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Local unitary representation of braids and N-qubit entanglements

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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 \(|\varPsi \rangle \), where the N-qubit state \(|\varPsi \rangle \) is obtained by applying the braiding operation on the natural basis. Specifically, we show that the separability of \(|\varPsi \rangle =\mathcal {B}|0\rangle ^{\otimes N}\) 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.

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Notes

  1. \(|0\rangle \) and \(|1\rangle \) are two eigenvectors of Pauli Z matrix \(\sigma ^z\) in \(\mathbb {C}^2\), where \(\sigma ^z|0\rangle =-|0\rangle \), \(\sigma ^z|1\rangle =|1\rangle \).

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Acknowledgements

The author would like to thank Professor Z. H. Wang for his helpful discussions and encouragements.

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Authors and Affiliations

  1. Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China

    Li-Wei Yu

  2. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Li-Wei Yu

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  1. Li-Wei Yu
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Correspondence to Li-Wei Yu.

Additional information

This work is in part supported by NSF of China (Grant No. 11475088) and China Scholarship Council (No. 201606205057).

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Yu, LW. Local unitary representation of braids and N-qubit entanglements. Quantum Inf Process 17, 44 (2018). https://doi.org/10.1007/s11128-018-1811-3

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