Frontiers of Physics

, 14:31601 | Cite as

Dynamical characteristic of measurement uncertainty under Heisenberg spin models with Dzyaloshinskii–Moriya interactions

  • Ying-Yue Yang
  • Wen-Yang Sun
  • Wei-Nan Shi
  • Fei Ming
  • Dong WangEmail author
  • Liu Ye
Research article


The dynamics of measurement’s uncertainty via entropy for a one-dimensional Heisenberg XYZ mode is examined in the presence of an inhomogeneous magnetic field and Dzyaloshinskii–Moriya (DM) interaction. It shows that the uncertainty of interest is intensively in connection with the filed’s temperature, the direction-oriented coupling strengths and the magnetic field. It turns out that the stronger coupling strengths and the smaller magnetic field would induce the smaller measurement’s uncertainty of interest within the current spin model. Interestingly, we reveal that the evolution of the uncertainty exhibits quite different dynamical behaviors in antiferromagnetic (Ji > 0) and ferromagnetic (Ji < 0) frames. Besides, an analytical solution related to the systematic entanglement (i.e., concurrence) is also derived in such a scenario. Furthermore, it is found that the DM-interaction is desirably working to diminish the magnitude of the measurement’s uncertainty in the region of high-temperature. Finally, we remarkably offer a resultful strategy to govern the entropy-based uncertainty through utilizing quantum weak measurements, being of fundamentally importance to quantum measurement estimation in the context of solid-state-based quantum information processing and computation.


measurement uncertainty concurrence Heisenberg XYZ chain weak measurement lower bound 



