Relation Between Stereographic Projection and Concurrence Measure in Bipartite Pure States

TL;DR

This paper explores how stereographic projection in quaternionic and octonionic fields relates to entanglement measures like concurrence in multi-qubit states, revealing that certain algebraic terms vanish for bi-separable states and are entanglement-sensitive.

Contribution

It establishes a novel connection between quaternionic/octonionic stereographic projections and entanglement measures, extending to higher-dimensional bipartite systems.

Findings

Quaternionic terms relate to concurrence in two-qubit states.

Octonionic terms vanish for bi-separable three-qubit states.

Stereographic projection terms are entanglement sensitive in higher dimensions.

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 \emph{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 \emph{octonionic} stereographic projection which means that if the state is bi-separable then, quaternionic and octonionic terms vanish. Finally, we generalize recent consequences to $2 \otimes N$ and $4\otimes N$ dimensional Hilbert spaces ($N\geq 2$) 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.