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New Z-eigenvalue inclusion theorem of tensors with application to the geometric measure of entanglement

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Abstract

The geometric measure of entanglement plays important role in quantum entanglement of multipartite cases. In this paper, with the eigenvalues of matrices, new Z-eigenvalue inclusion sets are given, some sufficient conditions for the positive definiteness of fourth-order tensors are presented based on the Z-eigenvalue inclusion sets, and then, upper and lower bounds of the geometric measure of entanglement for 4-qubit states are given. In addition, the upper bound of the geometric measure of entanglement is also applied to 4-qubit mixed state case. Finally, numerical experiments are reported to show the efficiency of the proposed new results.

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References

  1. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Qi, L., Luo Z.Y.: Tensor analysis spectral theory and special tensors. SIAM, USA(2017)

  3. Chang, K.C.: Tan Zhang: on the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors. J. Math. Anal. Appl. 408(2), 525–540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Che, M., Cichocki, A., Wei, Y.: Neural networks for computing best rank-one approximations of tensors and its applications. Neurocomputing 267, 114–133 (2017)

    Article  Google Scholar 

  5. Kolda, T.G., Mayo, J.R.: Shifted powermethod for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32, 1095 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, L., Han, L., Zhou, L.: Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 37, 290 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hillar, C.J., Lim, L.-H.: Most tensor problems are NP-hard. J. ACM 60(6), 1–39 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. He, J., Huang, T.: Upper bound for the largest \(Z\)-eigenvalue of positive tensors. Appl. Math. Lett. 38, 110 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, C., Li, Y., Kong, X.: New eigenvalue inclusion sets for tensors. Numer. Linear Algebra Appl. 21, 39–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhao, J.: A new Z-eigenvalue localization set for tensors. J. Inequal. Appl. 85, 2017 (2017)

    MathSciNet  Google Scholar 

  11. Sang, C.: A new brauer-type Z-eigenvalue inclusion set for tensors. Numer. Algorithms 1, 1–14 (2018)

    Google Scholar 

  12. Shimony, A.: Degree of entanglement. Ann. N. Y. Acad. Sci. 755, 675 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  13. Wei, T.C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)

    Article  ADS  Google Scholar 

  14. Wei, T.C., Das, D., Mukhopadyay, S., et al.: Global entanglement and quantum criticality in spin chains. Phys. Rev. A 71, 362 (2005)

    Article  Google Scholar 

  15. Yoshifumi, N., Damian, M., Mio, M.: Thermal robustness of multipartite entanglement of the 1-d spin 1/2 \(xy\) model. Phys. Rev. A 79, 126 (2009)

    Google Scholar 

  16. Hu, S., Qi, L., Zhang, G.: Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 93, 012304 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  17. Qi, L., Zhang, G., Ni, G.: How entangled can a multi-party system possibly be? Phys. Lett. A 382, 1465 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Wang, G., Zhou, G., Caccetta, L.: Z-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst. Ser. B 22, 187–198 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Chang, K., Pearson, K., Zhang, T.: Some variational principles for \(Z\)-eigenvalues of nonnegative tensors. Linear Algebra Appl. 438, 4166 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. He, J., Liu, Y., Tian, J., Liu, X.: Upper bounds for signless Laplacian Z-spectral radius of uniform hypergraphs. Front. Math. China 14(1), 17–24 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the Guizhou Province Natural Science Foundation in China (Qian Jiao He KY [2020]094, [2022]017), the Science and Technology Foundation of Guizhou Province, China (Qian Ke He Ji Chu ZK[2021]Yi Ban 014), the Key Laboratory of Evolutionary Artificial Intelligence in Guizhou (Qian Jiaoji [2022] No. 059) and the Key Talens Program in digital economy of Guizhou Province.

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Authors and Affiliations

  1. School of mathematics, Zunyi Normal College, Zunyi, 563006, Guizhou, People’s Republic of China

    Jun He, Yanmin Liu & Qingyu Zeng

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  1. Jun He
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  2. Yanmin Liu
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  3. Qingyu Zeng
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Contributions

Formal analysis and investigation: Jun He and Qingyu Zeng; Writing - original draft preparation: Jun He; Writing - review and editing: Yanmin Liu and Qingyu Zeng; Funding acquisition: Jun He; Supervision: Yanmin Liu.

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Correspondence to Yanmin Liu.

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He, J., Liu, Y. & Zeng, Q. New Z-eigenvalue inclusion theorem of tensors with application to the geometric measure of entanglement. Quantum Inf Process 22, 134 (2023). https://doi.org/10.1007/s11128-023-03890-w

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