Skip to main content
SpringerLink
Log in

Unified monogamy relation of entanglement measures

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

The monogamy of quantum entanglement captures the property of limitation in the distribution of entanglement. Various monogamy relations exist for different entanglement measures that are important in quantum information processing. Our goal in this work is to propose general monogamy inequalities for several entanglement measures on entangled qubit systems. The present results provide a unified model for various entanglement measures including the concurrence, the negativity, the entanglement of formation, Tsallis-q entropy, Rényi-q entropy, and Unified-(q, s) entropy. We then propose tightened monogamy inequalities for multipartite qubit systems and derive upper bounds of aforementioned entanglement measures for multipartite pure state under generalized bipartitions. We finally prove a generic result that most of tripartite high-dimensional entangled pure states have no entanglement monogamy. These results are useful for exploring the entanglement theory, quantum information processing and secure quantum communication.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. 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)

    Article  ADS  MathSciNet  MATH  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  MATH  Google Scholar 

  3. Hillery, M., BuŽek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145–195 (2001)

    Article  ADS  MATH  Google Scholar 

  5. Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.: Quantifying entanglement. Phys. Rev. Lett. 78, 2275–2279 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022–5025 (1997)

    Article  ADS  Google Scholar 

  7. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Landsberg, P.T., Vedral, V.: Distributions and channel capacities in generalized statistical mechanics. Phys. Lett. A 247, 211–217 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Kim, J.S.: Tsallis entropy and entanglement constraints in multiqubit systems. Phys. Rev. A 81, 062328 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  12. Horodecki, R., Horodecki, P., Horodecki, M.: Quantum \(\alpha \)-entropy inequalities: independent condition for local realism? Phys. Lett. A 210, 377–381 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Gour, G., Bandyopadhyay, S., Sanders, B.C.: Dual monogamy inequality for entanglement. J. Math. Phys. 48, 012108 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Kim, J.S., Sanders, B.C.: Monogamy of multi-qubit entanglement using Rényi entropy. J. Phys. A Math. Theor. 43, 445305 (2010)

    Article  MATH  Google Scholar 

  15. Kim, J.S., Sanders, B.C.: Unified entropy, entanglement measures and monogamy of multi-party entanglement. J. Phys. A Math. Theor. 44, 295303 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  17. Terhal, B.: Is entanglement monogamous? IBM J. Res. Dev. 48, 71–78 (2004)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  19. 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)

    Article  ADS  Google Scholar 

  20. de Oliveira, T.R., Cornelio, M.F., Fanchini, F.F.: Monogamy of entanglement of formation. Phys. Rev. A 89, 034303 (2014)

    Article  ADS  Google Scholar 

  21. 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 

  22. Bai, Y.K., Xu, Y.F., Wang, Z.D.: Hierarchical monogamy relations for the squared entanglement of formation in multipartite systems. Phys. Rev. A 90, 062343 (2014)

    Article  ADS  Google Scholar 

  23. Luo, Y., Tian, T., Shao, L.H., Li, Y.M.: General monogamy of Tsallis \(q\)-entropy entanglement in multiqubit systems. Phys. Rev. A 93, 062340 (2016)

    Article  ADS  Google Scholar 

  24. Song, W., Bai, Y.K., Yang, M., Cao, Z.L.: General monogamy relation of multi-qubit system in terms of squared Rényi-\(\alpha \) entanglement. Phys. Rev. A 93, 022306 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  25. Khan, A., ur Rehman, J., Wang, K., Shin, H.: Unified monogamy relations of multipartite entanglement. Sci. Rep. 9, 16419 (2019)

    Article  ADS  Google Scholar 

  26. Luo, Y., Li, Y.M.: Monogamy of \(\alpha \)th power entanglement measurement in qubit system. Ann. Phys. 362, 511–520 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  29. Jin, Z.X., Fei, S.M.: Tighter generalized monogamy and polygamy relations for multiqubit systems. Quantum Inf. Proc. 19, 1–15 (2020)

    ADS  MathSciNet  Google Scholar 

  30. Kim, J.S.: Negativity and tight constraints of multiqubit entanglement. Phys. Rev. A 97, 012334 (2018)

    Article  ADS  Google Scholar 

  31. Kim, J.S.: Hamming weight and tight constraints of multi-qubit entanglement in terms of unified entropy. Sci. Rep. 8, 12245 (2018)

    Article  ADS  Google Scholar 

  32. Jin, Z.X., Fei, S.M.: Tighter entanglement monogamy relations of qubit systems. Quantum Inf. Proc. 16, 77 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)

    Article  ADS  MATH  Google Scholar 

  34. Rungta, P., BuŽek, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  35. Gour, G., Sanders, B.C.: Remote preparation and distribution of bipartite entangled states. Phys. Rev. Lett. 93, 260501 (2004)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  37. Tian, T., Luo, Y., Li, Y.M.: Generalised monogamy relation of convex-roof extended negativity in multi-level systems. Sci. Rep. 6, 36700 (2016)

    Article  ADS  Google Scholar 

  38. 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 

  39. Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479–487 (1988)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Tsallis, C., Lloyd, S., Baranger, M.: Peres criterion for separability through nonextensive entropy. Phys. Rev. A 63, 042104 (2001)

    Article  ADS  Google Scholar 

  41. Rajagopal, A.K., Rendell, R.W.: Classical statistics inherent in a quantum density matrix. Phys. Rev. A 72, 022322 (2005)

    Article  ADS  Google Scholar 

  42. Yuan, G.M., Song, W., Yang, M., Li, D.C., Zhao, J.L., Cao, Z.L.: Monogamy relation of multi-qubit systems for squared Tsallis-q entanglement. Sci. Rep. 6, 28719 (2016)

    Article  ADS  Google Scholar 

  43. Rathie, P.N., Taneja, I.J.: Unified (r, s)-entropy and its bivariate measures. Inf. Sci. 54, 23–39 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gao, L.M., Yan, F.L., Gao, T.: Tighter monogamy and polygamy relations of multiparty quantum entanglement. Quantum Inf. Proc. 19, 276 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  45. Santos, E., Ferrero, M.: Linear entropy and Bell inequalities. Phys. Rev. A 62, 024101 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  46. Zhang, C.J., Gong, Y.X., Zhang, Y.S., Guo, G.C.: Observable estimation of entanglement for arbitrary finite-dimensional mixed states. Phys. Rev. A 78, 042308 (2008)

    Article  ADS  Google Scholar 

  47. Rastegin, A.E.: Some general properties of unified entropies. J. Stat. Phys. 143, 1120 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. Ou, Y.: Violation of monogamy inequality for higher-dimensional objects. Phys. Rev. A 75, 723–727 (2007)

    Article  Google Scholar 

  49. Chen, K., Albeverio, S., Fei, S.M.: Concurrence-based entanglement measure for Werner States. Rep. Math. Phys. 58, 325–334 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Ren, X.J., Jiang, W.: Entanglement monogamy inequality in a 2-2-4 system. Phys. Rev. A 81, 024305 (2010)

    Article  ADS  Google Scholar 

  51. Cécilia, L., Sara, D.M., Huber, M., Piani, M., Adesso, G., Winter, A.: Should entanglement measures be monogamous or faithful? Phys. Rev. Lett. 117, 060501 (2016)

    Article  Google Scholar 

  52. Osborne, T.J.: Entanglement measure for rank-2 mixed states. Phys. Rev. A 72, 656–665 (2005)

    Article  MathSciNet  Google Scholar 

  53. Guillaume, A., Stanisław, S., Elisabeth, W.: Hastings’s additivity counterexample via Dvoretzky’s theorem. Commun. Math. Phys. 305, 85–97 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Hayden, P., Leung, D.W., Winter, A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Lubkin, Elihu: Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028–1031 (1978)

    Article  ADS  MATH  Google Scholar 

  56. Acín, A., Andrianov, A., Costa, L., Jané, 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 

  57. Kempe, J., Bacon, D., Lidar, D.A., Whaley, K.B.: Theory of decoherence-free fault-tolerant universal quantum computation. Phys. Rev. A 63, 392–396 (2000)

    Google Scholar 

  58. Zhou, Y.S., Li, X., Deng, Y., Li, H.R., Luo, M.X.: Generation of hybrid four-qubit entangled Decoherence-free state assisted by the cavity-QED system. Opt. Commun. 366, 397–403 (2016)

    Article  ADS  Google Scholar 

  59. Stockton, J.K., Geremia, J.M., Doherty, A.C., Mabuchi, H.: Characterizing the entanglement of symmetric many-particle spin-1/2 systems. Phys. Rev. A 67, 426–430 (2003)

    Article  Google Scholar 

  60. Karmakar, S., Sen, A., Bhar, A., Sarkar, D.: Strong monogamy conjecture in a four-qubit system. Phys. Rev. A 93, 012327 (2016)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 61772437), Sichuan Youth Science and Technique Foundation (No. 2017JQ0048), and Fundamental Research Funds for the Central Universities (No. 2018GF07).

Author information

Authors and Affiliations

  1. The School of Information Science and Technology, Southwest Jiaotong University, Chengdu, 610031, China

    Xue Yang & Ming-Xing Luo

Authors
  1. Xue Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ming-Xing Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ming-Xing Luo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, X., Luo, MX. Unified monogamy relation of entanglement measures. Quantum Inf Process 20, 108 (2021). https://doi.org/10.1007/s11128-021-03041-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-021-03041-z

Keywords

Search

Navigation