The Physical Realizability of Quantum Systems with Applications to Coherent Quantum Control

Quantum versions of control problems are typically more difficult than their classical counterparts because of additional constraints imposed by quantum mechanics. In coherent quantum controller synthesis and optimization problems, the controller is required to be a physically realizable quantum system. This means that the equations describing the synthesized quantum controller must correspond to a meaningful quantum system. A tractable approach to such problems is to use established classical controller synthesis methods to obtain a classical solution and then modify this solution by introducing quantum vacuum noise channels to obtain a physically realizable quantum controller. Quantum vacuum noise channels in the controller place limits on the achievable controller performance. Hence it is desirable to minimize the number of these noises.The aim of this thesis is to improve on existing methods for implementing physically realizable quantum systems. We give a method to implement a strictly proper, linear time invariant system as a physically realizable quantum system. This method introduces a minimal number of quantum vacuum noises. We also give a condition, under which a strictly proper transfer function can be implemented as a physically realizable quantum system, where the number of introduced quantum noises is equal to the output dimension.Singular perturbation approximations are closely related to adiabatic elimination, a common technique within the physics literature for modeling quantum systems. We give two results concerning the physical realizability of the reduced dimension approximate system obtained from singular perturbation approximation. Coherent quantum observers represent an important building block in developing systematic and tractable approaches to coherent control problems. We give three algorithms for the design of coherent quantumobservers and incorporate novel refinements in an attempt to improve performance. In the coherent quantum linear quadratic Gaussian problem, the separation principle of combining an optimal state estimator and an optimal regulator to obtain an optimal controller does not apply. This is due to the quantum noises introduced to obtain a physically realizable controller. Taking this into account, we give an algorithm to obtain a suboptimal solution for the coherent quantum linear quadratic Gaussian problem