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Tighter monogamy relations of the \(S^{t}\) and \(T^{t}_q\)-entropy entanglement measures based on dual entropy

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Abstract

Monogamy of entanglement is the fundamental property of quantum systems. By using two new entanglement measures based on dual entropy, the \(S^{t}\)-entropy entanglement and \(T^{t}_q\)-entropy entanglement measures, we present the general monogamy relations in multi-qubit quantum systems. We show that these newly derived monogamy inequalities are tighter than the existing ones. Based on these general monogamy relations, we construct the set of multipartite entanglement indicators for N-qubit states, which are shown to work well even for the cases that the usual concurrence-based indicators do not work. Detailed examples are presented to illustrate our results.

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References

  1. Jafarpour, M., Hasanvand, F.K., Afshar, D.: Dynamics of entanglement and measurement-induced disturbance for a hybrid qubit-qutrit system interacting with a spin-chain environment: a mean field approach. Commun. Theor. Phys. 67, 27 (2017). https://doi.org/10.1088/0253-6102/67/1/27

    Article  ADS  Google Scholar 

  2. Wang, M.Y., Xu, J.Z., Yan, F.L., Gao, T.: Entanglement concentration for polarization-spatial-time-bin hyperentangled Bell states. Europhys. Lett. 123, 60002 (2018). https://doi.org/10.1209/0295-5075/123/60002

    Article  ADS  Google Scholar 

  3. Huang, H.L., Goswami, A.K., Bao, W.S., Panigrahi, P.K.: Demonstration of essentiality of entanglement in a Deutsch-like quantum algorithm. Sci. China-Phys. Mech. Astron. 61, 060311 (2018). https://doi.org/10.1007/s11433-018-9175-2

    Article  ADS  Google Scholar 

  4. Deng, F.G., Ren, B.C., Li, X.H.: Quantum hyperentanglement and its applications in quantum information processing. Sci. Bull. 62, 46 (2017). https://doi.org/10.1016/j.scib.2016.11.007

    Article  Google Scholar 

  5. Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022–5025 (1997). https://doi.org/10.1103/PhysRevLett.78.5022

    Article  ADS  Google Scholar 

  6. 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). https://doi.org/10.1103/physreva.54.3824

    Article  ADS  MathSciNet  Google Scholar 

  7. Horodecki, R., Horodecki, P., Horodecki, M.: Quantum \(\alpha \)-entropy inequalities: independent condition for local realism? Phys. Lett. A 210, 377–381 (1996). https://doi.org/10.1016/0375-9601(95)00930-2

    Article  ADS  MathSciNet  Google Scholar 

  8. Gour, G., Bandyopadhyay, S., Sanders, B.C.: Dual monogamy inequality for entanglement. J. Math. Phys. 48, 012108 (2007). https://doi.org/10.1063/1.2435088

    Article  ADS  MathSciNet  Google Scholar 

  9. Kim, J.S., Sanders, B.C.: Monogamy of multi-qubit entanglement using Rényi entropy. J. Phys. A Math. Theor. 43, 445305 (2010). https://doi.org/10.1088/1751-8113/43/44/445305

    Article  Google Scholar 

  10. Landsberg, P.T., Vedral, V.: Distributions and channel capacities in generalized statistical mechanics. Phys. Lett. A 247, 211–217 (1998). https://doi.org/10.1016/S0375-9601(98)00500-3

    Article  ADS  MathSciNet  Google Scholar 

  11. Kim, J.S.: Tsallis entropy and entanglement constraints in multiqubit systems. Phys. Rev. A 81, 062328 (2010). https://doi.org/10.1103/PhysRevA.81.062328

    Article  ADS  MathSciNet  Google Scholar 

  12. Kim, J.S., Sanders, B.C.: Unified entropy, entanglement measures and monogamy of multi-party entanglement. J. Phys. A Math. Theor. 44, 295303 (2011). https://doi.org/10.1088/1751-8113/44/29/295303

    Article  MathSciNet  Google Scholar 

  13. Yang, X., Yang, Y.H., Zhao, L.M., Luo, M.X.: A new entanglement measure based dual entropy. Eur. Phys. J. Plus 138, 654 (2023). https://doi.org/10.1140/epjp/s13360-023-04259-9

    Article  Google Scholar 

  14. Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 138, 052306 (2000). https://doi.org/10.1103/PhysRevA.61.052306

    Article  ADS  Google Scholar 

  15. Terhal, B.: Is entanglement monogamous? IBM J. Res. Dev. 48, 71–78 (2004). https://doi.org/10.1147/rd.481.0071

    Article  Google Scholar 

  16. Osborne, T.J., Verstraete, F.: General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006). https://doi.org/10.1103/PhysRevLett.96.220503

    Article  ADS  Google Scholar 

  17. de Oliveira, T.R., Cornelio, M.F., Fanchini, F.F.: Monogamy of entanglement of formation. Phys. Rev. A 89, 034303 (2014). https://doi.org/10.1103/10.1103/PhysRevA.89.034303

    Article  ADS  Google Scholar 

  18. 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). https://doi.org/10.1103/PhysRevLett.113.100503

    Article  ADS  Google Scholar 

  19. 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). https://doi.org/10.1103/10.1103/PhysRevA.90.062343

    Article  ADS  Google Scholar 

  20. 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). https://doi.org/10.1103/PhysRevA.93.022306

    Article  ADS  MathSciNet  Google Scholar 

  21. 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). https://doi.org/10.1103/10.1103/PhysRevA.93.062340

    Article  Google Scholar 

  22. Khan, A., ur Rehman, J., Wang, K., Shin, H.: Unified monogamy relations of multipartite entanglement. Sci. Rep. 9, 16419 (2019). https://doi.org/10.1038/s41598-019-52817-y

    Article  ADS  Google Scholar 

  23. Christandl, M., Winter, A.: “Squashed entanglement’’: an additive entanglement measure. J. Math. Phys. 45, 829–840 (2004). https://doi.org/10.1063/1.1643788

    Article  ADS  MathSciNet  Google Scholar 

  24. Zhu, X.N., Fei, S.M.: Entanglement monogamy relations of qubit systems. Phys. Rev. A 90, 024304 (2014). https://doi.org/10.1103/PhysRevA.90.024304

    Article  ADS  Google Scholar 

  25. Luo, Y., Li, Y.M.: Monogamy of \(\alpha \)-th power entanglement measurement in qubit system. Ann. Phys. 362, 511–520 (2015). https://doi.org/10.1016/j.aop.2015.08.022

    Article  ADS  MathSciNet  Google Scholar 

  26. Jin, Z.X., Fei, S.M.: Tighter entanglement monogamy relations of qubit systems. Quantum Inf. Process. 16, 77 (2017). https://doi.org/10.1007/s11128-017-1520-3

    Article  ADS  MathSciNet  Google Scholar 

  27. Jin, Z.X., Li, J., Li, T., Fei, S.M.: Tighter monogamy relations in multiqubit systems. Phys. Rev. A 97, 032336 (2018). https://doi.org/10.1103/PhysRevA.97.032336

    Article  ADS  MathSciNet  Google Scholar 

  28. Walter, M., Doran, B., Gross, D., Christandl, M.: Entanglement polytopes: multiparticle entanglement from single-particle information. Science 340, 1205 (2013). https://doi.org/10.1126/science.1232957

    Article  ADS  MathSciNet  Google Scholar 

  29. Seevinck, M. P.: Monogamy of correlations versus monogamy of entanglement. Quantum Inf. Process. 9, 273 (2010). https://doi.org/10.1007/s11128-009-0161-6

    Article  ADS  MathSciNet  Google Scholar 

  30. Ma, X.S., Dakic, B., Naylor, W., Zeilinger, A., Walther, P.: Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems. Nat. Phys. 7, 399 (2011). https://doi.org/10.1038/NPHYS1919

    Article  Google Scholar 

  31. Verlinde, E., Verlinde, H.: Black hole entanglement and quantum error correction. J. High Energy Phys. 1310, 107 (2013). https://doi.org/10.1007/JHEP10(2013)107

    Article  ADS  MathSciNet  Google Scholar 

  32. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948). https://doi.org/10.1002/j.1538-7305.1948.tb00917.x

    Article  MathSciNet  Google Scholar 

  33. Lad, F., Sanfilippo, G., Agro, G.: Extropy: complementary dual of entropy. Stat. Sci. 30, 40–58 (2015). https://doi.org/10.1214/14-STS430

    Article  MathSciNet  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). https://doi.org/10.1103/PhysRevA.64.042315

    Article  ADS  MathSciNet  Google Scholar 

  35. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998). https://doi.org/10.1103/PhysRevLett.80.2245

    Article  ADS  Google Scholar 

  36. Jin, Z.X., Qiao, C.F.: Monogamy and polygamy relations of multiqubit entanglement based on unified entropy. Chin. Phys. B 29, 020305 (2020). https://doi.org/10.1088/1674-1056/ab6720

    Article  ADS  Google Scholar 

  37. Tao, Y.H., Zheng, K., Jin, Z.X., Fei, S.M.: Tighter monogamy relations for concurrence and negativity in multiqubit systems. Mathematics 11(5), 1159 (2023). https://doi.org/10.3390/math11051159

    Article  Google Scholar 

  38. 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). https://doi.org/10.1103/PhysRevLett.85.1560

    Article  ADS  Google Scholar 

  39. Gao, X.H., Fei, S.M.: Estimation of concurrence for multipartite mixed states. Eur. Phys. J. Spec. Top. 159, 71 (2008). https://doi.org/10.1140/epjst/e2008-00694-x

    Article  Google Scholar 

  40. Karmakar, S., Sen, A., Bhar, A., Sarkar, D.: Strong monogamy conjecture in a four-qubit system. Phys. Rev. A 93, 012327 (2016). https://doi.org/10.1103/PhysRevA.93.012327

    Article  ADS  Google Scholar 

  41. 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). https://doi.org/10.1038/srep28719

    Article  ADS  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants 12075159, 12171044 and 12301582; the specific research fund of the Innovation Platform for Academicians of Hainan Province; the Start-up Funding of Dongguan University of Technology No. 221110084.

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Authors and Affiliations

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

    Zhong-Xi Shen, Kang-Kang Yang, Zhi-Xi Wang & Shao-Ming Fei

  2. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, China

    Zhi-Xiang Jin

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Writing-original draft, Zhong-Xi Shen, Kang-Kang Yang; Writing-review and editing, Zhi-Xiang Jin, Zhi-Xi Wang and Shao-Ming Fei. All authors have read and agreed to the published version of the manuscript.

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Shen, ZX., Yang, KK., Jin, ZX. et al. Tighter monogamy relations of the \(S^{t}\) and \(T^{t}_q\)-entropy entanglement measures based on dual entropy. Quantum Inf Process 23, 274 (2024). https://doi.org/10.1007/s11128-024-04481-z

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