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 \(9\times9\) matrix \(U\) with \(d=\sqrt{3}\). Via Yang-Baxterization approach, we obtain a unitary \( \breve{R}(\theta ,\varphi_{1},\varphi_{2})\)-matrix, a solution of the Yang-Baxter Equation. This \(9\times9\) Yang-Baxter matrix is universal for quantum computing.