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Cambridge University Press eBooks

Introduction to Quantum Probability Theory

Enrico G. Beltrametti, Gianni Cassinelli, Peter Carruthers

Published Dec 28, 1984• 10 citations
Publisher
DOI
10.1017/cbo9781107340725.033

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This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum Itô calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics.
These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.