Bloch sphere representation of geodesics and null phase curves of higher-dimensional state space

The geometrical representation of the state space of an n-level quantum system is essential in characterizing the system. One possible way to achieve that is to understand the structure of geodesics and null phase curves in the state space. The null phase curves are the paths along which there is no geometric phase accumulation, and geodesics give the shortest distance between any given two points and are special cases of null phase curves. The state-space for the 2-level system is the (Bloch) sphere, and geodesics are the great circles. However, finding geodesics is not trivial in higher-level systems. Here, in this work, we propose a consistent way to construct geodesics and a class of null phase curves in d-level systems using Majorana star representation which maps a pure quantum state of an n-level system to the symmetric subspace of n-1 2-level systems. This work can be instrumental in studying the topological phases in the systems with three or more band structures.