Temperley-Lieb Algebra, Yang-Baxterization and Universal Gate

TL;DR

This paper presents a method to construct solutions to the Temperley-Lieb algebra, derives a unitary Yang-Baxter matrix, and explores its entanglement and geometric properties, contributing to quantum information and algebraic studies.

Contribution

It introduces a new construction method for Temperley-Lieb algebra solutions and derives a novel unitary Yang-Baxter matrix with applications in quantum entanglement.

Findings

Constructed $n^2 imes n^2$ solutions with loop parameter $d=\sqrt{n}$.

Derived a specific $9 imes 9$ Yang-Baxter matrix with $d=\sqrt{3}$.

Analyzed entanglement and Berry phase properties of the Yang-Baxter system.

Abstract

A method of constructing $n^{2}\times n^{2}$ matrix solutions(with $n^{3}$ matrix elements) of Temperley-Lieb algebra relation is presented in this paper. The single loop of these solutions are $d=\sqrt{n}$. Especially, a $9\times9-$matrix solution with single loop d=$\sqrt{3}$ is discussed in detail. An unitary Yang-Baxter $\breve{R}(\theta,q_{1},q_{2})$ matrix is obtained via the Yang-Baxterization process. The entanglement property and geometric property (\emph{i.e.} Berry Phase) of this Yang-Baxter system are explored.