Temperley–Lieb algebra, Yang-Baxterization and universal gate

A method of constructing n2 × n2 matrix realization of Temperley–Lieb algebras is presented. The single loop of these realizations are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d=\sqrt{n}}$$\end{document}. In particular, a 9 × 9-matrix realization with single loop \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d=\sqrt{3}}$$\end{document} is discussed. A unitary Yang–Baxter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\breve{R}\theta,q_{1},q_{2})}$$\end{document} matrix is obtained via the Yang-Baxterization process. The entanglement properties and geometric properties (i.e., Berry Phase) of this Yang–Baxter system are explored.