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arXiv:math-ph/0002049 · Journal of Mathematical Physics

Classical and quantum probability

R. F. Streater

Published Jun 1, 2000• 85 citations
arXivHTMLPDF

Abstract

We follow the development of probability theory from the beginning of the last century, emphasizing that quantum theory is really a generalization of this theory. The great achievements of probability theory, such as the theory of processes, generalized random fields, estimation theory, and information geometry, are reviewed. Their quantum versions are then described.

DOI
10.1063/1.533322

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We follow the development of probability theory from the beginning of the last century, emphasizing that quantum theory is really a generalization of this theory. The great achievements of probability theory, such as the theory of processes, generalized random fields, estimation theory, and information geometry, are reviewed. Their quantum versions are then described.
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The mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933.Quantum theory as nonclassical probability theory was incorporated into the beginnings of noncommutative measure theory by von Neumann in the early thirties, as well.To precisely this end, von Neumann initiated the study of what are now called von Neumann algebras and, with Murray, made a first classification of such algebras into three types.The nonrelativistic quantum theory of systems with finitely many degrees of freedom deals exclusively with type I algebras.However, for the description of further quantum systems, the other types of von Neumann algebras are indispensable.The paper reviews quantum probability theory in terms of general von Neumann algebras, stressing the similarity of the conceptual structure of classical and noncommutative probability theories and emphasizing the correspondence between the classical and quantum concepts, though also indicating the nonclassical nature of quantum probabilistic predictions.In addition, differences between the probability theories in the type I, II and III settings are explained.A brief description is given of quantum systems for which probability theory based on type I algebras is known to be insufficient.These illustrate the physical significance of the previously mentioned differences.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi--Dirac anticommutation relations (see the introduction).
In this thesis I present a short review of ideas in quantum information theory. The first chapter contains introductory material, sketching the central ideas of probability and information theory. Quantum mechanics is presented at the level of advanced undergraduate knowledge, together with some useful tools for quantum mechanics of open systems. In the second chapter I outline how classical information is represented in quantum systems and what this means for agents trying to extract information from these systems. The final chapter presents a new resource: quantum information. This resource has some bewildering applications which have been discovered in the last ten years, and continually presents us with unexpected insights into quantum theory and the universe. The treatment is pedagogical and suitable for beginning graduates in the field.