Two-Qubit and Three-Qubit Geometry and Hopf Fibrations

Zhang Chi Liu

Rémy Mosseri

This paper reviews recent attempts to describe the two- and three-qubit Hilbert space geometries with the help of Hopf fibrations. In both cases, it is shown that the associated Hopf map is strongly sensitive to states entanglement content. In the two-qubit case, a generalization of the one-qubit celebrated Bloch sphere representation is described.

Rémy Mosseri, Rossen Dandoloff

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 S 3 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 7 , 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.

Т.Э. Кренкель

Lech Jakóbczyk, M. Siennicki

Hoshang Heydari

We establish relations between conifold, Segre variety, Hopf fibration, and separable sets of pure two-qubit states. Moreover, we investigate the geometry and topology of separable sets of pure multi-qubit states based on the complex multi-projective Segre variety and higher order Hopf fibration. We show that the Segre variety and Hopf fibration give a unified geometrical and topological picture of multi-qubit states. We also construct entanglement monotones for multi-qubit states.