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arXiv:quant-ph/0201014 · Quantum Information Processing

Qubit Geometry and Conformal Mapping

Jae Weon Lee, Chang Ho Kim, Eok Kyun Lee, Jaewan Kim, Soonchil Lee

Published Apr 1, 2002• 19 citations
arXivHTMLPDF
DOI
10.1023/a:1019645000745

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In this paper the geometry of two-qubit systems under local unitary group SO(2)⊗SU(2)SO(2)\otimes SU(2)SO(2)⊗SU(2) is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of Sp(2)Sp(2)Sp(2) and the quaternionic Möbius transformation which is rather a generalization of the results of Lee et al (2002 Quantum Inf. Process. 1 129).
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)SU(2)SU(2). We show that the second Hopf fiberation intertwines between local unitary group SU(2)⊗SU(2)SU(2)\otimes SU(2)SU(2)⊗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)Sp(2)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 Riemannian sphere under the action of group SU(2)SU(2)SU(2). We show that the quaternionic stereographic projection intertwines between local unitary group SU(2)⊗SU(2)SU(2)\otimes SU(2)SU(2)⊗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)Sp(2)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.