# Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates. (arXiv:1703.06102v2 [quant-ph] UPDATED)

We report a study of the Majorana geometrical representation of a qutrit,

where a pair of points on a unit sphere represents its quantum states. A

canonical form for qutrit states is presented, where every state can be

obtained from a one-parameter family of states via $SO(3)$ action. The notion

of spin-1 magnetization which is invariant under $SO(3)$ is geometrically

interpreted on the Majorana sphere. Furthermore, we describe the action of

several quantum gates in the Majorana picture and experimentally implement

these gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystalline

environment. We study the dynamics of the pair of points representing a qutrit

state under various useful quantum operations and connect them to different NMR

operations. Finally, using the Gell Mann matrix picture we experimentally

implement a scheme for complete qutrit state tomography.