Studies of Small Systems in Quantum Information

I study two topics in quantum information theory from the perspective of algebra and geometry. The first relates to exploring the geometry of unitary operators for small quantum systems, specifically three-level systems. Such an understanding of the space over which quantum systems evolve is central to understanding the detailed dynamics of quantum systems and to understand the correlation properties of subsystems that compose a given quantum system. The geometry of unitary operators also allows for the calculation of path-dependent phases called geometric phases. These geometric phases are central to understanding a variety of experiments. I present a general technique, called unitary integration to handle operator equations and employ it to study various physical systems in quantum optics and quantum information. Unitary integration employs an inductive program to solve for the time-evolution of a system in terms of a unitary integration solution of smaller systems. The solution to the smallest system involving just a phase is easily solved, hence truncating the program and providing a solution to the initial problem. Unitary integration is developed in chapters 2 and 3 and this technique is applied to three-level systems in chapter 4. The second topic involves quantum systems involving many subsystems. Understanding the correlation properties of the subsystems that compose such systems has been of interest in the recent past. A useful tool in furthering this understanding has been parametrized families of states. Such states depend on a smaller set of parameters than a general state in the system and hence are easier to study and manipulate. I will present an iterative procedure to define such a parametric family of states called X states. I discuss the algebraic characterization for such states and develop a geometric picture for the algebra of such states. This geometric picture involves generalizations of triangles called ``simplexes\u27. X states are developed along with their algebraic characterization and connections to geometry in chapters 5 and 6. The central theme that is common to both topics is the use of algebraic and geometric concepts to solve for various specific problems in quantum information iteratively. While the first topic deals with the iterative decomposition of operator equations, the second topic deals with the iterative definition of parametrized familes of quantum states