Two and Three-Qubits Geometry, Quaternionic and Octonionic Conformal Maps, and Intertwining Stereographic Projection

Abstract

In this paper the geometry of two and three-qubit states under local unitary groups is discussed. We first review the one qubit geometry and its relation with Riemannian sphere under the action of group $SU(2)$. We show that the quaternionic stereographic projection intertwines between local unitary group $SU(2)\otimes SU(2)$ and quaternionic M\"{o}bius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining stereographic projection for much more global group $Sp(2)$, generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic M\"{o}bius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps are entanglement sensitive.