Geometry of quantum noise

Open quantum systems refer to systems that are affected by interaction with the environment. The effects of these unwanted interactions, called \emph{quantum noise}, are studied using dynamical maps. We study the geometry of these maps in this work. We review the canonical representations of dynamical maps such as reduced dynamics, $A$ and $B$ forms and operator sum representation. We develop a framework for simplifying the action of dynamical maps in terms of their action on the coherence vector associated with the density matrix. We use the framework to describe the geometry of depolarization, dephasing and dissipation in the domain of complete positivity. We give a geometric picture of how two-, three- and four-level systems are affected by these common forms of quantum noises. We show useful similarities between two- and four-level depolarizing maps and give a generalization for $n$-qubits. We also derive important results that restrict dephasing and dissipation.