We study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of Lebesgue's dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence 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 explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
We study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of Lebesgue's dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence 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 explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
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.
It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
Martingales are fundamental stochastic process used to model the concept of fair game. They have a multitude of applications in the real world that include, random walks, Brownian motion, gamblers fortunes and survival analysis, Just as commutative integration theory may be realised as a special case of the more general non-commutative theory for integrals, so too, we find classical probability may be realised as a limiting, special case of quantum probability theory. In this thesis we are concerned with the development of multiparameter quantum stochastic integrals extending non-commutative constructions to the general n parameter case, these being multiparameter quantum stochastic integrals over the positive n - dimensional plane, employing martingales as integrator. The thesis extends previous analogues of type one, and type two stochastic integrals, for both Clifford and quasi free representations. As with one and two dimensional parameter sets, the stochastic integrals constructed form orthogonal, centred L2 - martingales, obeying isometry properties. We further explore analogues for weakly adapted processes, properties relating to the resulting quantum stochastic integrals, develop analogues to Fubini’s theorem, and explore applications for quantum stochastic integrals in a security setting.
Open Quantum Random Walks, as developed in \cite{APSS}, are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on $\ZZ^d$ we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on $\ZZ^d$ times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a Central Limit Theorem for the sequence of recordings of the quantum trajectories associated to any completely positive map. This is what we show and develop as an application of our result. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.
Open Quantum Random Walks are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on nets, their distribution seems to always converges to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z^d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z^d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.
Abstract A quantization of a Hamiltonian system is an ambiguous procedure. Accordingly, we introduce the notion of random quantization, related random variables with values in the set of self-adjoint operators, and random processes with values in the group of unitary operators. The procedures for the averaging of random unitary groups and averaging of random self-adjoint operators are defined. The generalized weak convergence of a sequence of measures and the corresponding generalized convergence in distribution of a sequence of random variables are introduced. The generalized convergence in distribution for some sequences of compositions of random mappings is obtained. In the case of a sequence of compositions of shifts by independent random vectors of Euclidean space, the obtained convergence coincides with the statement of the central limit theorem for a sum of independent random vectors. The results are applied to the dynamics of quantum systems arising in random quantization of a Hamiltonian system.