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Detection of Multipartite Entanglement Based on Heisenberg-Weyl Representation of Density Matrices

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Abstract

We study entanglement and genuine entanglement of tripartite and four-partite quantum states by using Heisenberg-Weyl (HW) representation of density matrices. Based on the correlation tensors in HW representation, we present criteria to detect entanglement and genuine tripartite and four-partite entanglement. Detailed examples show that our method can detect more entangled states than previous criteria.

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All data generated or analysed during this study are available from the corresponding author on reasonable request.

References

  1. Ekert, A.K.: Quantum cryptogaphy based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  2. Bennett, C.H., Wiesner, S.J.: Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bennett, C.H., Brassard, G., Jozsa, R.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  4. Peres, A.: . Phys. Rev. Lett. 77, 1413 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  5. Terhal, B.: . Phys. Lett. A 271, 319 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  6. Chen, K., Wu, L.A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3, 193 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Rudolph, O.: Some properties of the computable cross-norm criterion for separability. Phys. Rev. A 67, 032312 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  8. Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4, 219 (2005)

    Article  MathSciNet  Google Scholar 

  9. Li, M., Wang, J., Shen, S., Chen, Z., Fei, S.M.: Detection and measure of genuine tripartite entanglement with partial transposition and realignment of density matrices. Sci. Rep. 7, 17274 (2018)

    Article  ADS  Google Scholar 

  10. Chen, Z.H., Ma, Z.H., Chen, J.L., Severini, S.: Improved lower bounds on genuine-multipartite-entanglement concurrence. Phys. Rev. A 785, 062320 (2012)

    Article  ADS  Google Scholar 

  11. Zhao, H., Fei, S.M., Fan, J., Wang, Z.X.: Inequalities detecting entanglement for arbitrary bipartite systems. Int. J. Quantum Inform. 12, 1450013 (2014)

    Article  MathSciNet  Google Scholar 

  12. Huber, M., Sengupta, R.: Witnessing genuine multipartite entanglement with positive maps. Phys. Rev. Lett. 113, 100501 (2014)

    Article  ADS  Google Scholar 

  13. Shen, S.Q., Yu, J., Li, M., Fei, S.M.: . Sci. Rep. 6, 28850 (2016)

    Article  ADS  Google Scholar 

  14. Vicente, J.: Separability criteria based on the bloch representation of density matrices. Quantum Inf. Comput. 7, 624 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Bertlmann, R.A., Krammer, P.: Bloch vectors for qudits. J. Phys. A: Math. Theor. 41, 235303 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  16. Asadian, A., Erker, P., Huber, M., Klöckl, C.: Heisenberg-weyl Observables: Bloch vectors in phase space. Phys. Rev. A 94, 010301 (2016)

    Article  ADS  Google Scholar 

  17. Chang, J., Cui, M., Zhang, T., Fei, S. M.: Separability criteria based on Heisenberg-Weyl representation of density matrices. Chinese Physics B 27, 030302 (2018)

    Article  ADS  Google Scholar 

  18. Zhao, H., Zhang, M.M., Jing, N., Wang, Z.X.: Separability criteria based on Bloch representation of density matrice. Quantum Inf. Process. 19, 1 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  19. Wang, J., Li, M., Li, H., Fei, S.M., Li-Jost, X.: Bounds on multipartite concurrence and tangle. Quantum Inf. Process. 15, 4211 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  20. Li, M., Wang, Z., Wang, J., Shen, S., Fei, S.M.: The norms of Bloch vectors and classification of four-qudits quantum states. EPL Europhys. Lett. 125, 20006 (2019)

    Article  ADS  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under grant nos. 12075159, 12126351 and 12171044, Simons Foundation under grant no. 523868, Beijing Natural Science Foundation (grant no. Z190005), Academy for Multidisciplinary Studies, Capital Normal University, and Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (no. SIQSE202001), and the Academician Innovation Platform of Hainan Province.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, 100124, China

    Hui Zhao & Yu Yang

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Naihuan Jing

  3. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Zhi-Xi Wang & Shao-Ming Fei

  4. Max-Planck-Institute for Mathematics in the Sciences, Leipzig, 04103, Germany

    Shao-Ming Fei

Authors
  1. Hui Zhao
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  2. Yu Yang
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  3. Naihuan Jing
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  4. Zhi-Xi Wang
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  5. Shao-Ming Fei
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Correspondence to Hui Zhao.

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Zhao, H., Yang, Y., Jing, N. et al. Detection of Multipartite Entanglement Based on Heisenberg-Weyl Representation of Density Matrices. Int J Theor Phys 61, 136 (2022). https://doi.org/10.1007/s10773-022-05123-9

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  • DOI: https://doi.org/10.1007/s10773-022-05123-9

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