Geometrical description of the dynamics of entangled two-qubit states under U(2) x U(2) local unitary operations

We realize the second Hopf fibration for a two-qubit system without using the quaternionic language. In this respect, we explore the geometrical features emerging from this Hopf fibration. Further, we investigate the metric tensor and the SO(4) non-abelian gauge field defined on the S-4-base in terms of the entanglement quantified by the Wootters concurrence on the associated Hopf bundle. Finally, by transforming an entangled two-qubit state in the Schmidt form, we examine the different quantum phases acquired by this state under U(2) x U(2) local unitary operations in relation to the entanglement as well as the geometry of the corresponding state manifold.