Two- and three-qubit 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 $$SU(2)\otimes SU(2)$$SU(2)?SU(2) and quaternionic Mobius 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 two-level systems. Subsequently, we introduce octonionic stereographic projection and octonionic conformal map (or octonionic Mobius 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.