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

2015/01/24 by G. Najarbashi, Najarbashi, G., B. Seifi +3
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.1501.06013

openalex publication_date 2015/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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)⊗ 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.

Citations

• Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?
• Entanglement of Formation of an Arbitrary State of Two Qubits
• Separability Criterion for Density Matrices

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