The representations of Temperley-Lieb algebras and entanglement in a Yang-Baxter system

Abstract

A method of constructing Temperley-Lieb algebras(TLA) representations has been introduced in [Xue \emph{et.al} arXiv:0903.3711]. Using this method, we can obtain another series of $n^2\times n^2$ matrices $U$ which satisfy the TLA with the single loop $d=\sqrt{n}$. Specifically, we present a $9\times 9$ matrix $U$ with $d=\sqrt{3}$. Via Yang-Baxterization approach, we obtain a unitary $\breve{R}(\theta ,{\phi }_1,{\phi }_2)$-matrix, a solution of the Yang-Baxter Equation. This $9\times 9$ Yang-Baxter matrix is universal for quantum computing.