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The geometric measure of entanglement of multipartite states and the Z-eigenvalue of tensors

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Abstract

It is not easy to compute the entanglement of multipartite pure or mixed states, because it usually involves complex optimization. In this paper, we are devoted to the geometric measure of entanglement of multipartite pure or mixed state by the means of real tensor spectrum theory. On the basis of Z-eigenvalue inclusion theorem and the estimation of weakly symmetric nonnegative tensor Z-spectrum radius, we propose some theoretical upper and lower bounds of the geometric measure of entanglement for weakly symmetric pure state with nonnegative amplitudes for two kinds of geometric measures with different definitions, respectively. In addition, the upper bound of the geometric measure of entanglement is also applied to multipartite mixed state case.

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Acknowledgements

The authors are grateful to the anonymous referees for their helpful suggestions and comments. This work was supported by the National Natural Science Foundation of China (Grant11971413,11571292) and Science and Technology Foundation of Shenzhen City (No. JCYJ20190808174211224).

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Authors and Affiliations

  1. College of Physics and Optoelectronic Engineering and Key Laboratory of Optoelectronic Devices and Systems, Shenzhen University, Shenzhen, 518060, China

    Liang Xiong & Qi Qin

  2. School of Mathematics and Computational Science and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, China

    Liang Xiong & Jianzhou Liu

Authors
  1. Liang Xiong
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  2. Jianzhou Liu
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  3. Qi Qin
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Correspondence to Jianzhou Liu or Qi Qin.

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Xiong, L., Liu, J. & Qin, Q. The geometric measure of entanglement of multipartite states and the Z-eigenvalue of tensors. Quantum Inf Process 21, 102 (2022). https://doi.org/10.1007/s11128-022-03434-8

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  • DOI: https://doi.org/10.1007/s11128-022-03434-8

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