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IntechOpen eBooks

An Alternative Approach to Probability in Quantum Information Science

Christian Jansson

Published May 8, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.5772/intechopen.1005070

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De Gruyter eBooks•Oct 24, 2019•0 citations
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.
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 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.
When quantum mechanics is understood as a new generalized theory of probability - to be called the quantum probability theory - mysteries and controversies regarding quantum mechanics are dissolved. In the classical probability theory, that a measurement of some system requires an additional measurement apparatus is of insignificant importance - in the quantum probability theory, this comes to change. For one central single reason around a particular classical probability equation, the generalized probability view has not gained much traction, despite the fact that this essentially echoes (and provides logical underpinnings of) the conventional wisdom that `quantum mechanics just works as it is.' A classical probability axiom is just an initial intuition - there is no reason why we have to dogmatically cling onto axioms that can clearly be generalized. Issues with the principle of indifference in the classical probability theory are emphasized, along with the quantum reconstruction project of deriving quantum mechanics from epistemic requirements and potential quantum gravity consequences from the principle of maximum entropy.
Born’s probabilistic interpretation of quantum mechanics is commonly assumed to rely on classical probability theory in the sense of Kolmogorov. In this paper we argue that this assumption is methodologically unjustified. We show that quantum probabilities naturally arise from an analytic method fundamentally different from the synthetic approach underlying classical probability theory. In the analytic framework, the whole (the totality of alternatives) is primary, while probabilities of individual alternatives are derived quantities. Interference appears at the level of the realization probability of the whole, whereas conditional probabilities of alternatives remain additive within a given context. We demonstrate that many well-known quantum paradoxes originate from the inappropriate application of synthetic probabilistic reasoning. The proposed formulation is consistent with the Born rule, its extensions developed in our earlier works, and with Birkhoff–von Neumann quantum logic. Classical probability theory appears only as a degenerate formal limit and is not methodologically universal.
In this paper we propose a unified conceptual framework in which quantum probability, quantum logic, and classical probability theory are understood as different structural levels of a single theoretical hierarchy. Starting from a statistical interpretation of quantum interference, previously developed by the authors, we argue that quantum probability is not an alternative to classical Kolmogorov probability, but a more general structure. Classical probability theory emerges as a limiting case corresponding to the disappearance of interference and the effective decomposition of the whole into independent elements. Within this framework, the Birkhoff–von Neumann quantum logic appears as a natural logical expression of quantum probability rather than as an abstract algebraic construction detached from physical meaning. The priority of the whole over its parts plays a central role in both quantum probability and quantum logic, while classical logic and probability arise when this priority becomes trivial. We emphasize that the proposed interpretation does not reject existing quantum formalism, but reinterprets its logical and probabilistic foundations. This approach may have implications for the conceptual understanding of quantum algorithms and quantum information processing.
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.