中国物理B ›› 2021, Vol. 30 ›› Issue (3): 30303-.doi: 10.1088/1674-1056/abc2bc

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  • 收稿日期:2020-06-13 修回日期:2020-09-25 接受日期:2020-10-20 出版日期:2021-02-22 发布日期:2021-02-22

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

Yu-Guo Su(苏玉国)1, Fei Yao(姚飞)1, Hong-Bin Liang(梁宏宾)1, Yan-Ming Che(车彦明)1, Li-Bin Fu(傅立斌)2, and Xiao-Guang Wang(王晓光)1,†   

  1. 1 Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics, Zhejiang University, Hangzhou 310027, China; 2 National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
  • Received:2020-06-13 Revised:2020-09-25 Accepted:2020-10-20 Online:2021-02-22 Published:2021-02-22
  • Contact: Corresponding author. E-mail: xgwang1208@zju.edu.cn
  • Supported by:
    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).

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.

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

中图分类号:  (Quantum systems with finite Hilbert space)

  • 03.65.Aa
02.40.Dr (Euclidean and projective geometries) 74.25.Op (Mixed states, critical fields, and surface sheaths)