With the birth of quantum mechanics, the two disciplines that Hilbert proposed to axiomatize, probability and mechanics, became entangled and a new probabilistic model arose in addition to the classical one. Thus, to meet Hilbert's challenge, an axiomatization should account deductively for the basic features of all three disciplines. This goal was achieved within the framework of quantum probability. The present paper surveys the quantum probabilistic axiomatization.This article is part of the themed issue 'Hilbert's sixth problem'.
With the birth of quantum mechanics, the two disciplines that Hilbert proposed to axiomatize, probability and mechanics, became entangled and a new probabilistic model arose in addition to the classical one. Thus, to meet Hilbert's challenge, an axiomatization should account deductively for the basic features of all three disciplines. This goal was achieved within the framework of quantum probability. The present paper surveys the quantum probabilistic axiomatization.This article is part of the themed issue 'Hilbert's sixth problem'.
This chapter provides a probabilistic framework for formulating classical probability theory, quantum probability, thermodynamics, diffusion, and the Wiener integral using a set of four axioms or principles. It explains everything that conventional quantum information theory and classical probability theory achieve. We want to emphasize that this framework is not an interpretation of quantum mechanics such as “Many-Worlds,” “Bohm’s Theory,” or the “Copenhagen interpretation.” It is much more general and can be viewed as a probability algorithm that calculates probabilities of future events. As a result, previously perplexing paradoxes find resolution. In particular, the superposition principle takes on a new meaning. Our probabilistic framework stands apart from the Hilbert space formalism. It relies solely on elementary set theory, classical logic, and complex numbers. Consequently, this theory is accessible for instruction in educational settings. This framework can be regarded as an axiomatic approach to probability in the sense of Hilbert. In his sixth of the twenty-three open problems presented at the International Congress of Mathematicians in Paris in 1900, Hilbert called for an axiomatic probability theory.
Quantum entanglement, a fundamental phenomenon in quantum mechanics, presents a unique opportunity to revolutionize computation. This paper explores the integration of quantum entanglement and probability theory to develop a novel architecture for quantum computing, aiming to achieve computational capabilities exceeding those of classical systems. We propose a probabilistic model predicated on quantum entanglement, aiming to enhance the efficiency and expand the scope of quantum computation. The core mechanism centers on a synergistic combination of these two disciplines, offering a potential pathway to a new era of computational paradigms. This work provides a foundational investigation into the intersection of quantum mechanics and probability theory, potentially opening doors to entirely new algorithms and computational frameworks.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
Classical probability theory, as axiomatized in 1933 by Andrey Kolmogorov, has provided a useful and almost universally accepted theory for describing and quantifying uncertainty in scientific applications outside quantum mechanics. Recently, cognitive psychologists and mathematical economists have provided examples where classical probability theory appears inadequate but the probability theory underlying quantum mechanics appears effective. Formally, quantum probability theory is a generalization of classical probability. This article explores relationships between generalized probability theories, in particular quantum-like probability theories and those that do not have full complementation operators (e.g. event spaces based on intuitionistic logic), and discusses how these generalizations bear on important issues in the foundations of probability and the development of non-classical probability theories for the behavioural sciences.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.
After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.