilistic alternatives' can be obtained in classical (but contextual) probabilistic framework. Therefore we need not apply to wave arguments (or consider superposition principle) to get Quantum Statistics. In particular, interference could be a feature of experiments with purely corpuscular objects.
We provide frequency probabilistic analysis of perturbations of physical systems by preparation procedures. We obtained the classification of possible probabilistic transformations connecting input and output probabilities that can appear in physical experiments. We found that so called quantum probabilistic rule is just one of possible rules. Besides the well known trigonometric transformation (for example, for the polarization of light), there exist the hyperbolic transformation of probabilities. In fact, `hyperbolic polarization' have laready been observed in experiments with elementary particles. However, it was not interpreted in such a way. The situation is more complex with the hyperbolic interference of alternatives.
We demonstrate the quantum probabilistic rule (which differ from classical Bayes' formula by the cosinus factor) can be obtained on purely classical basis as a consequence of the perturbation effect of preparation procedures. In any case the interference can be easily simulated for macrosystems. Moreover, we obtain the classification of all possible transformations of probablities which can arise via perturbation effects. There are two main transformation classes: with trigonometric and hyperbolic ("interference") perturbations of classical Bayes' formula. Quantum probabilistic rule is just a particular case of trigonometric probablistic behaviour. Hyperbolic probablistic behaviour can be easily simulated. However, the question of its observation for "real physical phenomena" is open.
We study the following problem: Is it possible to explain the quantum interference of probabilities in the purely corpuscular model for elementary particles? We demonstrate that (by taking into account perturbation effects of measurement and preparation procedures) we can obtain cos⁡θ\cosθcosθ-perturbation (interference term) in probabilistic rule connecting preparation procedures for purely corpuscular objects. On one hand, our investigation demonstrated that there is nothing special in so called `quantum probabilities': the right choice of statistical ensembles gives the possibility to escape all `pathologies'. On the other hand, we found that the standard trigonometric interference of alternatives (observed, in particular, in quantum mechanics) is not the unique possibility to extend (disturb) the conventional probabilistic rule for addition of alternatives. There exist two other probabilistic rules that connect three preparation procedures: hyperbolic and hyper-trigonometric interferences.
We continue the development of a so-called contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric cos-interference, there exist contexts producing the hyperbolic cos-interference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born’s rule. Incompatible observables are represented by noncommutative operators. This paper can be considered as the first step towards hyperbolic quantum probability. We also discuss possibilities of experimental verification of hyperbolic quantum mechanics: in physics of elementary particles, string theory as well as in experiments with nonphysical systems, e.g., in psychology, cognitive sciences, and economy.