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The effects of mixedness and entanglement on the properties of the entropic uncertainty in Heisenberg model with Dzyaloshinski–Moriya interaction

  • Xiao Zheng
  • Guo-Feng Zhang
Article

Abstract

The effects of mixedness and entanglement on the lower bound and tightness of the entropic uncertainty in the Heisenberg model with Dzyaloshinski–Moriya (DM) interaction have been investigated. It is found that the mixedness can reflect the essence of the entropic uncertainty better than the entanglement. Meanwhile, the uncertainty of measurement results will be reduced by the entanglement and improved by the mixedness. The entanglement can destroy the tightness of the uncertainty, while the tightness will be improved with the increase in the mixedness. In addition, the tightness of the uncertainty in Heisenberg model can be expressed as a function of the magnetic properties, the strength of the DM interaction as well as the mixedness of the state and the functional form has no relationship with temperature. What’s more, the entropic uncertain inequality becomes uncertain equality when the mixedness of the system reaches the minimum value. For a given mixedness, the tightness will be reduced with the increase in the strength of DM interaction at the antiferromagnetic case while the situation is just the opposite for the ferromagnetic case.

Keywords

Entropic uncertainty Mixedness Tightness Lower bound 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11574022 and 11174024) and the Open Project Program of State Key Laboratory of Low-Dimensional Quantum Physics (Tsinghua University) Grants No. KF201407 and supported by the exploratory study of State Key Laboratory of Software Development Environment Grants No. SKLSDE-2015ZX-10.

References

  1. 1.
    Heisenberg, W.: Überden anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172 (1927)ADSCrossRefGoogle Scholar
  2. 2.
    Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163 (1929)ADSCrossRefGoogle Scholar
  3. 3.
    Kraus, K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070 (1987)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    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 (2011)CrossRefGoogle Scholar
  5. 5.
    Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659 (2010)CrossRefGoogle Scholar
  6. 6.
    Renes, J.M., Boileau, J.C.: Conjectured strong complementary information tradeoff. Phys. Rev. Lett. 103, 020402 (2009)ADSCrossRefGoogle Scholar
  7. 7.
    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 (2011)CrossRefGoogle Scholar
  8. 8.
    Cui, J., Gu, M., Kwek, L.C., Santos, M.F., Fan, H., Vedral, V.: Quantum phases with differing computational power. Nat. Commun. 3, 812 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Tomamichel, M., Renner, R.: Uncertainty relation for smooth entropies. Phys. Rev. Lett. 106, 110506 (2011)ADSCrossRefGoogle Scholar
  10. 10.
    Coles, P.J., Colbeck, R., Yu, L., Zwolak, M.: Uncertainty relations from simple entropic properties. Phys. Rev. Lett. 108, 210405 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Coles, P.J.: Collapse of the quantum correlation hierarchy links entropic uncertainty to entanglement creation. Phys. Rev. A 86, 062334 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    Xu, Z.Y., Yang, W.L., Feng, M.: Quantum-memory-assisted entropic uncertainty relation under noise. Phys. Rev. A 86, 012113 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Giovanetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330 (2004)ADSCrossRefGoogle Scholar
  14. 14.
    Busch, P., Heinonen, T., Lahti, P.J.: Heisenberg’s uncertainty principle. Phys. Rep. 452, 155 (2007)ADSCrossRefGoogle Scholar
  15. 15.
    Wehner, S., Winter, A.: Entropic uncertainty relations–A survey. New J. Phys. 12, 025009 (2010)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103 (1988)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Dzyaloshinskii, I.: A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241 (1958)ADSCrossRefGoogle Scholar
  18. 18.
    Moriya, T.: New mechanism of anisotropic superexchange interaction. Phys. Rev. Lett. 4, 228 (1960)ADSCrossRefGoogle Scholar
  19. 19.
    Zhang, G.F.: Thermal entanglement and teleportation in a two-qubit Heisenberg chain with Dzyaloshinski-Moriya anisotropic antisymmetric interaction. Phys. Rev. A 75, 034304 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    Tsallis, C.: Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52, 479 (1988)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Boukobza, E., Tannor, D.J.: Entropy exchange and entanglement in the Jaynes-Cummings model. Phys. Rev. A 71, 063821 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    Hu, M.L., Fan, H.: Quantum-memory-assisted entropic uncertainty principle, teleportation, and entanglement witness in structured reservoirs. Phys. Rev. A 86, 032338 (2007)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Key Laboratory of Micro-Nano Measurement-Manipulation and Physics (Ministry of Education), School of Physics and Nuclear Energy EngineeringBeihang UniversityBeijingChina
  2. 2.State Key Laboratory of Low-Dimensional Quantum PhysicsTsinghua UniversityBeijingChina

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