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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. Acanonical form for qutrit states is presented, where every state can beobtained from a one-parameter family of states via $SO(3)$ action. The notionof spin-1 magnetization which is invariant under $SO(3)$ is geometricallyinterpreted on the Majorana sphere. Furthermore, we describe the action ofseveral quantum gates in the Majorana picture and experimentally implementthese gates on a spin-1 system (an NMR qutrit) oriented in a liquid crystallineenvironment. We study the dynamics of the pair of points representing a qutritstate under various useful quantum operations and connect them to different NMRoperations. Finally, using the Gell Mann matrix picture we experimentallyimplement a scheme for complete qutrit state tomography.

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates