Relation Between Stereographic Projection and Concurrence Measure in Bipartite Pure States

Abstract

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.

Comparison of qubit and qutrit like entangled squeezed and coherent states of light

“…In order to compare this results with ECS we also display the results of τ (c) ABD calculated in Ref. [44]. Figure 11 shows that in general τ…”

“…is also plotted in figure 1 (for further details see [44]). A comparison between full and dashed line in figure 1 shows that for f > 0 the concurrence of squeezed state is less than the concurrence of coherent state.…”

“…In Ref. [44] the production and entanglement properties of the generalized balanced N-mode Glauber coherent states of the form…”

Quantum phase transition in the Dzyaloshinskii–Moriya interaction with inhomogeneous magnetic field: Geometric approach

“…Now we are define the second Hopf fibration by the map as the composition of a stereographic projection P from S 7 to R 4 + {∞}, followed by an inverse stereographic projection S from 10) or explicitly the fibration P maps the state |ψ Q as…”

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“…See e.g. L. V. Ahlfors [2] (page 19) and E. Hille [31] (pages [38][39][40][41][42][43][44] for derivations. In 1881, Poincare proposed a different mapping where the fixed projection point coincides with the center of the sphere.…”