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Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection. (English) Zbl 1333.81039

Summary: 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 \(\mathrm{SU}(2)\). We show that the quaternionic stereographic projection intertwines between local unitary group \(\mathrm{SU}(2)\otimes\mathrm{SU}(2)\) 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 \(\mathrm{Sp}(2)\), generalizing the familiar Bloch sphere in two-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.

MSC:

81P40 Quantum coherence, entanglement, quantum correlations
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