Limiting Operations for Quantum Random Variables and a Quantum Martingale Convergence Theorem

Kyler Scott Bridgen Johnson

Published Apr 1, 2014• 0 citations

Abstract

This thesis builds on the notion of quantum, or operator valued, probability as discussed in Farenick, Plosker, and Smith (J. Math. Phys, 2011) as well as Farenick and Kozdron (J. Math. Phys, 2012) by generalizing classical limiting results to the quantum setting. Mimicking the classical setting, we prove a continuity of quantum expectation result, which is the quantum analogue of Lebesgue's dominated convergence theorem and use it extensively to prove other limiting results. With the quantum limiting results in place, we de ne a quantum martingale and prove a quantum martingale convergence theorem. This quantum martingale theorem is of particular interest since it exhibits non-classical behaviour; even though the limit of the martingale exists and is unique, it is not even identi able. In order to understand the limit it is necessary to understand the space of all mean zero quantum random variables. However, we are able to provide a partial classi cation of the limit through a study of the space of all mean zero quantum random variables.

DOI: 10.82465/6657

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