Majorana stellar representation for mixed-spin (s, 1/2) systems |
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Authors: | Yu-Guo Su Fei Yao Hong-Bin Liang Yan-Ming Che Li-Bin Fu Xiao-Guang Wang |
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Affiliation: | (Zhejiang Province Key Laboratory of Quantum Technology and Device,Department of Physics,Zhejiang University,Hangzhou 310027,China;National Laboratory of Science and Technology on Computational Physics,Institute of Applied Physics and Computational Mathematics,Beijing 100088,China) |
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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 highdimensional 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(2 s + 1) +(2 s-1) + 1 = 4 s + 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. |
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