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

arXiv:quant-ph/0103065

Quantum statistics via perturbation effects of preparation procedures

Andrei Yu. Khrennikov

Published Mar 13, 2001• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0103065

Related papers

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 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

Interference of probabilities in purely corpuscular model

Andrei Yu. Khrennikov

We show that, in spite of a rather common opinion, interference of probabilities can be easily obtained in purely corpuscular statistical model. All distinguishing features of the quantum probabilistic model (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, classical model should be considered as a contextual model (in the sense that all probabilities are determined by contexts-complexes of physical conditions). One of the main consequences of our contextual probabilistic reconsideration of foundations of quantum mechanics is that interference can be explained without wave-particle duality.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 1, 2005•0 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

Ensemble fluctuations and the origin of quantum probabilistic rule

Andrei Yu. Khrennikov

We demonstrate that the origin of the so-called quantum probabilistic rule (which differs from the classical Bayes’ formula by the presence of cos θ-factor) might be explained in the framework of ensemble fluctuations which are induced by preparation procedures. In particular, quantum rule for probabilities (with nontrivial cos θ-factor) could be simulated for macroscopic physical systems via preparation procedures producing ensemble fluctuations of a special form. We discuss preparation and measurement procedures which may produce probabilistic rules which are neither classical nor quantum; in particular, hyperbolic “quantum theory.”

Journal of Mathematical Physics•Feb 1, 2002•37 citations
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 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 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 show that, in spite of a rather common opinion, interference of probabilities can be easily obtained in purely corpuscular statistical model. All distinguishing features of the quantum probabilistic model (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, classical model should be considered as a contextual model (in the sense that all probabilities are determined by contexts-complexes of physical conditions). One of the main consequences of our contextual probabilistic reconsideration of foundations of quantum mechanics is that interference can be explained without wave-particle duality.
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.
We demonstrate that the origin of the so-called quantum probabilistic rule (which differs from the classical Bayes’ formula by the presence of cos θ-factor) might be explained in the framework of ensemble fluctuations which are induced by preparation procedures. In particular, quantum rule for probabilities (with nontrivial cos θ-factor) could be simulated for macroscopic physical systems via preparation procedures producing ensemble fluctuations of a special form. We discuss preparation and measurement procedures which may produce probabilistic rules which are neither classical nor quantum; in particular, hyperbolic “quantum theory.”