Advertisement

Entropic uncertainty relations in the Heisenberg XXZ model and its controlling via filtering operations

  • Fei Ming
  • Dong Wang
  • Wei-Nan Shi
  • Ai-Jun Huang
  • Wen-Yang Sun
  • Liu Ye
Article

Abstract

The uncertainty principle is recognized as an elementary ingredient of quantum theory and sets up a significant bound to predict outcome of measurement for a couple of incompatible observables. In this work, we develop dynamical features of quantum memory-assisted entropic uncertainty relations (QMA-EUR) in a two-qubit Heisenberg XXZ spin chain with an inhomogeneous magnetic field. We specifically derive the dynamical evolutions of the entropic uncertainty with respect to the measurement in the Heisenberg XXZ model when spin A is initially correlated with quantum memory B. It has been found that the larger coupling strength \( J \) of the ferromagnetism (\( J < 0 \)) and the anti-ferromagnetism (\( J > 0 \)) chains can effectively degrade the measuring uncertainty. Besides, it turns out that the higher temperature can induce the inflation of the uncertainty because the thermal entanglement becomes relatively weak in this scenario, and there exists a distinct dynamical behavior of the uncertainty when an inhomogeneous magnetic field emerges. With the growing magnetic field \( \left| B \right| \), the variation of the entropic uncertainty will be non-monotonic. Meanwhile, we compare several different optimized bounds existing with the initial bound proposed by Berta et al. and consequently conclude Adabi et al.’s result is optimal. Moreover, we also investigate the mixedness of the system of interest, dramatically associated with the uncertainty. Remarkably, we put forward a possible physical interpretation to explain the evolutionary phenomenon of the uncertainty. Finally, we take advantage of a local filtering operation to steer the magnitude of the uncertainty. Therefore, our explorations may shed light on the entropic uncertainty under the Heisenberg XXZ model and hence be of importance to quantum precision measurement over solid state-based quantum information processing.

Keywords

Entropic uncertainty relation Lower bound Mixedness Filtering operation 

Notes

Acknowledgements

This work was supported by the National Science Foundation of China under Grant Nos. 61601002 and 11575001, Anhui Provincial Natural Science Foundation (Grant No. 1508085QF139), and the fund from CAS Key Laboratory of Quantum Information (Grant No. KQI201701).

References

  1. 1.
    Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172 (1927)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Robertson, H.P.: Violation of Heisenberg’s uncertainty principle. Phys. Rev. 34, 163 (1929)ADSCrossRefGoogle Scholar
  3. 3.
    Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys. 44, 326 (1927)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103 (1988)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070 (1987)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Maccone, L., Pati, A.K.: Stronger uncertainty relations for the sum of variances. Phys. Rev. Lett. 113, 260401 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    Wang, K.K., Zhan, X., Bian, Z.H., Li, J., Zhang, Y.S., Xue, P.: Experimental investigation of the stronger uncertainty relations for all incompatible observables. Phys. Rev. A 93, 052108 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Renes, J.M., Boileau, J.C.: Conjectured strong complementary information tradeoff. Phys. Rev. Lett. 103, 020402 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659–662 (2010)CrossRefGoogle Scholar
  11. 11.
    Prevedel, R., Hamel, D.R., Colbeck, R., Fisher, K., Resch, K.J.: Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement. Nat. Phys. 7, 757–761 (2011)CrossRefGoogle Scholar
  12. 12.
    Li, C.F., Xu, J.S., Xu, X.Y., Li, K., Guo, G.C.: Experimental investigation of the entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752–756 (2011)CrossRefGoogle Scholar
  13. 13.
    Nielson, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2002)Google Scholar
  14. 14.
    Pati, A.K., Wilde, M.M., Devi, A.U., Rajagopal, A.K., Sudha, : Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory. Phys. Rev. A 86, 042105 (2012)ADSCrossRefGoogle Scholar
  15. 15.
    Coles, P.J., Piani, M.: Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A 89, 022112 (2014)ADSCrossRefGoogle Scholar
  16. 16.
    Adabi, F., Salimi, S., Haseli, S.: Tightening the entropic uncertainty bound in the presence of quantum memory. Phys. Rev. A 93, 062123 (2016)ADSCrossRefGoogle Scholar
  17. 17.
    Pramanik, T., Mal, S., Majumdar, A.S.: Lower bound of quantum uncertainty from extractable classical information. Quantum Inf. Process. 15, 981–999 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Hu, M. L., Hu, X. Y., Wang, J. C., Peng, Y., Zhang, Y. R and Fan, H.: Quantum coherence and quantum correlations. arXiv: 1703.01852 (2017)Google Scholar
  20. 20.
    Hu, M.L., Fan, H.: Upper bound and shareability of quantum discord based on entropic uncertainty relations. Phys. Rev. A 88, 014105 (2013)ADSCrossRefGoogle Scholar
  21. 21.
    Dupuis, F., Fawzi, O., Wehner, S.: Entanglement sampling and applications. IEEE Trans. Inf. Theory 61, 1093 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Koenig, R., Wehner, S., Wullschleger, J.: Unconditional security from noisy quantum storage. IEEE Trans. Inf. Theory 58, 1962–1984 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vallone, G., Marangon, D.G., Tomasin, M., Villoresi, P.: Quantum randomness certified by the uncertainty principle. Phys. Rev. A 90, 052327 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using d-level systems. Phys. Rev. Lett. 88, 127902 (2002)ADSCrossRefGoogle Scholar
  25. 25.
    Grosshans, F., Cerf, N.J.: Continuous-variable quantum cryptography is secure against non-Gaussian attacks. Phys. Rev. Lett. 92, 047905 (2004)ADSCrossRefGoogle Scholar
  26. 26.
    Jarzyna, M., Demkowicz-Dobrzański, R.: True precision limits in quantum metrology. New J. Phys. 17, 013010 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    Jia, L.J., Tian, Z.H., Jing, J.L.: Entropic uncertainty relation in de Sitter space. Ann. Phys. 353, 37–47 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zou, H.M., Fang, M.F., Yang, B.Y., Guo, Y.N., He, W., Zhang, S.Y.: The quantum entropic uncertainty relation and entanglement witness in the two-atom system coupling with the non-Markovian environments. Phys. Scr. 89, 115101 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Yao, C.M., Chen, Z.H., Ma, Z.H., Severini, S., Serafini, A.: Entanglement and discord assisted entropic uncertainty relations under decoherence. Sci. China Phys. Mech. Astron. 57, 1703–1711 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Zhang, Y.L., Fang, M.F., Kang, G.D., Zhou, Q.P.: Reducing quantum-memory-assisted entropic uncertainty by weak measurement and weak measurement reversal. Int. J. Quantum Inf. 13, 1550037 (2015)CrossRefzbMATHGoogle Scholar
  31. 31.
    Xu, Z.Y., Yang, W.L., Feng, M.: Quantum-memory-assisted entropic uncertainty relation under noise. Phys. Rev. A 86, 012113 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Wang, D., Ming, F., Huang, A.J., Sun, W.Y., Shi, J.D., Ye, L.: Exploration of quantum-memory-assisted entropic uncertainty relations in a noninertial frame. Laser Phys. Lett. 14, 055205 (2017)ADSCrossRefGoogle Scholar
  33. 33.
    Feng, J., Zhang, Y.Z., Gould, M.D., Fan, H.: Entropic uncertainty relations under the relativistic motion. Phys. Lett. B 726, 527–532 (2013)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Huang, A.J., Shi, J.D., Wang, D., Ye, L.: Steering quantum-memory-assisted entropic uncertainty under unital and nonunital noises via filtering operations. Quantum Inf. Process. 16, 46 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zheng, X., Zhang, G.F.: The effects of mixedness and entanglement on the properties of the entropic uncertainty in Heisenberg model with Dzyaloshinski–Moriya interaction. Quantum Inf. Process. 16, 1 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Huang, A.J., Wang, D., Wang, J.M., Shi, J.D., Sun, W.Y., Ye, L.: Exploring entropic uncertainty relation in the Heisenberg XX model with inhomogeneous magnetic field. Quantum Inf. Process. 16, 204 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Asoudeh, M., Karimipour, V.: Thermal entanglement of spins in an inhomogeneous magnetic field. Phys. Rev. A 71, 022308 (2005)ADSCrossRefGoogle Scholar
  38. 38.
    Liang, Q.: Quantum correlations in a two-qubit Heisenberg XX model under intrinsic decoherence. Commun. Theor. Phys. 60, 391 (2013)CrossRefGoogle Scholar
  39. 39.
    Zhang, G.F., Li, S.S.: Thermal entanglement in a two-qubit Heisenberg XXZ spin chain under an inhomogeneous magnetic field. Phys. Rev. A 72, 034302 (2005)ADSCrossRefGoogle Scholar
  40. 40.
    Wang, D., Ming, F., Huang, A.J., Sun, W.Y., Ye, L.: Entropic uncertainty for spin-1/2 XXX chains in the presence of inhomogeneous magnetic fields and its steering via weak measurement reversals. Laser Phys. Lett. 14, 095204 (2017)ADSCrossRefGoogle Scholar
  41. 41.
    Sumana, K., Ajoy, S., Amit, B., Debasis, S.: Effect of local filtering on freezing phenomena of quantum correlation. Quantum Inf. Process. 14, 2517–2533 (2015)CrossRefzbMATHGoogle Scholar
  42. 42.
    Michael, S., Ali, A.K.: Defeating entanglement sudden death by a single local filtering. Phys. Rev. A 86, 032304 (2012)CrossRefGoogle Scholar
  43. 43.
    Peters, N.A., Wei, T.C., Kwiat, P.G.: Mixed-state sensitivity of several quantum-information benchmarks. Phys. Rev. A 70, 052309 (2004)ADSCrossRefGoogle Scholar
  44. 44.
    Singh, U., Bera, M.N., Dhar, H.S., Pati, A.K.: Maximally coherent mixed states: complementarity between maximal coherence and mixedness. Phys. Rev. A 91, 052115 (2015)ADSCrossRefGoogle Scholar
  45. 45.
    Wang, S.C., Yu, Z.W., Wang, X.B.: Protecting quantum states from decoherence of finite temperature using weak measurement. Phys. Rev. A 89, 022318 (2014)ADSCrossRefGoogle Scholar
  46. 46.
    Zhang, S.Y., Fang, M.F., Yu, M.: Controlling of entropic uncertainty in qubits system under the generalized amplitude damping channel via weak measurements. Int. J. Theor. Phys. 55, 1824–1832 (2016)CrossRefzbMATHGoogle Scholar
  47. 47.
    Zhang, S.Y., Fang, M.F., Zhang, Y.L., Guo, Y.N., Zhao, Y.J., Tang, W.W.: Reduction of entropic uncertainty in entangled qubits system by local PT-symmetric operation. Chin. Phys. B 24, 090304 (2015)ADSCrossRefGoogle Scholar
  48. 48.
    Flavien, H., Marco, T.Q., Joseph, B., Nicolas, B.: Genuine hidden quantum nonlocality. Phys. Rev. Lett. 111, 160402 (2013)CrossRefGoogle Scholar
  49. 49.
    Sun, Q.Q., Al-Amri, M., Davidovich, L., Suhail Zubairy, M.: Reversing entanglement change by a weak measurement. Phys. Rev. A 82, 052323 (2010)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Physics and Material ScienceAnhui UniversityHefeiChina
  2. 2.CAS Key Laboratory of Quantum InformationUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations