6 research outputs found
Hamilton's turns as visual tool-kit for designing of single-qubit unitary gates
Unitary evolutions of a qubit are traditionally represented geometrically as
rotations of the Bloch sphere, but the composition of such evolutions is
handled algebraically through matrix multiplication [of SU(2) or SO(3)
matrices]. Hamilton's construct, called turns, provides for handling the latter
pictorially through the as addition of directed great circle arcs on the unit
sphere S, resulting in a non-Abelian version of the
parallelogram law of vector addition of the Euclidean translation group. This
construct is developed into a visual tool-kit for handling the design of
single-qubit unitary gates. As an application, it is shown, in the concrete
case wherein the qubit is realized as polarization states of light, that all
unitary gates can be realized conveniently through a universal gadget
consisting of just two quarter-wave plates (QWP) and one half-wave plate (HWP).
The analysis and results easily transcribe to other realizations of the qubit:
The case of NMR is obtained by simply substituting and pulses
respectively for QWPs and HWPs, the phases of the pulses playing the role of
the orientation of fast axes of these plates.Comment: 16 Pages, 14 Figures, Published versio
The structure of states and maps in quantum theory
The structure of statistical state spaces in the classical and quantum theories are compared in an interesting and novel manner. Quantum state spaces and maps on them have rich convex structures arising from the superposition principle and consequent entanglement. Communication channels (physical processes) in the quantum scheme of things are in one-to-one correspondence with completely positive maps. Positive maps which are not completely positive do not correspond to physical processes. Nevertheless they prove to be invaluable mathematical tools in establishing or witnessing entanglement of mixed states. We consider some of the recent developments in our understanding of the convex structure of states and maps in quantum theory, particularly in the context of quantum information theory