Geometry of entangled states, Bloch spheres and Hopf fibrations

Abstract

Abstract. 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 3. The S 2 base space of a suitably oriented S 3 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 7, which also allows for a Hopf fibration, with S 3 fibres and a S 4 base. A main striking result is that suitably oriented S 7 Hopf fibrations are entanglement sensitive. The relation with the standard Schmidt decomposition is also discussed 1.