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Finer distribution of quantum correlations among multiqubit systems

  • Zhi-Xiang JinEmail author
  • Shao-Ming Fei
Article

Abstract

We study the distribution of quantum correlations characterized by monogamy relations in multipartite systems. By using the Hamming weight of the binary vectors associated with the subsystems, we establish a class of monogamy inequalities for multiqubit entanglement based on the \(\alpha \)th (\(\alpha \ge 2\)) power of concurrence, and a class of polygamy inequalities for multiqubit entanglement in terms of the \(\beta \)th (\(0\le \beta \le 2\)) power of concurrence and concurrence of assistance. Moveover, we give the monogamy and polygamy inequalities for general quantum correlations. Application of these results to quantum correlations like squared convex-roof extended negativity, entanglement of formation and Tsallis-q entanglement gives rise to either tighter inequalities than the existing ones for some classes of quantum states or less restrictions on the quantum states. Detailed examples are presented.

Keywords

Monogamy relation Hamming weight Polygamy inequality Multiqubit systems 

Notes

Acknowledgements

This work is supported by the NSF of China under Grant No. 11675113, and Beijing Municipal Commission of Education under No. KZ201810028042.

References

  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  2. 2.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Mintert, F., Kuś, M., Buchleitner, A.: Concurrence of mixed bipartite quantum states in arbitrary dimensions. Phys. Rev. Lett. 92, 167902 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    Chen, K., Albeverio, S., Fei, S.M.: Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett. 95, 040504 (2005)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Breuer, H.P.: Separability criteria and bounds for entanglement measures. J. Phys. A: Math. Gen. 39, 11847 (2006)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Breuer, H.P.: Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett. 97, 080501 (2006)ADSCrossRefGoogle Scholar
  7. 7.
    de Vicente, J.I.: Lower bounds on concurrence and separability conditions. Phys. Rev. A 75, 052320 (2007)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    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)ADSCrossRefGoogle Scholar
  9. 9.
    Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881 (1992)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Pawlowski, M.: Security proof for cryptographic protocols based only on the monogamy of bells inequality violations. Phys. Rev. A 82, 032313 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Pati, A.K.: Minimum classical bit for remote preparation and measurement of a qubit. Phys. Rev. A 63, 014302 (2000)ADSCrossRefGoogle Scholar
  13. 13.
    Raussendorf, R., Briegel, H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001)ADSCrossRefGoogle Scholar
  14. 14.
    Koashi, M., Winter, A.: Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Osborne, T.J., Verstraete, F.: General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    Bai, Y.K., Ye, M.Y., Wang, Z.D.: Entanglement monogamy and entanglement evolution in multipartite systems. Phys. Rev. A 80, 044301 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    de Oliveira, T.R., Cornelio, M.F., Fanchini, F.F.: Monogamy of entanglement of formation. Phys. Rev. A 89, 034303 (2014)ADSCrossRefGoogle Scholar
  18. 18.
    Adesso, G., Illuminati, F.: Strong monogamy of bipartite and genuine multipartite entanglement: the Gaussian case. Phys. Rev. Lett. 99, 150501 (2007)ADSCrossRefGoogle Scholar
  19. 19.
    Hiroshima, T., Adesso, G., Illuminati, F.: Monogamy inequality for distributed gaussian entanglement. Phys. Rev. Lett. 98, 050503 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    Adesso, G., Illuminati, F.: Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems. New J. Phys. 8, 15 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    Christandl, M., Winter, A.: Squashed entanglement: an additive entanglement measure. J. Math. Phys. 45, 829 (2004)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Yang, D., et al.: Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof. IEEE Trans. Inf. Theory 55, 3375 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ou, Y.C., Fan, H.: Monogamy inequality in terms of negativity for three-qubit states. Phys. Rev. A 75, 062308 (2007)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Kim, J.S., Das, A., Sanders, B.C.: Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extend negativity. Phys. Rev. A 79, 012329 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    He, H., Vidal, G.: Disentangling theorem and monogamy for entanglement negativity. Phys. Rev. A 91, 012339 (2015)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Choi, J.H., Kim, J.S.: Negativity and strong monogamy of multiparty quantum entanglement beyond qubits. Phys. Rev. A 92, 042307 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    Luo, Y., Li, Y.: Monogamy of \(\alpha \)-th power entanglement measurement in qubit system. Ann. Phys. 362, 511 (2015)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Kim, J.S.: Tsallis entropy and entanglement constraints in multiqubit systems. Phys. Rev. A 81, 062328 (2010)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kim, J.S.: Generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Ann. Phys. 373, 197–206 (2016)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Kim, J.S., Sanders, B.C.: Monogamy of multi-qubit entanglement using Rényi entropy. J. Phys. A: Math. Theory 43, 445305 (2010)CrossRefGoogle Scholar
  31. 31.
    Cornelio, M.F., de Oliveira, M.C.: Strong superadditivity and monogamy of the Renyi measure of entanglement. Phys. Rev. A 81, 032332 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    Wang, Y.X., Mu, L.Z., Vedral, V., Fan, H.: Entanglement Rényi-entropy. Phys. Rev. A 93, 022324 (2016)ADSCrossRefGoogle Scholar
  33. 33.
    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)ADSCrossRefGoogle Scholar
  34. 34.
    Uhlmann, A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Rungta, P., Buzek, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Albeverio, S., Fei, S.M.: A note on invariants and entanglements. J. Opt. B: Quantum Semiclass Opt. 3, 223 (2001)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Laustsen, T., Verstraete, F., Van Enk, S.J.: Local vs. joint measurements for the entanglement of assistance. Quantum Inf. Comput. 3, 64 (2003)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Yu, C.S., Song, H.S.: Entanglement monogamy of tripartite quantum states. Phys. Rev. A 77, 032329 (2008)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Goura, G., Bandyopadhyayb, S., Sandersc, B.C.: Dual monogamy inequality for entanglement. J. Math. Phys. 48, 012108 (2007)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Zhu, X.N., Fei, S.M.: Entanglement monogamy relations of qubit systems. Phys. Rev. A 90, 024304 (2014)ADSCrossRefGoogle Scholar
  41. 41.
    Jin, Z.X., Fei, S.M.: Tighter entanglement monogamy relations of qubit systems. Quantum Inf. Process. 16, 77 (2017)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Jin, Z.X., Li, J., Li, T., Fei, S.M.: Tighter monogamy relations in multiqubit systems. Phys. Rev. A 97, 032336 (2018)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Kumar, A., Prabhu, R., De Sen, A., Sen, U.: Effect of a large number of parties on the monogamy of quantum correlations. Phys. Rev. A 91, 012341 (2015)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Adesso, G., Serafini, A., Illuminati, F.: Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: quantification, sharing structure, and decoherence. Phys. Rev. A 73, 032345 (2006)ADSCrossRefGoogle Scholar
  46. 46.
    Giorgi, G.L.: Monogamy properties of quantum and classical correlations. Phys. Rev. A 84, 054301 (2011)ADSCrossRefGoogle Scholar
  47. 47.
    Prabhu, R., Pati, A.K., De Sen, A., Sen, U.: Conditions for monogamy of quantum correlations: Greenberger–Horne–Zeilinger versus \(W\) states. Phys. Rev. A 85, 040102(R) (2012)ADSCrossRefGoogle Scholar
  48. 48.
    Salini, K., Prabhu, R., De Sen, A., Sen, U.: Monotonically increasing functions of any quantum correlation can make all multiparty states monogamous. Ann. Phys. 348, 297–305 (2014)ADSCrossRefGoogle Scholar
  49. 49.
    Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A. 65, 032314 (2002)ADSCrossRefGoogle Scholar
  50. 50.
    Kim, J.S.: Negativity and tight constraints of multiqubit entanglement. Phys. Rev. A 97, 012334 (2018)ADSCrossRefGoogle Scholar
  51. 51.
    Kim, J.S., Sanders, B.C.: Generalized \(W\)-class state and its monogamy relation. J. Phys. A 41, 495301 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  2. 2.School of PhysicsUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

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