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

## TL;DR

This paper explores the significance of the $ ext{L}_1$-norm in three-qubit quantum systems constrained by the Yang-Baxter equation, revealing its connection to quantum entanglement types like GHZ and W states.

## Contribution

It demonstrates how the $ ext{L}_1$-norm's extreme values correspond to key entangled states in three-qubit systems under Yang-Baxter constraints, extending understanding beyond the traditional $ ext{L}_2$-norm.

## Key findings

- Extreme $ ext{L}_1$-norm values yield GHZ and W states.
- The work links $ ext{L}_1$-norm to braid matrices and entanglement classification.
- Provides evidence for the role of $ ext{L}_1$-norm in quantum mechanics.

## Abstract

Usually the $\ell_2$-norm plays vital roles in quantum physics, acting as the probability of states. In this paper, we show the important roles of $\ell_1$-norm in Yang-Baxter quantum system, in connection with both the braid matrix and quantum entanglements. Concretely, we choose the 2-body and 3-body S-matrices, constrained by Yang-Baxter equation. It has been shown that for 2-body case, the extreme values of $\ell_1$-norm lead to two types of braid matrices and 2-qubit Bell states. Here we show that for the 3-body case, due to the 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, which cover all 3-qubit genuine entanglements for pure states under SLOCC. This is a more convincing proof for the roles of $\ell_1$-norm in quantum mechanics.