  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.