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


  1. 1.
    W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys. 43(3–4), 172 (1927)ADSzbMATHGoogle Scholar
  2. 2.
    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2000zbMATHGoogle Scholar
  3. 3.
    I. Bialynicki-Birula, Rényi entropy and the uncertainty relations, AIP Conf. Proc. 889, 52 (2007)ADSzbMATHGoogle Scholar
  4. 4.
    E. H. Kennard, Zur Quantenmechanik einfacher Bewegungstypen, Z. Phys. 44(4–5), 326 (1927)ADSzbMATHGoogle Scholar
  5. 5.
    H. P. Robertson, The uncertainty principle, Phys. Rev. 34(1), 163 (1929)ADSGoogle Scholar
  6. 6.
    L. Maccone and A. K. Pati, Stronger Uncertainty Relations for All Incompatible Observables., Phys. Rev. Lett. 113(26), 260401 (2014)ADSGoogle Scholar
  7. 7.
    K. K. Wang, X. Zhan, Z. H. Bian, J. Li, Y. S. Zhang, and P. Xue, Experimental investigation of the stronger uncertainty relations for all incompatible observables, Phys. Rev. A 93(5), 052108 (2016)ADSGoogle Scholar
  8. 8.
    K. Kraus, Complementary observables and uncertainty relations, Phys. Rev. D 35(10), 3070 (1987)ADSMathSciNetGoogle Scholar
  9. 9.
    H. Maassen and J. B. M. Uffink, Generalized entropic uncertainty relations, Phys. Rev. Lett. 60(12), 1103 (1988)ADSMathSciNetGoogle Scholar
  10. 10.
    A. E. Rastegin, Entropic uncertainty relations for successive measurements of canonically conjugate observables, Ann. Phys. (Berlin) 528(11–12), 835 (2016)ADSzbMATHGoogle Scholar
  11. 11.
    A. Ghasemi, M. R. Hooshmandasl, and M. K. Tavassoly, On the quantum information entropies and squeezing associated with the eigenstates of the isotonic oscillator, Phys. Scr. 84(3), 035007 (2011)ADSzbMATHGoogle Scholar
  12. 12.
    D. Wang, A. J. Huang, R. D. Hoehn, F. Ming, W. Y. Sun, J. D. Shi, L. Ye, and S. Kais, Entropic uncertainty relations for Markovian and non-Markovian processes under a structured bosonic reservoir, Sci. Rep. 7(1), 1066 (2017)ADSGoogle Scholar
  13. 13.
    J. M. Renes and J. C. Boileau, Conjectured strong complementary information tradeoff, Phys. Rev. Lett. 103(2), 020402 (2009)ADSGoogle Scholar
  14. 14.
    M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, The uncertainty principle in the presence of quantum memory, Nat. Phys. 6(9), 659 (2010)Google Scholar
  15. 15.
    R. Prevedel, D. R. Hamel, R. Colbeck, K. Fisher, and K. J. Resch, Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement, Nat. Phys. 7(10), 757 (2011)Google Scholar
  16. 16.
    C. F. Li, J. S. Xu, X. Y. Xu, K. Li, and G. C. Guo, Experimental investigation of the entanglement-assisted entropic uncertainty principle, Nat. Phys. 7(10), 752 (2011)Google Scholar
  17. 17.
    Z. Jin, S. L. Su, A. D. Zhu, H. F. Wang, and S. Zhang, Engineering multipartite steady entanglement of distant atoms via dissipation, Front. Phys. 13(5), 134209 (2018)Google Scholar
  18. 18.
    P. J. Coles, R. Colbeck, L. Yu, and M. Zwolak, Uncertainty Relations from Simple Entropic Properties, Phys. Rev. Lett. 108(21), 210405 (2012)ADSGoogle Scholar
  19. 19.
    Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. 85(2), 623 (2013)ADSGoogle Scholar
  20. 20.
    M. J. W. Hall and H. M. Wiseman, Heisenberg-style bounds for arbitrary estimates of shift parameters including prior information, New J. Phys. 14(3), 033040 (2012)ADSGoogle Scholar
  21. 21.
    C. S. Yu, Quantum coherence via skew information and its polygamy, Phys. Rev. A 95(4), 042337 (2017)ADSGoogle Scholar
  22. 22.
    P. J. Coles and M. Piani, Complementary sequential measurements generate entanglement, Phys. Rev. A 89(1), 010302 (2014)ADSGoogle Scholar
  23. 23.
    M. L. Hu and H. Fan, Upper bound and shareability of quantum discord based on entropic uncertainty relations, Phys. Rev. A 88(1), 014105 (2013)ADSGoogle Scholar
  24. 24.
    X. Y. Chen, L. Z. Jiang, and Z. A. Xu, Precise detection of multipartite entanglement in four-qubit Greenberger–Horne–Zeilinger diagonal states, Front. Phys. 13(5), 130317 (2018)Google Scholar
  25. 25.
    X. M. Liu, W. W. Cheng, and J. M. Liu, Renormalization-group approach to quantum Fisher information in an XY model with staggered Dzyaloshinskii–Moriya interaction, Sci. Rep. 6, 19359 (2016)ADSGoogle Scholar
  26. 26.
    X. M. Liu, Z. Z. Du, and J. M. Liu, Quantum Fisher information for periodic and quasiperiodic anisotropic XY chains in a transverse field, Quantum Inform. Process. 15(4), 1793 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Security of quantum key distribution using d-level systems, Phys. Rev. Lett. 88(12), 127902 (2002)ADSGoogle Scholar
  28. 28.
    F. Grosshans and N. J. Cerf, Continuous-variable quantum cryptography is secure against non-Gaussian attacks, Phys. Rev. Lett. 92(4), 047905 (2004)ADSGoogle Scholar
  29. 29.
    F. Dupuis, O. Fawzi, and S. Wehner, Entanglement Sampling and Applications, IEEE Trans. Inf. Theory 61(2), 1093 (2015)MathSciNetzbMATHGoogle Scholar
  30. 30.
    R. Konig, S. Wehner, and J. Wullschleger, Unconditional security from noisy quantum storage, IEEE Trans. Inf. Theory 58(3), 1962 (2012)MathSciNetzbMATHGoogle Scholar
  31. 31.
    G. Vallone, D. G. Marangon, M. Tomasin, and P. Villoresi, Quantum randomness certified by the uncertainty principle, Phys. Rev. A 90(5), 052327 (2014)ADSGoogle Scholar
  32. 32.
    C. A. Miller and Y. Shi, Proceedings of ACM STOC, New York: ACM Press, 2014, pp 417–426Google Scholar
  33. 33.
    D. Mondal, S. Bagchi, and A. K. Pati, Tighter uncertainty and reverse uncertainty relations, Phys. Rev. A 95(5), 052117 (2017)ADSGoogle Scholar
  34. 34.
    A. Riccardi, C. Macchiavello, and L. Maccone, Tight entropic uncertainty relations for systems with dimension three to five, Phys. Rev. A 95(3), 032109 (2017)ADSMathSciNetGoogle Scholar
  35. 35.
    Z. Y. Xu, W. L. Yang, and M. Feng, Quantum-memoryassisted entropic uncertainty relation under noise, Phys. Rev. A 86(1), 012113 (2012)ADSGoogle Scholar
  36. 36.
    Z. Y. Zhang, D. X. Wei, and J. M. Liu, Entropic uncertainty relation of a two-qutrit Heisenberg spin model in nonuniform magnetic fields and its dynamics under intrinsic decoherence, Laser Phys. Lett. 15(6), 065207 (2018)ADSGoogle Scholar
  37. 37.
    M. Yu and M. F. Fang, Controlling the quantummemory-assisted entropic uncertainty relation by quantum-jump-based feedback control in dissipative environments, Quantum Inform. Process. 16(9), 213 (2017)ADSzbMATHGoogle Scholar
  38. 38.
    Y. L. Zhang, M. F. Fang, G. D. Kang, and Q. P. Zhou, Reducing quantum-memory-assisted entropic uncertainty by weak measurement and weak measurement reversal, Int. J. Quant. Inf. 13(05), 1550037 (2015)zbMATHGoogle Scholar
  39. 39.
    H. M. Zou, M. F. Fang, B. Y. Yang, Y. N. Guo, W. He, and S. Y. Zhang, The quantum entropic uncertainty relation and entanglement witness in the two-atom system coupling with the non-Markovian environments, Phys. Scr. 89(11), 115101 (2014)ADSGoogle Scholar
  40. 40.
    L. J. Jia, Z. H. Tian, and J. L. Jing, Entropic uncertainty relation in de Sitter space, Ann. Phys. 353, 37 (2015)ADSMathSciNetzbMATHGoogle Scholar
  41. 41.
    A. J. Huang, J. D. Shi, D. Wang, and L. Ye, Steering quantum-memory-assisted entropic uncertainty under unital and nonunital noises via filtering operations, Quantum Inform. Process. 16(2), 46 (2017)ADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    X. Zheng and G. F. Zhang, The effects of mixedness and entanglement on the properties of the entropic uncertainty in Heisenberg model with Dzyaloshinski-Moriya interaction, Quantum Inform. Process. 16(1), 1 (2017)ADSMathSciNetzbMATHGoogle Scholar
  43. 43.
    D. Wang, F. Ming, A. J. Huang, W. Y. Sun, J. D. Shi, and L. Ye, Exploration of quantum-memory-assisted entropic uncertainty relations in a noninertial frame, Laser Phys. Lett. 14(5), 055205 (2017)ADSGoogle Scholar
  44. 44.
    D. Wang, W. N. Shi, R. D. Hoehn, F. Ming, W. Y. Sun, S. Kais, and L. Ye, Effects of Hawking radiation on the entropic uncertainty in a Schwarzschild space-time, Ann. Phys. (Berlin) 530(9), 1800080 (2018)Google Scholar
  45. 45.
    Z. M. Huang, Dynamics of entropic uncertainty for atoms immersed in thermal fluctuating massless scalar field, Quantum Inform. Process. 17(4), 73 (2018)ADSMathSciNetzbMATHGoogle Scholar
  46. 46.
    Z. Y. Zhang, J. M. Liu, Z. F. Hu, and Y. Z. Wang, Entropic uncertainty relation for dirac particles in Garfinkle-Horowitz-Strominger dilation space-time, Ann. Phys. (Berlin) 530(11), 1800208 (2018)ADSGoogle Scholar
  47. 47.
    L. M. Yang, B. Chen, S. M. Fei, and Z. X. Wang, Dynamics of coherence-induced state ordering under Markovian channels, Front. Phys. 13(5), 130310 (2018)Google Scholar
  48. 48.
    J. W. Zhou, P. F. Wang, F. Z. Shi, P. Huang, X. Kong, X. K. Xu, Q. Zhang, Z. X. Wang, X. Rong, and J. F. Du, Quantum information processing and metrology with color centers in diamonds, Front. Phys. 9(5), 587 (2014)Google Scholar
  49. 49.
    P. F. Yu, J. G. Cai, J. M. Liu, and G. T. Shen, Teleportation via a two-qubit Heisenberg XYZ model in the presence of phase decoherence, Physica A 387(18), 4723 (2008)ADSGoogle Scholar
  50. 50.
    R. Daneshmand and M. K. Tavassoly, The generation and properties of new classes of multipartite entangled coherent squeezed states in a conducting cavity, Ann. Phys. (Berlin) 529(5), 1600246 (2017)ADSGoogle Scholar
  51. 51.
    M. Qin, X. Wang, Y. B. Li, Z. Bai, and S. J. Lin, Effects of inhomogeneous magnetic fields and different Dzyaloshinskii–Moriya interaction on entanglement and teleportation in a two-qubit Heisenberg XYZ chain, Chin. Phys. C 37(11), 113102 (2013)ADSGoogle Scholar
  52. 52.
    G. Bowen and S. Bose, Teleportation as a depolarizing quantum channel, relative entropy, and classical capacity, Phys. Rev. Lett. 87(26), 267901 (2001)ADSGoogle Scholar
  53. 53.
    W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80(10), 2245 (1998)ADSzbMATHGoogle Scholar
  54. 54.
    Y. Aharonov, D. Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100, Phys. Rev. Lett. 60(14), 1351 (1988)ADSGoogle Scholar
  55. 55.
    A. N. Korotkov, Continuous quantum measurement of a double dot, Phys. Rev. B 60(8), 5737 (1999)ADSGoogle Scholar
  56. 56.
    A. N. Korotkov and A. N. Jordan, Undoing a weak quantum measurement of a solid-state qubit, Phys. Rev. Lett. 97(16), 166805 (2006)ADSGoogle Scholar
  57. 57.
    X. P. Liao, M. S. Rong, and M. F. Fang, Protecting and enhancing spin squeezing from decoherence using weak measurements, Laser Phys. Lett. 14(6), 065201 (2017)ADSGoogle Scholar
  58. 58.
    R. Y. Yang and J. M. Liu, Enhancing the fidelity of remote state preparation by partial measurements, Quantum Inform. Process. 16(5), 125 (2017)ADSMathSciNetzbMATHGoogle Scholar
  59. 59.
    A. N. Korotkov and K. Keane, Decoherence suppression by quantum measurement reversal, Phys. Rev. A 81(4), 040103 (2010)ADSGoogle Scholar
  60. 60.
    S. C. Wang, Z. W. Yu, W. J. Zou, and X. B. Wang, Protecting quantum states from decoherence of finite temperature using weak measurement, Phys. Rev. A 89(2), 022318 (2014)ADSGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Ying-Yue Yang
    • 1
  • Wen-Yang Sun
    • 1
  • Wei-Nan Shi
    • 1
  • Fei Ming
    • 1
  • Dong Wang
    • 1
    • 2
    Email author
  • Liu Ye
    • 1
  1. 1.School of Physics & Material ScienceAnhui UniversityHefeiChina
  2. 2.CAS Key Laboratory of Quantum InformationUniversity of Science and Technology of ChinaHefeiChina

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