The relationship between single-particle commuting observables lz, sz entangled states and the spin–orbit coupling

Quantum entanglement is a unique phenomenon which can be described in quantum mechanics. There are two categories of quantum entanglement: one is based on a single-body system in various freedom degrees and another is found in the multi-body system. In recent years, the spin–orbit coupling effect has been widely concerned. Various electronic devices based on the spin–orbit coupling effect have been emerging in an endless stream and bringing great practical application value. At present, the relationship between single-particle commuting observables lz,sz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ l_{z} ,s_{z} $$\end{document} entangled states and the spin–orbit coupling is rarely reported. In this paper, based on the diverse freedom degrees of the quantum entanglement states in a single-body system, the correlation between the sole particle commuting observables lz,sz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ l_{z} ,s_{z} $$\end{document} entangled states and the spin–orbit coupling has been analyzed by MATLAB software and the degree of entanglement is measured according to von Neumann entropy. This work provides innovation to further understand quantum entanglement and demonstrates that there is a close relationship between the degree of entanglement and the spin–orbit coupling coefficient j/2-1/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {j \mathord{\left/ {\vphantom {j 2}} \right. \kern-0pt} 2} - {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4} $$\end{document}. As j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ j $$\end{document} increases, the degree of the maximum entanglement firstly decreases, then increases, afterward decreases and then increases repeatedly, with the maximum value of entanglement degree being 0.6828. When j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ j $$\end{document} is beyond 85.5, the degree of entanglement will not make sense any longer.