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

## TL;DR

This paper explores the Majorana geometric representation of qutrits, introduces a canonical form, interprets spin-1 magnetization geometrically, and demonstrates experimental implementation of qutrit gates and state tomography using NMR techniques.

## Contribution

It provides a novel geometric framework for qutrit states, details the implementation of qutrit gates in NMR, and demonstrates complete state tomography experimentally.

## Key findings

- Majorana representation offers a clear geometric visualization of qutrit states.
- Quantum gates can be effectively implemented and visualized in the Majorana picture.
- Complete qutrit state tomography is experimentally achieved using NMR techniques.

## Abstract

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.