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 Möbius 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 Möbius 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.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 Möbius 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 Möbius 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.

arXiv:1501.06013Jan 24, 20156 citations

Two and Three-Qubits Geometry, Quaternionic and Octonionic Conformal maps, and Intertwining Hopf Fibration

G. Najarbashi, B. Seifi, Sevda Mirzaei

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 Riemanian sphere under the action of group $SU(2)$ . We show that the second Hopf fiberation intertwines between local unitary group $SU(2)\otimes SU(2)$ and quaternionic M\obius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining Hopf map for much more global group $Sp(2)$ , generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce third Hopf fiberation and octonionic conformal map (or octonionic M\obius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps have entanglement sensitive.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 Riemanian sphere under the action of group $SU(2)$. We show that the second Hopf fiberation intertwines between local unitary group $SU(2)\otimes SU(2)$ and quaternionic M\obius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining Hopf map for much more global group $Sp(2)$, generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce third Hopf fiberation and octonionic conformal map (or octonionic M\obius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps have entanglement sensitive.

Jan 24, 20150 citations

Four-Qubit Geometry

Zhang Chi Liu

Sep 1, 20260 citations

Two-Qubit and Three-Qubit Geometry and Hopf Fibrations

Rémy Mosseri

Springer series in solid-state sciencesJan 1, 200614 citations

SU(2)

. We show that the quaternionic stereographic projection intertwines between local unitary group

SU(2)\otimes SU(2)

and quaternionic Möbius 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 Möbius 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.

SU(2)

. We show that the second Hopf fiberation intertwines between local unitary group

SU(2)\otimes SU(2)

and quaternionic M\obius transformation. The invariant term appearing in this operation is related to concurrence measure. Yet, there exists the same intertwining Hopf map for much more global group

Sp(2)

, generalizing the familiar Bloch sphere in 2-level systems. Subsequently, we introduce third Hopf fiberation and octonionic conformal map (or octonionic M\obius maps) for three-qubit states and find evidence that they may have invariant terms under local unitary operations which shows that both maps have entanglement sensitive.