In this dissertation an explicit parameterization for the groups SU(N) and U(N) is given in terms of the Euler-Rodriguez angles. The corresponding Haar measure and invariant volume element specific to this parameterization is then calculated. The importance of these calculations is then manifested in the development of a generalized representation of any N by N density matrix and its corresponding pure and mixed state volume measure, in terms of the N2 - 1 parameters defining SU(N). Applications to quantum information theory are then presented through the explicit calculation of the manifold of unitary transformations which entangle two two-state systems, otherwise known as two qubit systems, and one two-state system with one three-state system, otherwise known as a qubit/qutrit system. Hypothesized volumes of the corresponding set of entangled two qubit and qubit/qutrit states are then presented.
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