Geometry of Entangled States , Bloch Spheres and Hopf Fibrations

We discuss a generalization to 2 qubits of the standard Bloch sphere representation for a single qubit, in the framework of Hopf fibrations of high dimensional spheres by lower dimensional spheres. The single qubit Hilbert space is the 3-dimensional sphere S. The S base space of a suitably oriented S Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the qubit overall phase degree of freedom. For the two qubits case, the Hilbert space is a 7-dimensional sphere S, which also allows for a Hopf fibration, with S fibres and a S base. A main striking result is that suitably oriented S Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed