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.PACs Index: 03.67.a 03.65.Ud
One-qubit pure states, living on the surface of Bloch sphere, can be mapped onto the usual complex plane by using stereographic projection. In this paper, after reviewing the entanglement of two-qubit pure state, it is shown that the quaternionic stereographic projection is related to concurrence measure. This is due to the fact that every two-qubit state, in ordinary complex field, corresponds to the one-qubit state in quaternionic skew field, called quaterbit. Like the one-qubit states in complex field, the stereographic projection maps every quaterbit onto a quaternion number whose complex and quaternionic parts are related to Schmidt and concurrence terms respectively. Rather, the same relation is established for three-qubit state under octonionic stereographic projection which means that if the state is bi-separable then, quaternionic and octonionic terms vanish.Finally, we generalize recent consequences to 2 ⊗ N and 4 ⊗ N dimensional Hilbert spaces (N ≥ 2) and show that, after stereographic projection, the quaternionic and octonionic terms are entanglement sensitive. These trends are easily confirmed by direct computation for general multi-particle W-and GHZ-states.
Abstract-Numerical simulation of the deformation process during flat rolling of multifilament HTS tapes has been investigated using a commercial FEM program, ELFEN. The numerical models were built up in 2D and 3D using a DruckerPragerKap model for the powder. Three different roll diameters (024 mm, 0 8 5 mm and 0126 mm) have been investigated.It is found that it is possible to perform numerical simulation with 3D models of flat rolling of multifilament wire. Two 3D models have been used; 3D pressing with rolls and 3D rolling. 3D pressing with rolls have the advance that the simulation time is lower than in the 3D rolling. The 3D models have the advantage compared to 2D pressing that they can predict the 3 dimensional flow in the flat rolling, which has been showed to be very imported for the super conduction properties.Indm Terms-Finite Element Modelling, superconductor, flat r o I I i n g .
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