$\ell_1$-norm and entanglement in screening out braiding from Yang-Baxter equation associated with $\mathbb{Z}_3$ parafermion

Abstract

The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, We investigate the applications of 2-qutrit entanglement in the braiding associated with $\mathbb{Z}_3$ parafermion. The 2-qutrit entangled state $|\Psi(\theta)\rangle$, generated by acting the localized unitary solution $\breve{R}(\theta)$ of YBE on 2-qutrit natural basis, achieves its maximal $\ell_1$-norm and maximal von Neumann entropy simultaneously at $\theta=\pi/3$. Meanwhile, at $\theta=\pi/3$, the solutions of YBE reduces braid matrices, which implies the role of $\ell_1$-norm and entropy plays in determining real physical quantities. On the other hand, we give a new realization of 4-anyon topological basis by qutrit entangled states, then the $9\times9$ localized braid representation in 4-qutrit tensor product space $(\mathbb{C}^3)^{\otimes 4}$ are reduced to Jones representation of braiding in the 4-anyon topological basis. Hence, we conclude that the entangled states are powerful tools in analysing the characteristics of braiding and $\breve{R}$-matrix.