A method of constructing n 2 × n 2 matrix realization of Temperley---Lieb algebras is presented. The single loop of these realizations are $${d=\sqrt{n}}$$ . In particular, a 9 × 9-matrix realization with single loop $${d=\sqrt{3}}$$ is discussed. A unitary Yang---Baxter $${\breve{R}\theta,q_{1},q_{2})}$$ 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.
['Gangcheng Wang', 'Kang Xue', 'Chunfang Sun', 'Chengcheng Zhou', 'Taotao Hu', 'Qingyong Wang']
Temperley---Lieb algebra, Yang-Baxterization and universal gate