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Necessary conditions for classifying \(\mathbf {m}\)-separability of multipartite entanglements

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Abstract

We study the norms of the Bloch vectors for arbitrary n-partite quantum states. A tight upper bound of the norms is derived for n-partite systems with different individual dimensions. These upper bounds are used to deal with the separability problems. Necessary conditions are presented for \(\mathbf {m}\)-separable states in n-partite quantum systems. Based on the upper bounds, classification of multipartite entanglement is illustrated with detailed examples.

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References

  1. Ekert, A.K.: Quantum cryptography based on Bells 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., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and EPR channels. Phys. Rev. Lett. 70, 1895 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  4. Dür, W., Cirac, J.I., Tarrach, R.: Separability and distillability of multiparticle quantum systems. Phys. Rev. Lett. 83, 3562 (1999)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. de Vicente, J.I., Huber, M.: Multipartite entanglement detection from correlation tensors. Phys. Rev. A 84, 062306 (2011)

    Article  ADS  Google Scholar 

  7. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  8. Jakóbczyk, L., Siennicki, M.: Geometry of Bloch vectors in two-qubit system. Phys. Lett. A 286, 383 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  9. Kimura, G.: The Bloch vector for N-level systems. Phys. Lett. A 314, 339 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  10. de Vicente, J.I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. 7, 624 (2007)

    MathSciNet  MATH  Google Scholar 

  11. de Vicente, J.I.: Further results on entanglement detection and quantification from the correlation matrix criterion. J. Phys. A Math. Theor. 41, 065309 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  12. Hassan, A.S.M., Joag, P.S.: Separability criterion for multipartite quantum states based on the Bloch representation of density matrices. Quantum Inf. Comput. 8, 773 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Li, M., Wang, J., Fei, S.M., Li-Jost, X.Q.: Quantum separability criteria for arbitrary-dimensional multipartite states. Phys. Rev. A 89, 022325 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Tănăsescu, A., Popescu, P.: Bloch vector norms of separable multi-partite quantum systems. Europhys. Lett. A 126, 60003 (2019)

    Article  Google Scholar 

  16. Hassan, A.S.M., Joag, P.S.: An experimentally accessible geometric measure for entanglement in N-qubit pure states. Phys. Rev. A 77, 062334 (2008)

    Article  ADS  Google Scholar 

  17. Hassan, A.S.M., Joag, P.S.: Geometric measure for entanglement in N-qudit pure states. Phys. Rev. A 80, 042302 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. Yu, B., Jing, N.H., Li-Jost, X.Q.: Distribution of spin correlation strengths in multipartite systems. Quantum Inf. Process. 18, 344 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  19. van Loock, P., Furusawa, A.: Detecting genuine multipartite continuous-variable entanglement. Phys. Rev. A 67, 052315 (2003)

    Article  ADS  Google Scholar 

  20. Zhao, M.J., Zhang, T.G., Li-Jost, X.Q., Fei, S.M.: Identification of three-qubit entanglement. Phys. Rev. A 87, 012316 (2013)

    Article  ADS  Google Scholar 

  21. Li, M., Wang, J., Shen, S.Q., Chen, Z.H., Fei, S.M.: Detection and measure of genuine tripartite entanglement with partial transposition and realignment of density matrices. Sci. Rep. 7, 17274 (2017)

    Article  ADS  Google Scholar 

  22. Bancal, J.D., Gisin, N., Liang, Y.C., Pironio, S.: Device-independent witnesses of genuine multipartite entanglement. Phys. Rev. Lett. 106, 250404 (2011)

    Article  ADS  Google Scholar 

  23. Jungnitsch, B., Moroder, T., Gühne, O.: Entanglement witnesses for graph states: general theory and examples. Phys. Rev. A 84, 032310 (2011)

    Article  ADS  Google Scholar 

  24. Wu, J.Y., Kampermann, H., Bruß, D., Klockl, C., Huber, M.: Determining lower bounds on a measure of multipartite entanglement from few local observables. Phys. Rev. A 86, 022319 (2012)

    Article  ADS  Google Scholar 

  25. Zhao, H., Zhang, M.M., Jing, N.H., Wang, Z.X.: Separability criteria based on Bloch representation of density matrices. Quantum Inf. Process. 19, 14 (2020)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the NSF of China under Grant Nos. 11571119 and 11675113, the Key Project of Beijing Municipal Commission of Education (Grant No. KZ201810028042), and Beijing Natural Science Foundation (Z190005). It is a pleasure to thank Jin-Wei Huang for helpful discussion.

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Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, 510640, People’s Republic of China

    Wen Xu & Zhu-Jun Zheng

  2. College of Mathematics and Physics Science, Hunan University of Arts and Science, Changde, 415000, People’s Republic of China

    Chuan-Jie Zhu

  3. Department of Physics, Renmin University of China, Beijing, 100872, People’s Republic of China

    Chuan-Jie Zhu

  4. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Shao-Ming Fei

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

    Shao-Ming Fei

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  1. Wen Xu
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  2. Chuan-Jie Zhu
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  3. Zhu-Jun Zheng
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  4. Shao-Ming Fei
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Correspondence to Zhu-Jun Zheng.

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Xu, W., Zhu, CJ., Zheng, ZJ. et al. Necessary conditions for classifying \(\mathbf {m}\)-separability of multipartite entanglements. Quantum Inf Process 19, 200 (2020). https://doi.org/10.1007/s11128-020-02705-6

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