Communications in Mathematical Physics

, Volume 349, Issue 1, pp 285–327 | Cite as

Landauer’s Principle in Repeated Interaction Systems

  • Eric P. Hanson
  • Alain Joye
  • Yan PautratEmail author
  • Renaud Raquépas


We study Landauer’s Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system \({\mathcal{S}}\) in contact with a structured environment \({\mathcal{E}}\) made of a chain of independent quantum probes; \({\mathcal{S}}\) interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer’s lower bound, which relates the energy variation of the environment \({\mathcal{E}}\) to a decrease of entropy of the system \({\mathcal{S}}\) during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment \({\mathcal{E}}\) displaying small variations of order \({T^{-1}}\) between the successive probes encountered by \({\mathcal{S}}\), after \({n \simeq T}\) interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of \({\mathcal{S}}\) in this regime, in order to tackle the adiabatic limit of Landauer’s bound. We find that saturation of Landauer’s bound is related to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauer’s bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abou Salem W., Fröhlich J.: Adiabatic theorems and reversible isothermal processes. Lett. Math. Phys. 72, 153–163 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Attal, S., Joye, A., Pillet, C.-A. (eds.): Open Quantum Systems. I, Lecture Notes in Mathematics, vol. 1880. Springer, Berlin (2006). The Hamiltonian approach, Lecture notes from the Summer School held in Grenoble, June 16–July 4, 2003Google Scholar
  3. 3.
    Attal, S., Joye, A., Pillet, C.-A. (eds.): Open quantum systems. II, Lecture Notes in Mathematics, vol. 1881. Springer, Berlin (2006). The Markovian approach, Lecture notes from the Summer School held in Grenoble, June 16–July 4, 2003Google Scholar
  4. 4.
    Attal, S., Joye, A., Pillet, C.-A. (eds.): Open quantum systems. III, Lecture Notes in Mathematics, vol. 1882. Springer, Berlin (2006). Recent developments, Lecture notes from the Summer School held in Grenoble, June 16–July 4, 2003Google Scholar
  5. 5.
    Attal S., Pautrat Y.: From repeated to continuous quantum interactions. Ann. H. Poincaré 7(1), 59–104 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Avron J.E., Elgart A.: Adiabatic theorem without a gap condition. Commun. Math. Phys. 203(2), 445–463 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Avron J.E., Fraas M., Graf G.M., Grech P.: Adiabatic theorems for generators of contracting evolutions. Commun. Math. Phys. 314(1), 163–191 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Avron J.E., Seiler R., Yaffe L.G.: Adiabatic theorems and applications to the quantum hall effect. Commun. Math. Phys. 110, 33–49 (1987)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Born M., Fock V.: Beweis des Adiabatensatzes. Z. Phys. 51, 165–180 (1928)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Bruneau L., Joye A., Merkli M.: Asymptotics of repeated interaction quantum systems. J. Funct. Anal. 239(1), 310–344 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bruneau L., Joye A., Merkli M.: Random repeated interaction quantum systems. Commun. Math. Phys. 284(2), 553–581 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bruneau L., Joye A., Merkli M.: Repeated interactions in open quantum systems. J. Math. Phys. 55(7), 075204 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bruneau L., Pillet C.-A.: Thermal relaxation of a QED cavity. J. Stat. Phys. 134(5–6), 1071–1095 (2009)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Cipriani, F.: Dirichlet forms on noncommutative spaces. In: Quantum Potential Theory, Lecture Notes in Mathematics, vol. 1954, pp. 161–276. Springer, Berlin (2008)Google Scholar
  15. 15.
    Crooks G.E.: Quantum operation time reversal. Phys. Rev. A 77, 034101 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Dereziński, J., Früboes, R.: Fermi golden rule and open quantum systems. In: Open Quantum Systems. III, Lecture Notes in Mathematics, vol. 1882, pp. 67–116. Springer, Berlin (2006)Google Scholar
  17. 17.
