Monogamy properties of qubit systems


We investigate monogamy relations related to quantum entanglement for n-qubit quantum systems. General monogamy inequalities are presented to the \(\beta \hbox {th}\,(\beta \in (0,2))\) power of concurrence, negativity and the convex-roof extended negativity, as well as the \(\beta \hbox {th}\,(\beta \in (0,\sqrt{2}))\) power of entanglement of formation. These monogamy relations are complementary to the existing ones with different regions of parameter \(\beta \). In additions, new monogamy relations are also derived which include the existing ones as special cases.

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  1. 1.

    Mintert, F., Kuś, M., Buchleitner, A.: Concurrence of mixed bipartite quantum states in arbitrary dimensions. Phys. Rev. Lett. 92, 167902 (2004)

    Article  ADS  Google Scholar 

  2. 2.

    Chen, K., Albeverio, S., Fei, S.M.: Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett. 95, 040504 (2005)

    MathSciNet  Article  ADS  Google Scholar 

  3. 3.

    Breuer, H.P.: Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett. 97, 080501 (2006)

    Article  ADS  Google Scholar 

  4. 4.

    de Vicente, J.I.: Lower bounds on concurrence and separability conditions. Phys. Rev. A 75, 052320 (2007)

    MathSciNet  Article  ADS  Google Scholar 

  5. 5.

    Zhang, C.J., Zhang, Y.S., Zhang, S., Guo, G.C.: Optimal entanglement witnesses based on local orthogonal observables. Phys. Rev. A 76, 012334 (2007)

    Article  ADS  Google Scholar 

  6. 6.

    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  7. 7.

    Pawlowski, M.: Security proof for cryptographic protocols based only on the monogamy of Bell’s inequality violations. Phys. Rev. A 82, 032313 (2010)

    Article  ADS  Google Scholar 

  8. 8.

    Koashi, M., Winter, A.: Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)

    MathSciNet  Article  ADS  Google Scholar 

  9. 9.

    Bai, Y.K., Ye, M.Y., Wang, Z.D.: Entanglement monogamy and entanglement evolution in multipartite systems. Phys. Rev. A 80, 044301 (2009)

    Article  ADS  Google Scholar 

  10. 10.

    Osborne, T.J., Verstraete, F.: General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006)

    Article  ADS  Google Scholar 

  11. 11.

    Bai, Y.K., Xu, Y.F., Wang, Z.D.: General monogamy relation for the entanglement of formation in multiqubit systems. Phys. Rev. Lett. 113, 100503 (2014)

    Article  ADS  Google Scholar 

  12. 12.

    Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)

    Article  ADS  Google Scholar 

  13. 13.

    Zhu, X.N., Fei, S.M.: Entanglement monogamy relations of qubit systems. Phys. Rev. A 90, 024304 (2014)

    Article  ADS  Google Scholar 

  14. 14.

    Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)

    Article  ADS  Google Scholar 

  15. 15.

    He, H., Vidal, G.: Disentangling theorem and monogamy for entanglement negativity. Phys. Rev. A 91, 012339 (2015)

    MathSciNet  Article  ADS  Google Scholar 

  16. 16.

    Jin, Z.X., Li, J., Li, T., Fei, S.M.: Tighter monogamy relations in multiqubit systems. Phys. Rev. A 97, 032336 (2018)

    MathSciNet  Article  ADS  Google Scholar 

  17. 17.

    Uhlmann, A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)

    MathSciNet  Article  ADS  Google Scholar 

  18. 18.

    Rungta, P., Bužek, V.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)

    MathSciNet  Article  ADS  Google Scholar 

  19. 19.

    Albeverio, S., Fei, S.M.: A note on invariants and entanglements. J. Opt. B Quantum Semiclass Opt. 3, 223 (2001)

    MathSciNet  Article  ADS  Google Scholar 

  20. 20.

    Acin, A., Andrianov, A., Costa, L., Jane, E., Latorre, J.I., Tarrach, R.: Generalized schmidt decomposition and classification of three-quantum-bit states. Phys. Rev. Lett. 85, 1560 (2000)

    Article  ADS  Google Scholar 

  21. 21.

    Gao, X.H., Fei, S.M.: Estimation of concurrence for multipartite mixed states. Eur. Phys. J. Spec. Top. 159, 71 (2008)

    Article  Google Scholar 

  22. 22.

    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)

    MathSciNet  Article  ADS  Google Scholar 

  23. 23.

    Lee, S., Chi, D.P., Oh, S.D., Kim, J.: Convex-roof extended negativity as an entanglement measure for bipartite quantum systems. Phys. Rev. A 68, 062304 (2003)

    Article  ADS  Google Scholar 

  24. 24.

    Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)

    Article  ADS  Google Scholar 

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This work is supported by NSFC under numbers 11675113, 11605083, and Beijing Municipal Commission of Education (KM201810011009).

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Correspondence to Xue-Na Zhu.

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Zhu, X., Fei, S. Monogamy properties of qubit systems. Quantum Inf Process 18, 23 (2019).

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  • Monogamy relations
  • Concurrence
  • Negativity
  • Entanglement of formation