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Usually the $\ell_2$-norm plays vital roles in quantum physics, acting as theprobability of states. In this paper, we show the important roles of$\ell_1$-norm in Yang-Baxter quantum system, in connection with both the braidmatrix and quantum entanglements. Concretely, we choose the 2-body and 3-bodyS-matrices, constrained by Yang-Baxter equation. It has been shown that for2-body case, the extreme values of $\ell_1$-norm lead to two types of braidmatrices and 2-qubit Bell states. Here we show that for the 3-body case, due tothe constraint of YBE, the extreme values of $\ell_1$-norm lead to both 3-qubit$|GHZ\rangle$ (local maximum) and $|W\rangle$ (local minimum) states, whichcover all 3-qubit genuine entanglements for pure states under SLOCC. This is amore convincing proof for the roles of $\ell_1$-norm in quantum mechanics.

$\ell_1$-norm in three-qubit quantum entanglement constrained by Yang-Baxter equation