Journal of Statistical Physics

, Volume 156, Issue 1, pp 55–65 | Cite as

Equivalent Definitions of the Quantum Nonadiabatic Entropy Production

Article

Abstract

The nonadiabatic entropy production is a useful tool for the thermodynamic analysis of continuously dissipating, nonequilibrium steady states. For open quantum systems, two seemingly distinct definitions for the nonadiabatic entropy production have appeared in the literature, one based on the quantum relative entropy and the other based on quantum trajectories. We show that these two formulations are equivalent. Furthermore, this equivalence leads us to a proof of the monotonicity of the quantum relative entropy under a special class of completely-positive, trace-preserving quantum maps, which circumvents difficulties associated with the noncommuntative structure of operators.

Keywords

Quantum nonequilibrium thermodynamics Nonadiabatic entropy production Quantum relative entropy monotonicity 

References

  1. 1.
    Parrondo, J.M.R., De Cisneros, B.J.: Energetics of brownian motors: a review. Appl. Phys. A 75, 179–191 (2002)ADSCrossRefGoogle Scholar
  2. 2.
    Seifert, U.: Stochastic thermodynamics, fluctuation theorems, and moleculer machines. Rep. Prog. Phys. 75, 126,001 (2012).Google Scholar
  3. 3.
    Esposito, M., Harbola, U., Mukamel, S.: Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665–1702 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Hatano, T., Sasa, S.I.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86(16), 3463–3466 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    Trepagnier, E.T., Jarzynski, C., Ritort, F., Crooks, G.E., Bustamante, C.J., Liphardt, J.: Experimental test of Hatano and Sasa’s nonequilibrium steady-state equality. Proc. Nat. Acad. Sci. USA 101, 15038–15041 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    Speck, T., Seifert, U.: Integral fluctuation theorem for the housekeeping heat. J. Phys. A: Math. Gen. 38(34), L581–L588 (2005)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Esposito, M., Van den Broeck, C.: Three detailed fluctuation theorems. Phys. Rev. Lett. 104(9), 090,601 (2010).Google Scholar
  8. 8.
    Ge, H., Qian, H.: Physical origins of entropy produciton, free energy dissipation, and their mathematical representations. Phys. Rev. E 81, 051,133 (2010).Google Scholar
  9. 9.
    Komatsu, T.S., Nakagawa, N., Sasa, S.I., Tasaki, H.: Steady-state thermodynamics for heat conduction: microscopic derivation. Phys. Rev. Lett. 100, 230602 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Sagawa, T., Hayakawa, H.: Geometrical expression of excess entropy production. Phys. Rev. E 84, 051110 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    Maes, C., Netocny, K.: A nonequilibrium extension of the Clausius heat theorem. http://arxiv.org/abs/1206.3423
  12. 12.
    Bertini, L., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Clausius inequality and optimality of quaisistatic transformations for nonequilibrium stationary states. Phys. Rev. Lett. 110, 020601 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    Mandal, D.: Nonequlibrium heat capacity. http://arxiv.org/abs/1311.7176v1
  14. 14.
    Spinney, R.E., Ford, I.J.: Nonequilibrium thermodynamics of stochastic systems with odd and even variables. Phys. Rev. Lett. 108, 170603 (2012)ADSCrossRefGoogle Scholar
  15. 15.
    Yukawa, S.: The second law of steady state thermodynamics for nonequilibrium quantum dynamics. http://arxiv.org/abs/cond-mat/0108421
  16. 16.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  17. 17.
    Hayashi, M.: Quantum Information: An Introduction. Springer-Verlag, Berlin (2006)Google Scholar
  18. 18.
    Sagawa, T.: Second law-like inequalities with quantum relative entropy: An introduction. http://arxiv.org/abs/1202.0983. In: Nakahara, M. (ed.) Lectures on quantum computing, thermodynamics and statistical physics, Kinki University Series on Quantum Computing, vol. 8, World Scientific, New Jersey (2012)
  19. 19.
    Horowitz, J.M., Parrondo, J.M.R.: Entropy production along nonequilibrium quantum jump trajectories. New J. Phys. 15, 085028 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley-Interscience, New York (2006)MATHGoogle Scholar
  21. 21.
    Lieb, E.H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math. 11(3), 267–288 (1973)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Lieb, E.H.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938 (1973)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lieb, E.H., Ruskai, M.B.: A fundamental property of quantum-mechanical entropy. Phys. Rev. Lett. 30(10), 434–436 (1973)ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    Headrick, M., Takayanagi, T.: Holographic proof of the strong subadditivity of entanglement entropy. Phys. Rev. D 76, 106013 (2007)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Nechita, I., Pellegrini, C.: Quantum trajectories in random environment: the statistical model for a heat bath. Confluentes Math. 1, 249–289 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Horowitz, J.M.: Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator. Phys. Rev. E 85, 031110 (2012)ADSCrossRefGoogle Scholar
  27. 27.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Claredon Press, Oxford (2009)Google Scholar
  28. 28.
    Chetrite, R., Mallick, K.: Quantum fluctuation relations for the Lindblad master equation. J. Stat. Phys. 148, 480–501 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Esposito, M., Van den Broeck, C.: Three faces of the second law. I. Master equation formulation. Phys. Rev. E 82, 011143 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    Van den Broeck, C., Esposito, M.: Three faces of the second law: II. Fokker-planck formulation. Phys. Rev. E 82, 011144 (2010)Google Scholar
  31. 31.
    Lindblad, G.: Expectations and entropy inequalities for finite quantum systems. Commun. Math. Phys. 39(2), 111–119 (1974)ADSCrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Lindblad, G.: Completely positive maps and entropy inequalitites. Commun. Math. Phys. 40(2), 147–151 (1975)ADSCrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Uhlmann, A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 54(1), 21–32 (1977)ADSCrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Nielsen, M.A., Petz, D.: A simple proof of the strong subadditivity inequality. Quantum Inf. Comput. 5, 507–513 (2005)MATHMathSciNetGoogle Scholar
  35. 35.
    Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23(1), 57–65 (1986)ADSCrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Petz, D.: Monotonicity of quantum relative entropy revisited. Rev. Math. Phys. 15, 79 (2003)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Brun, T.A.: A simple model of quantum trajectories. Am. J. Phys. 70(7), 719–737 (2002)ADSCrossRefGoogle Scholar
  38. 38.
    Jacobs, K., Steck, D.: A straightforward introduction to continuous quantum measurement. Contemp. Phys. 47, 279 (2006)ADSCrossRefGoogle Scholar
  39. 39.
    Wiseman, H.M.: Quantum trajectories and feedback. Ph.D. thesis, University of Queensland (1994).Google Scholar
  40. 40.
    Sakurai, J.: Modern Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  41. 41.
    Fagnola, F., Umanità, V.: Generators of detailed balance quantum markov semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(3), 335–363 (2007)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Harris, R.J., Schütz, G.M.: Fluctuation theorems for stochastic dynamics. J. Stat. Mech.: Theor. Exp. 07, P07020 (2007).Google Scholar
  43. 43.
    Sekimoto, K.: Stochastic Energetics. Lect. Notes Phys. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  44. 44.
    Liu, F.: Operator equality on entropy production in quantum Markovian master equations. http://arxiv.org/abs/1210.5798
  45. 45.
    Liu, F.: Equivalence of two Bochkov-Kuzovlev equalities in quantum two-level systems. http://arxiv.org/abs/1312.6570

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of Massachusetts at BostonBostonUSA
  2. 2.Department of Basic ScienceThe University of TokyoTokyoJapan

Personalised recommendations