From Hopf fiberation to duality-entanglement relation

We show that the entanglement-duality relation [Optica 5, 942 (2018)] can indeed be obtained, in its most generic form, from Hopf fiberation and stereographic geometry. We demonstrate that this relation is a natural consequence of the stereographic geometry, providing the first geometric approach to the notion of complementarity problem. We show that this geometry is duality-entanglement sensitive. This means that, it is sensitive to the wave nature, particle nature and entanglement nature of a single qubit.