The geometry and topology of entanglement: Conifold, Segre variety, and Hopf fibration

Abstract

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.