Quantitative conditions for time evolution in terms of the von Neumann equation

  • WenHua Wang
  • HuaiXin Cao
  • ZhengLi ChenEmail author
  • Lie Wang


The adiabatic theorem describes the time evolution of the pure state and gives an adiabatic approximate solution to the Schödinger equation by choosing a single eigenstate of the Hamiltonian as the initial state. In quantum systems, states are divided into pure states (unite vectors) and mixed states (density matrices, i.e., positive operators with trace one). Accordingly, mixed states have their own corresponding time evolution, which is described by the von Neumann equation. In this paper, we discuss the quantitative conditions for the time evolution of mixed states in terms of the von Neumann equation. First, we introduce the definitions for uniformly slowly evolving and δ-uniformly slowly evolving with respect to mixed states, then we present a necessary and sufficient condition for the Hamiltonian of the system to be uniformly slowly evolving and we obtain some upper bounds for the adiabatic approximate error. Lastly, we illustrate our results in an example.


quantitative condition time evolution uniformly slowly evolving δ-uniformly slowly evolving upper bound 


  1. 1.
    M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).zbMATHGoogle Scholar
  2. 2.
    L. I. Schiff, Quantum Mechanics (McGraw-Hill Book Co., Inc., New York, 1949).Google Scholar
  3. 3.
    T. Kato, J. Phys. Soc. Jpn. 5, 435 (1950).ADSCrossRefGoogle Scholar
  4. 4.
    B. Simon, Phys. Rev. Lett. 51, 2167 (1983).ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Chen, and Y. Li, Sci. China-Phys. Mech. Astron. 59, 640302 (2016).CrossRefGoogle Scholar
  6. 6.
    M. V. Berry, Proc. R. Soc. A-Math. Phys. Eng. Sci. 392, 45 (1984).ADSCrossRefGoogle Scholar
  7. 7.
    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, arXiv: quantph/0001106v1.Google Scholar
  8. 8.
    D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 150402 (2007).ADSCrossRefGoogle Scholar
  9. 9.
    J. D. Wu, M. S. Zhao, J. L. Chen, and Y. D. Zhang, arXiv: quantph/0706.0264.Google Scholar
  10. 10.
    A. Ambainis, and O. Regev, arXiv: quant-ph/0411152v2.Google Scholar
  11. 11.
    J. Sun, S. F. Lu, and F. Liu, Sci. China-Phys. Mech. Astron. 55, 1630 (2012).ADSCrossRefGoogle Scholar
  12. 12.
    H. X. Cao, Z. H. Guo, Z. L. Chen, and W. H. Wang, Sci. China-Phys. Mech. Astron. 56, 1401 (2013).ADSCrossRefGoogle Scholar
  13. 13.
    W. H. Wang, Z. H. Guo, and H. X. Cao, Sci. China-Phys. Mech. Astron. 57, 218 (2014).CrossRefGoogle Scholar
  14. 14.
    Z. H. Guo, H. X. Cao, and L. Lu, Sci. China-Phys. Mech. Astron. 57, 1835 (2014).ADSCrossRefGoogle Scholar
  15. 15.
    B. M. Yu, H. X. Cao, Z. H. Guo, and W. H. Wang, Sci. China-Phys. Mech. Astron. 57, 2031 (2014).ADSCrossRefGoogle Scholar
  16. 16.
    J. Jing, and L. A. Wu, Sci. Bull. 60, 328 (2015).CrossRefGoogle Scholar
  17. 17.
    W. H. Wang, H. X. Cao, L. Lu, and B. M. Yu, Sci. China-Phys. Mech. Astron. 58, 030001 (2015).Google Scholar
  18. 18.
    Z. Yin, and Z. Wei, Sci. Bull. 62, 741 (2017).CrossRefGoogle Scholar
  19. 19.
    H. Li, Y. Liu, and G. L. Long, Sci. China-Phys. Mech. Astron. 60, 080311 (2017).ADSCrossRefGoogle Scholar
  20. 20.
    G. F. Xu, Sci. China-Phys. Mech. Astron. 61, 010331 (2018).ADSCrossRefGoogle Scholar
  21. 21.
    M. Berman, and R. Kosloff, Comput. Phys. Commun. 63, 1 (1991).ADSCrossRefGoogle Scholar
  22. 22.
    S. J. Wang, D. Zhao, H. G. Luo, L. X. Cen, and C. L. Jia, Phys. Rev. A 64, 052102 (2001).ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    H. Tian, and G. H. Chen, J. Chem. Phys. 137, 204114 (2012).ADSCrossRefGoogle Scholar
  24. 24.
    K. S. Khalid, L. Schulz, and D. Schulz, IEEE Trans. Nanotechnol. 16, 1053 (2017).ADSCrossRefGoogle Scholar
  25. 25.
    J. da Providencia, M. Yamamura, and A. Kuriyama, Prog. Theor. Phys. 85, 939 (1991).ADSCrossRefGoogle Scholar
  26. 26.
    V. I. Man'ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, J. Russ. Laser Res. 20, 421 (1999).CrossRefGoogle Scholar
  27. 27.
    A. Uhlmann, Sci. China-Phys. Mech. Astron. 59, 630301 (2016).CrossRefGoogle Scholar
  28. 28.
    A. C. Aguiar Pinto, K. M. Fonseca Romero, and M. T. Thomaz, Physica A 311, 169 (2002).ADSCrossRefGoogle Scholar
  29. 29.
    K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, J. Chem. Phys. 133, 174119 (2010).ADSCrossRefGoogle Scholar
  30. 30.
    T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, arXiv: 1206.4197v2.Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • WenHua Wang
    • 1
  • HuaiXin Cao
    • 2
  • ZhengLi Chen
    • 2
    Email author
  • Lie Wang
    • 2
  1. 1.School of Ethnic EducationShaanxi Normal UniversityXi’anChina
  2. 2.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina

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