Information Flow Versus Divisibility for Non-invertible Dynamical Maps

  • Dariusz Chruściński
  • Ángel RivasEmail author
  • Sagnik Chakraborty
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 237)


We study the equivalence between information flow and completely positive divisibility—the two main approaches to Markovianity in quantum regime. Such equivalence is well known to hold for maps which are invertible. For non-invertible maps, the problem is more subtle. We show that for a class of so-called image non-increasing dynamical maps, the equivalence still holds true. Moreover, for qubit dynamics we show that the equivalence is universal, thus providing a comprehensive theory of quantum Markovianity at least for two dimensions. In the course of our proofs, we found certain mathematical restrictions on existence and impossibility of existence of completely positive trace-preserving projectors onto subspaces of finite-dimensional operator space. We illustrate our results with appropriate examples.


Open quantum systems Dynamical maps Divisible maps Markovian evolution 



D.C. was supported by the Polish National Science Centre project 2018/30/A/ST2/00837. A.R. acknowledges the Spanish MINECO grants FIS2015-67411 and FIS2017-91460-EXP, the CAM research consortium QUITEMAD + grant S2013/ICE-2801, and the US Army Research Office through Grant No. W911NF-14-1-0103 for partial financial support.


  1. 1.
    H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007)CrossRefGoogle Scholar
  2. 2.
    U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 2000)Google Scholar
  3. 3.
    A. Rivas, S.F. Huelga, Open Quantum Systems. An Introduction (Springer, Heidelberg, 2011)zbMATHGoogle Scholar
  4. 4.
    Á. Rivas, S.F. Huelga, M.B. Plenio, Rep. Prog. Phys. 77, 094001 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    H.-P. Breuer, E.-M. Laine, J. Piilo, B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    I. de Vega, D. Alonso, Rev. Mod. Phys. 89, 015001 (2017)ADSCrossRefGoogle Scholar
  7. 7.
    L. Li, M.J.W. Hall, H.M. Wiseman, Phys. Rep. 759, 1 (2018)Google Scholar
  8. 8.
    Á. Rivas, S.F. Huelga, M.B. Plenio, Phys. Rev. Lett. 105, 050403 (2010)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    H.-P. Breuer, E.-M. Laine, J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Chruściński, A. Kossakowski, Á. Rivas, Phys. Rev. A 83, 052128 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    F. Buscemi, N. Datta, Phys. Rev. A 93, 012101 (2016)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    B. Bylicka, M. Johansson, A. Acín, Phys. Rev. Lett. 118, 120501 (2017)ADSCrossRefGoogle Scholar
  13. 13.
    S. Chakraborty, Phys. Rev. A 97, 032130 (2018)ADSCrossRefGoogle Scholar
  14. 14.
    D. Chruściński, Á. Rivas, E. Størmer, Phys. Rev. Lett. 121, 080407 (2018)ADSCrossRefGoogle Scholar
  15. 15.
    A. Kossakowski, Rep. Math. Phys. 3, 247 (1972); Bull. Acad. Pol. Sci. Math. Ser. Math. Astron. 20, 1021 (1972)Google Scholar
  16. 16.
    D. Chruściński, S. Maniscalco, Phys. Rev. Lett. 112, 120404 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    V. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, Cambridge, 2003)Google Scholar
  18. 18.
    T. Heinosaari, M.A. Jivulescu, D. Reeb, M.M. Wolf, J. Math. Phys. 53, 102208 (2012)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    E. Størmer, Positive Linear Maps of Operator Algebras. Springer Monographs in Mathematics (2013)Google Scholar
  20. 20.
    A. Jencova, J. Math. Phys. 53, 012201 (2012)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    D. Braun, O. Giraud, I. Nechita, C. Pellegrini, M. Zńidaric, J. Phys. A 47, 135302 (2014)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    S. Chakraborty, D. Chruściński, Phys. Rev. A 99, 042105 (2019)ADSCrossRefGoogle Scholar
  23. 23.
    P. Alberti, A. Uhlmann, Rep. Math. Phys. 18, 163 (1980)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dariusz Chruściński
    • 1
  • Ángel Rivas
    • 2
    • 3
    Email author
  • Sagnik Chakraborty
    • 4
  1. 1.Institute of Physics, Faculty of Physics, Astronomy and InformaticsNicolaus Copernicus UniversityToruńPoland
  2. 2.Departamento de Física Teórica, Facultad de Ciencias FísicasUniversidad ComplutenseMadridSpain
  3. 3.CCS-Center for Computational SimulationCampus de Montegancedo UPMMadridSpain
  4. 4.Optics and Quantum Information GroupThe Institute of Mathematical SciencesTaramani, ChennaiIndia

Personalised recommendations