Majorana stellar representation for mixed-spin (s, 1/2) systems

doi:10.1088/1674-1056/abc2bc

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 the quantum system with high-dimensional Hilbert space. However, the representation of a two-spin coupling system on a Bloch sphere has not been solved satisfactorily yet. Here, a practical method is presented to resolve the problem for the mixed-spin (s, 1/2) system and describe the entanglement of the 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, any pure state may be written 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 initial state can be regarded as a state of a pseudo spin-1/2. In this way, the mixed spin decomposes into three spins. Therefore, the state can be represented by (2s+1)+(2s-1)+1=4s+1 sets of stars on a Bloch sphere. Finally, some examples are given to show symmetric patterns on the Bloch sphere and unveil the properties of the high-spin system by analyzing the trajectories of the Majorana stars on the Bloch sphere.

Keywords: Majorana's stellar representation Bloch sphere high-dimensional projective Hilbert space mixed-spin

Received: 13 June 2020
Revised: 25 September 2020
Accepted manuscript online: 20 October 2020

PACS:	03.65.Aa	(Quantum systems with finite Hilbert space)
	02.40.Dr	(Euclidean and projective geometries)
	74.25.Op	(Mixed states, critical fields, and surface sheaths)

Fund: Project supported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0304202 and 2017YFA0205700), the National Natural Science Foundation of China (Grant No. 11875231), and the Fundamental Research Funds for the Central Universities, China (Grant No. 2018FZA3005).

Corresponding Authors: ^†Corresponding author. E-mail: xgwang1208@zju.edu.cn

Cite this article:

Yu-Guo Su(苏玉国), Fei Yao(姚飞), Hong-Bin Liang(梁宏宾), Yan-Ming Che(车彦明), Li-Bin Fu(傅立斌), and Xiao-Guang Wang(王晓光) Majorana stellar representation for mixed-spin (s, 1/2) systems 2021 Chin. Phys. B 30 030303

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