Bloch sphere model for two-qubit pure states

Abstract: The two-qubit pure state is explicitly parameterized by three unit 2-spheres and a phase factor. For separable states, two of the three unit spheres are the Bloch spheres of each qubit. The third sphere parameterizes the degree and phase of concurrence, an entanglement measure. This sphere may be considered a variable complex imaginary unit where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable imaginary unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states. We argue that the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuth angles of the two quasi-Bloch spheres. On these three unit spheres and a phase factor, the two-qubit pure states and unitary gates can be geometrically represented. Accomplished by means of Hopf fibration, the complex amplitudes of a two-qubit pure state and the Bloch sphere parameters are related by a single quaternionic relation.