Annales Henri Poincaré

, Volume 19, Issue 6, pp 1787–1816 | Cite as

Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

  • Aurelian GheondeaEmail author


We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on \(C^*\)-algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Work supported by the grant PN-III-P4-ID-PCE-2016-0823 Dynamics and Differentiable Ergodic Theory from UEFISCDI, Romania.


  1. 1.
    Albeverio, S., Høegh-Krohn, R.: Frobenius theory for positive maps of von Neumann algebras. Commun. Math. Phys. 64, 83–94 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arias, A., Gheondea, A., Gudder, S.: Fixed points of quantum operations. J. Math. Phys. 43, 5872–5881 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arveson, W.: Subalgebras of \(C^*\)-algebras. Acta Math. 123, 141–224 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arveson, W.: Subalgebras of \(C^*\)-algebras II. Acta Math. 128, 271–308 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avron, J.E., Fraas, M., Graf, G.M.: Adiabatic response for Lindblad dynamics. J. Stat. Phys. 148, 800–823 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baez, J.C., Biamonte, J.: A course on quantum techniques for stochastic mechanics. arXiv:1209.3632 [quant-ph]
  7. 7.
    Baez, J.C., Fong, B.: A Noether theorem for Markov processes. J. Math. Phys. 54, 013301 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bartoszek, K., Bartoszek, W.: A Noether theorem for stochastic operators on Schatten classes. J. Math. Anal. Appl. 452, 1395–1412 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bratelli, O., Jørgensen, P., Kishimoto, A., Werner, R.: Pure states on \(O_d\). J. Oper. Theory 43, 97–143 (2000)zbMATHGoogle Scholar
  10. 10.
    Brown, N., Ozawa, N.: \(C^*\)-Algebras and Finite-Dimensional Approximations. American Mathematical Society, Providence (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Choi, M.-D.: A Schwarz type inequality for positive maps on \(C^*\)-algebras. Illinois J. Math. 18, 565–574 (1974)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Conway, J.B.: A Course in Operator Theory. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  13. 13.
    Chruscinski, D., Kossakowski, A.: On the structure of entanglement witnesses and new class of positive indecomposable maps. Open Syst. Inf. Dyn. 14, 275–294 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Davies, E.B.: Quantum stochastic processes II. Commun. Math. Phys. 19, 83–105 (1970)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Interscience Publishers, New York (1958)zbMATHGoogle Scholar
  16. 16.
    Evans, E.E., Hoegh-Krohn, R.: Spectral properties of positive maps on \(C^*\)-algebras. J. Lond. Math. Soc. 17, 345–355 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fagnola, F., Rebolledo, R.: Subharmonic projections for a quantum Markov semigroup. J. Math. Phys. 43, 1074–1082 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fagnola, F., Rebolledo, R.: Notes on the qualitative behaviour of quantum Markov semigroups. In: Attal, S., Joye, A. (eds.) Open Quantum Systems III. Recent Developments, pp. 161–205. Springer, Berlin (2006).Google Scholar
  19. 19.
    Frigerio, A., Verri, M.: Long-time asymptotic properties of dynamical semigroups on \(W^*\)-algebras. Math. Z. 180, 275–286 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821–825 (1976)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gough, J.E., Raţiu, T.S., Smolyanov, O.G.: Noether’s theorem for dissipative quantum dynamical systems. J. Math. Phys. 56, 022108 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gudder, S.: Sequential products of quantum measurements. Rep. Math. Phys. 60, 273–288 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hakeda, J., Tomiyama, J.: On some extension property of von Neumann algebras. Tohoku Math. J. 19, 315–323 (1967)CrossRefzbMATHGoogle Scholar
  24. 24.
    Hille, E., Phillips, R.S.: Functional Analysis on Semi-groups, vol. 31. Colloquium Publications - American Mathematical Society, Providence (1957)zbMATHGoogle Scholar
  25. 25.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kadison, R.: A generalized Schwarz inequality and algebraic invariants for operator algebras. Ann. Math. 56, 494–503 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kraus, K.: General state changes in quantum theory. Ann. Phys. 64, 311–335 (1971)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kraus, K.: States, Effects, and Operations. Springer, Berlin (1983)zbMATHGoogle Scholar
  29. 29.
    Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Noether, E.: Invariante Variationsprobleme. Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen, Math-phys. Klasse. 235–257 (1918)Google Scholar
  31. 31.
    Paulsen, V.I.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  32. 32.
    Phillips, R.S.: The adjoint semigroup. Pac. J. Math. 5, 269–283 (1955)CrossRefzbMATHGoogle Scholar
  33. 33.
    Schwartz, J.T.: Non-isomorphism of a pair of factors of type III. Commun. Pure Appl. Math. 16, 111–120 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Stinespring, W.F.: Positive functions on \(C^*\)-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Størmer, E.: Positive linear maps on operator algebras. Acta Math. 110, 233–278 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tomiyama, J.: On the projection of norm one in \(W^*\)-algebras. Proc. Jpn. Acad. 33, 608–612 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tomiyama, J.: Tensor Products and Projections of Norm One in von Neumann Algebras. Seminar Lecture Notes, University of Copenhagen, Copenhagen (1970)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBilkent UniversityBilkent, AnkaraTurkey
  2. 2.Institutul de Matematică al Academiei RomâneBucureştiRomânia

Personalised recommendations