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

Quantum Studies Mathematics and Foundations

Addendum: On the applicability of Kolmogorov’s theory of probability to the description of quantum phenomena. Part I: foundations

Maik Reddiger

Published Sep 17, 2026• 0 citations
Publisher
DOI
10.1007/s40509-026-00413-x

Related papers

Quantum Measurements and Kolmogorov's Theory of Probability

D. A. Slavnov

Theoretical and Mathematical Physics•Sep 1, 2003•7 citations

On the applicability of Kolmogorov’s theory of probability to the description of quantum phenomena. Part I: foundations

Maik Reddiger

Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.

arXiv:2405.05710•Quantum Studies Mathematics and Foundations•Oct 23, 2025•2 citations

On the applicability of Kolmogorov's theory of probability to the description of quantum phenomena. Part II: Bell inequalities

Maik Reddiger

Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions. This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation. Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.

quant-pharXiv:2609.31075•Sep 28, 2026•0 citations

Interpretations of Probability and Their p -Adic Extensions

Andrei Yu. Khrennikov

This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called p-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.

Theory of Probability and Its Applications•Jan 1, 2002•14 citations

Quantum theory and non-kolmogorovian probability

Luigi Accardi

Springer eBooks•Apr 6, 2008•5 citations

Non-Kolmogorov Probability and Quantum Physics

Andrei Yu. Khrennikov

Springer eBooks•Jan 1, 1997•3 citations
Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.
Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions. This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation. Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.
This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called p-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.