Local Unitary Representation of Braids and N-Qubit Entanglements

Abstract

In this paper, by utilizing the idea of stabilizer codes, we give some relationships between one local unitary representation of braid group in N-qubit tensor space and the corresponding entanglement properties of the N-qubit pure state $|\Psi\rangle$, where the N-qubit state $|\Psi\rangle$ is obtained by applying the braiding operation on the natural basis. Specifically, we show that the separability of $|\Psi\rangle=\mathcal{B}|0\rangle^{\otimes N}$ is closely related to the diagrammatic version of the braid operator $\mathcal{B}$. This may provide us more insights about the topological entanglement and quantum entanglement.