# Majorana stellar representation for mixed-spin $(s,\frac{1}{2})$ systems

## TL;DR

This paper introduces a geometric method using Majorana stellar representation to visualize and analyze mixed-spin systems, decomposing them into multiple spins and representing their states with star trajectories on a Bloch sphere.

## Contribution

The authors develop a practical approach to represent mixed $(s, 1/2)$ spin systems on a Bloch sphere, extending Majorana's stellar representation to high-spin coupled systems.

## Key findings

- Effective visualization of mixed-spin states using Majorana stars.
- Symmetric patterns observed in star trajectories on the Bloch sphere.
- Unveiled properties of high-spin systems through geometric analysis.

## Abstract

By describing the evolution of a quantum state with the trajectories of the Majorana stars on a Bloch sphere, Majorana's stellar representation provides an intuitive geometric perspective to comprehend a quantum system with high-dimensional Hilbert space. However, the problem of the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, we present a practical method to resolve the problem for the mixed-spin $(s, 1/2)$ system. The system can be decomposed into two spins: spin-$(s+1/2)$ and spin-$(s-1/2)$ at the coupling bases, which can be regarded as independent spins. Besides, we may write any pure state as a superposition of two orthonormal states with one spin-$(s+1/2)$ state and the other spin-$(s-1/2)$ state. Thus, the whole state can be regarded as a state of a pseudo spin-$1/2$. In this way, the mixed spin decomposes into three spins. Therefore, we can represent the state by $(2s+1)+(2s-1)+1=4s+1$ sets of stars on a Bloch sphere. Finally, to demonstrate our theory, we give some examples that indeed show laconic and symmetric patterns on the Bloch sphere, and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on a Bloch sphere.