Geometric characterization of mixed quantum states

Characterization of mixed quantum states represented by density operator is one of the most important task in quantum information processing. In this work we will present a geometric approach to characterize the density operator in terms of fiber bundle over a quantum phase space. The geometrical structure of the quantum phase space of an isospectral mixed quantum states can be realized as a co-adjoint orbit of a Lie group equipped with a specific K\"{a}hler structure. In particular we will briefly discuss the construction of a fiber bundle over the quantum phase space based on symplectic reduction and purification method. We will also show that the map is a Riemannian submersion which enable us to provide some applications of the geometric framework such as geometric phase and quantum speed limit.