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The relationships between quantum entangled states and braid matrices havebeen well studied in recent years. However, most of the results are based onqubits. In this paper, We investigate the applications of 2-qutrit entanglementin the braiding associated with $\mathbb{Z}_3$ parafermion. The 2-qutritentangled state $|\Psi(\theta)\rangle$, generated by acting the localizedunitary solution $\breve{R}(\theta)$ of YBE on 2-qutrit natural basis, achievesits maximal $\ell_1$-norm and maximal von Neumann entropy simultaneously at$\theta=\pi/3$. Meanwhile, at $\theta=\pi/3$, the solutions of YBE reducesbraid matrices, which implies the role of $\ell_1$-norm and entropy plays indetermining real physical quantities. On the other hand, we give a newrealization 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 inthe 4-anyon topological basis. Hence, we conclude that the entangled states arepowerful tools in analysing the characteristics of braiding and$\breve{R}$-matrix.

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