Quantum Measurements and Kolmogorov's Theory of Probability

Oleg Georgievich Smolyanov, Andrei Yu. Khrennikov

Andrei Yu. Khrennikov

We present comparative probabilistic analysis of the Greenburger-Horne-Zeilinger paradox in the frameworks of Kolmogorov's (measure-theoretical) and von Mises' (frequency) models of the probability theory. This analysis demonstrated that the GHZ paradox is merely a consequence of the use of Kolmogorov's probabilistic model. By using von Mises' frequency approach we escape the contradiction between the local realism and quantum formalism. The frequency approach implies automatically contextual interpretation of quantum formalism: different collectives induce different probability distributions. On the other hand, the formal use of Kolmogorov's model implies the identification of such distributions with one abstract Kolmogorov measure. In the measure-theoretical approach we can escape the paradox, if we do not suppose that probability distributions corresponding to different settings of measurement devices are equivalent. We discuss the connection between equivalence/singularity dichotomy in measure theory and the existence of compatible and noncompatible observables.

Luigi Accardi

Shiro Ishikawa

Recently, we proposed quantum language (or, measurement theory), which is characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. We believe that this language has a great powet of description, and therefore, even statistics can be described by quantum language. In this paper, we show that ANOVA (analysis of variance (one-way and two-way)) can be formulated in quantum language. Since quantum language is suited for theoretical arguments, we believe that our results are visible and understandable. For example, we can answer the question "What kind of role does Kolmogorov's probability theory play in ANOVA?" That is, the readers find that Kolmogorov's probability theory is merely used in order to calculate multi-dimenstional Gauss integrals, and thus, they can avoid to confuse the relation between Kolmogorov's probability theory and statistics.

Andrei Yu. Khrennikov

Revista, Zen, MATH, 10

This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.