    Dranov A., Kellendonk J., Seiler R.: Discrete time adiabatic theorems for quantum mechanical systems. J. Math. Phys. 39(3), 1340–1349 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Evans D.E., Høegh-Krohn R.: Spectral properties of positive maps on C*-algebras. J. Lond. Math. Soc. (2) 17(2), 345–355 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Fagnola F., Umanità V.: Generators of kms symmetric markov semigroups on \({\mathcal{B}(h)}\) symmetry and quantum detailed balance. Commun. Math. Phys. 298(2), 523–547 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Goldstein S., Lindsay J.M.: Kms-symmetric markov semigroups. Math. Z. 219(1), 591–608 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Groh U.: The peripheral point spectrum of Schwarz operators on C*-algebras. Math. Z. 176(3), 311–318 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Horowitz J.M., Parrondo J.M.R.: Entropy production along nonequilibrium quantum jump trajectories. New J. Phys. 15(8), 085028 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Jakšić, V., Ogata, Y., Pautrat, Y., Pillet, C.-A.: Entropic fluctuations in quantum statistical mechanics. an introduction. In: Quantum Theory from Small to Large Scales, pp. 213–410 (2012)Google Scholar
  24. 24.
    Jakšić V., Pillet C.-A.: A note on the Landauer principle in quantum statistical mechanics. J. Math. Phys. 55(7), 075210 (2014)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Jakšić, V., Pillet, C.-A., Westrich, M.: Entropic fluctuations of quantum dynamical semigroups. J. Stat. Phys. 154(1–2), 153–187 (2014)Google Scholar
  26. 26.
    Joye A.: General adiabatic evolution with a gap condition. Commun. Math. Phys. 275, 139–162 (2007)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kato T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Japan 5, 435 (1950)ADSCrossRefGoogle Scholar
  28. 28.
    Kato T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1976)CrossRefGoogle Scholar
  29. 29.
    Kümmerer, B.: Quantum Markov processes and applications in physics. In: Quantum Independent Increment Processes. II, Lecture Notes in Mathematics, vol. 1866, pp. 259–330. Springer, Berlin (2006)Google Scholar
  30. 30.
    Landauer R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183–191 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Marden, M.: The Geometry of the Zeros of a Polynomial in a Complex Variable. Mathematical Surveys, No. 3. Amer. Math. Soc., New York (1949)Google Scholar
  32. 32.
    Nenciu G.: On the adiabatic theorem of quantum mechanics. J. Phys. A Math. Gen. 13, 15–18 (1980)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Ohya M., Petz D.: Quantum Entropy and Its Use. Texts and Monographs in Physics. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rastegin A.: Relations for certain symmetric norms and anti-norms before and after partial trace. J. Stat. Phys. 148, 1040–1053 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Reeb D., Wolf M.M.: An improved Landauer principle with finite-size corrections. New J. Phys. 16(10), 103011 (2014)ADSCrossRefGoogle Scholar
  36. 36.
    Russo B., Dye H.: A note on unitary operators in C*-algebras. Duke Math. J. 33, 413–416 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Schmid, J.: Adiabatic theorems with and without spectral gap condition for non- semisimple spectral values. In: Exner, P., König, W., Neidhardt, H. (eds.) Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference. World Scientific Publishing, Singapore (2014). arXiv:1401.0089
  38. 38.
    Schrader, R.: Perron–Frobenius theory for positive maps on trace ideals. In: Mathematical Physics in Mathematics and Physics (Siena, 2000), Fields Inst. Commun., vol. 30, pp. 361–378. Amer. Math. Soc., Providence (2001)Google Scholar
  39. 39.
    Tanaka, A.: Adiabatic theorem for discrete time evolution. J. Phys. Soc. Japan 80(12) (2011)Google Scholar
  40. 40.
    Teufel S.: A note on the adiabatic theorem without gap condition. Lett. Math. Phys. 58, 261–266 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Wolf, M.M.: Quantum channels and operations: Guided tour. Lecture notes based on a course given at the Niels-Bohr Institute (2012)

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada
  2. 2.Université Grenoble Alpes CNRS, Institut FourierGrenobleFrance
  3. 3.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

Personalised recommendations