# Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection

Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining... 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 $$SU(2)\otimes SU(2)$$ S U ( 2 ) ⊗ S U ( 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 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection

, Volume 15 (1) – Nov 13, 2015
20 pages

/lp/springer_journal/two-and-three-qubit-geometry-quaternionic-and-octonionic-conformal-1ghk00bC3z
Publisher
Springer Journals
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1172-0
Publisher site
See Article on Publisher Site

### 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 $$SU(2)\otimes SU(2)$$ S U ( 2 ) ⊗ S U ( 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 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.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Nov 13, 2015

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