Who's afraid of the Unit Quaternion ? (November 2016) - [download pdf]
Far from being just a 2-level Quantum system the Qubit is a Unit Quaternion, also known as a Spinor. Hence the t-parameterized Qubit is a 4-dimensional vector which traces a path on the surface of the unit 3-sphere. This is the meaning of the global phase.
Vectors, Spinors and Galilean Frames (December 2015) - [download pdf]
In this article we derive the closed form expressions for the intrinsic parameters of the unit spinor. The intrinsic parameters are the global, geometric and dynamic phases which are presented as functions of the elements of the Hamiltonian and Bloch vector. We show unequivocally that spinors, generated by the SU(2) group, and vectors, generated by the SO(3) group, are equivalent mathematical objects, as the vector possesses the same intrinsic parameters as the spinor, albeit hidden.
Parallel Transport, Quaternions and the Bloch Sphere (August 2015) - [download pdf]
The global phase is the characteristic phase of the qubit which identifies it as a spinor. In this analysis we show that despite common belief, the global phase is only loosely defined and poorly understood. We detail the nuances of this conundrum with recourse to Hamilton’s quaternions, parallel transport, moving frames, and the contrast between the SU(2) and SO(3) description of the qubit. A formalism for the global and geometric phases are presented as functions of the elements of the Hamiltonian and Bloch vector.
The Geometric Phase of the Qubit (July 2014) - [download pdf]
Through a direct analogy with classical parallel transport, the geometric phase of a two level quantum system, or qubit, is defined. The qubit`s position on the surface of the Bloch sphere, is specified by the time dependent azimuthal and polar angles. When the qubit completes a closed loop path, the orientation of the qubit`s tangent vector has changed due to the curvature of the spherical surface. The geometric phase of the qubit is defined as the angular change in the orientation of the tangent vector, and is shown to be a contour integral involving the time parameterized azimuthal and polar angles.