Geometry of entangled states, Bloch spheres and Hopf fibrations

We discuss a generalization of the standard Bloch sphere representation for a single qubit to two qubits, in the framework of Hopf fibrations of high-dimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is the three-dimensional sphere S-3. The S-2 base space of a suitably oriented S3 Hopf fibration is nothing but the Bloch sphere, while the circular fibres represent the overall qubit phase degree of freedom. For the two-qubits case, the Hilbert space is a seven-dimensional sphere S-1, which also allows for a Hopf fibration, with S-3 fibres and a S-4 base. The most striking result is that suitably oriented S-7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed.