Skip to main content
SpringerLink
Log in

Fluctuation Theorems for Quantum Maps

  • Chapter
  • First Online:
Book cover Thermodynamics and Synchronization in Open Quantum Systems

Part of the book series: Springer Theses ((Springer Theses))

  • 798 Accesses

Abstract

Part II of the thesis has been devoted to the study of dynamical properties of dissipative quantum many-body systems, where we employed open quantum system theory and its related quantum information tools in order to predict the emergence of synchronization phenomena and unveil the behavior of quantum correlations. In the remaining of the thesis we will turn our view to the nonequilibrium thermodynamic properties of open quantum systems, using the full methods introduced in Chaps. 1 and 2.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Most of the results in the chapter have been published in Ref. [15].

  2. 2.

    Notice however that we are changing the nomenclature with respect to Ref. [18], in which the map \(\tilde{\mathcal E}_{k}\) is called the dual or time-reversed map. The reasons for this change will be specified in the next chapter of this thesis.

  3. 3.

    F. Fagnola, private communication.

References

  1. M. Campisi, P. Talkner, P. Hänggi, Fluctuation theorems for continuously monitored quantum fluxes. Phys. Rev. Lett. 105, 140601 (2010)

    Google Scholar 

  2. M. Esposito, U. Harbola, S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665–1702 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  3. B.P. Venkatesh, G. Watanabe, P. Talkner, Transient quantum fluctuation theorems and generalized measurements. New J. Phys. 16, 015032 (2014)

    Article  MathSciNet  Google Scholar 

  4. G. Watanabe, B.P. Venkatesh, P. Talkner, Generalized energy measurements and modified transient quantum fluctuation theorems. Phys. Rev. E 89, 052116 (2014)

    Article  ADS  Google Scholar 

  5. P. Hänggi, P. Talkner, The other QFT. Nat. Phys. 11, 108–110 (2015)

    Article  ADS  Google Scholar 

  6. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)

    MATH  Google Scholar 

  7. K. Kraus, A. Böhm, J.D. Dollard, W.H. Wootters, States, Effects, and Operations: Fundamental Notions of Quantum Theory. Lecture Notes in Physics (Springer, Berlin, 1983)

    Google Scholar 

  8. H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, New York, 2002)

    MATH  Google Scholar 

  9. H.M. Wiseman, G.J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2010)

    MATH  Google Scholar 

  10. S. Yukawa, The Second Law of Steady State Thermodynamics for Nonequilibrium Quantum Dynamics (2001), arXiv:0108421v2

  11. T. Sagawa, Second law-like inequalitites with quantum relative entropy: an introduction, in Lectures on Quantum Computing, Thermodynamics and Statistical Physics, vol. 8, ed. by M. Nakahara. Kinki University Series on Quantum Computing (World Scientific, New Jersey, USA, 2013)

    Google Scholar 

  12. J. Anders, V. Giovannetti, Thermodynamics of discrete quantum processes. New J. Phys. 15, 033022 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  13. J.M. Horowitz, T. Sagawa, Equivalent definitions of the quantum nonadiabatic entropy production. J. Stat. Phys. 156, 55–65 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  14. F. Binder, S. Vinjanampathy, K. Modi, J. Goold, Quantum thermodynamics of general quantum processes. Phys. Rev. E 91, 032119 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  15. G. Manzano, J.M. Horowitz, J.M.R. Parrondo, Nonequilibrium potential and fluctuation theorems for quantum maps. Phys. Rev. E 92, 032129 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  16. M. Campisi, P. Talkner, P. Hünggi, Influence of measurements on the statistics of work performed on a quantum system. Phys. Rev. E 83, 041114 (2011)

    Article  ADS  Google Scholar 

  17. G. Watanabe, B.P. Venkatesh, P. Talkner, M. Campisi, P. Hünggi, Quantum fluctuation theorems and generalized measurements during the force protocol. Phys. Rev. E 89, 032114 (2014)

    Article  ADS  Google Scholar 

  18. G.E. Crooks, Quantum operation time reversal. Phys. Rev. A 77, 034101 (2008)

    Article  ADS  Google Scholar 

  19. J.M. Horowitz, J.M.R. Parrondo, Entropy production along nonequilibrium quantum jump trajectories. New. J. Phys 15, 085028 (2013)

    Article  ADS  Google Scholar 

  20. R. Chetritie, K. Mallick, Quantum fluctuation relations for the Lindblad master equation. J. Stat. Phys. 148, 480–501 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  21. F. Liu, Equivalence of two Bochkov-Kuzovlev equalities in quantum two-level systems. Phys. Rev. E 89, 042122 (2014)

    Article  ADS  Google Scholar 

  22. F. Liu, Calculating work in adiabatic two-level quantum Markovian master equations: a characteristic function method. Phys. Rev. E 90, 032121 (2014)

    Article  ADS  Google Scholar 

  23. T. Albash, D.A. Lidar, M. Marvian, P. Zanardi, Fluctuation theorems for quantum processes. Phys. Rev. E 88, 032146 (2013)

    Article  ADS  Google Scholar 

  24. A.E. Rastegin, Non-equilibirum equalities with unital quantum channels. J. Stat. Mech.: Theor. Exp. 59, P06016 (2013)

    Article  MathSciNet  Google Scholar 

  25. D. Kafri, S. Deffner, Holevo’s bound from a gernal quantum fluctuation theorem. Phys. Rev. A 86, 044302 (2012)

    Article  ADS  Google Scholar 

  26. A.E. Rastegin, K. Życzkowski, Jarzynski equality for quantum stochastic maps. Phys. Rev. E 89, 012127 (2014)

    Article  ADS  Google Scholar 

  27. J. Goold, M. Paternostro, K. Modi, Nonequilibrium quantum Landauer principle. Phys. Rev. Lett. 114, 060602 (2015)

    Article  ADS  Google Scholar 

  28. U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95, 040602 (2005)

    Article  ADS  Google Scholar 

  29. M.O. Scully, M.S. Zubairy, G.S. Agarwal, H. Walther, Extracting work from a single heat bath via vanishing quantum coherence. Science 299, 862–864 (2003)

    Article  ADS  Google Scholar 

  30. R. Dillenschneider, E. Lutz, Energetics of quantum correlations. Europhys. Lett. 88, 50003 (2009)

    Article  ADS  Google Scholar 

  31. J. Roßnagel, O. Abah, F. Schmidt-Kaler, K. Singer, E. Lutz, Nanoscale heat engine beyond the Carnot limit. Phys. Rev. Lett. 112, 030602 (2014)

    Article  ADS  Google Scholar 

  32. G. Manzano, F. Galve, R. Zambrini, J.M.R. Parrondo, Entropy production and thermodynamic power of the squeezed thermal reservoir. Phys. Rev. E 93, 052120 (2016)

    Article  ADS  Google Scholar 

  33. T. Hatano, S.-I. Sasa, Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463 (2001)

    Article  ADS  Google Scholar 

  34. H. Spohn, Entropy production for quantum dynamical semigroups. J. Math. Phys. 19, 1227–1230 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  35. J. Prost, J.F. Joanny, J.M.R. Parrondo, Generalized fluctuation- dissipation theorem for steady-state systems. Phys. Rev. Lett. 103, 090601 (2009)

    Article  ADS  Google Scholar 

  36. F. Haake, Quantum Signatures of Chaos, 3rd edn. Springer Series in Synergetics (Springer, Berlin, 2010)

    Book  Google Scholar 

  37. M. Campisi, P. Hünggi, P. Talkner, Colloquium: quantum fluctuation relations: foundations and applications. Rev. Mod. Phys. 83, 771–791 (2011)

    Article  ADS  Google Scholar 

  38. J. Anders, Thermal state entanglement in harmonic lattices. Phys. Rev. A 77, 062102 (2008)

    Article  ADS  Google Scholar 

  39. C. Maes, The fluctuation theorem as a Gibbs property. J. Stat. Phys. 95, 367–392 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  40. P.I. Hurtado, C. Perez-Espigares, J.J. del Pozo, P.L. Garrido, Symmetries in fluctuations far from equilibrium. Proc. Natl. Acad. Sci. 108, 7704–7709 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  41. D. Lacoste, P. Gaspard, Isometric fluctuation relations for equilibrium states with broken symmetry. Phys. Rev. Lett. 113, 240602 (2014)

    Article  ADS  Google Scholar 

  42. J.M. Horowitz, Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator. Phys. Rev. E 85, 031110 (2012)

    Article  ADS  Google Scholar 

  43. P. Faist, J. Oppenheim, R. Renner, Gibbs-preserving maps outperform thermal operations in the quantum regime. New J. Phys. 17, 043003 (2015)

    Article  ADS  Google Scholar 

  44. E.B. Davies, Quantum Theory of Open Systems (Academic Press, London, 1976)

    MATH  Google Scholar 

  45. R. Alicki, D.A. Lidar, P. Zanardi, Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum Markovian limit. Phys. Rev. A 73, 052311 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  46. K. Szczygielski, D. Gelbwaser-Klimovsky, R. Alicki, Markovian master equation and thermodynamics of a two-level system in a strong laser field. Phys. Rev. E 87, 012120 (2013)

    Article  ADS  Google Scholar 

  47. A. Rivas, S.F. Huelga, Open Quantum Systems: An Introduction (Springer, Berlin, 2012)

    Book  Google Scholar 

  48. R. Kawai, J.M.R. Parrondo, C. Van den Broeck, Dissipation: the phase-space perspective. Phys. Rev. Lett. 98, 080602 (2007)

    Article  ADS  Google Scholar 

  49. M. Esposito, C. Van den Broeck, Three detailed fluctuation theorems. Phys. Rev. Lett. 104, 090601 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  50. M. Esposito, C. Van den Broeck, Three faces of the second law. I. Master equation formulation. Phys. Rev. E 82, 011143 (2010)

    Article  ADS  Google Scholar 

  51. C. Van den Broeck, M. Esposito, Three faces of the second law. II. Fokker-Planck formulation. Phys. Rev. E 82, 011144 (2010)

    Article  Google Scholar 

  52. V.Y. Chernyak, M. Chertkov, C. Jarzynsk, Path-integral analysis of fluctuation theorems for general Langevin processes. J. Stat. Mech.: Theor. Exp. P08001 (2006)

    Google Scholar 

  53. T. Speck, U. Seifert, Integral fluctuation theorem for the housekeeping heat. J. Phys. A: Math. Gen. 38, L581–L588 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  54. T. Monnai, Unified treatment of the quantum fluctuation theorem and the Jarzynski equality in terms of microscopic reversibility. Phys. Rev. E 72, 027102 (2005)

    Article  ADS  Google Scholar 

  55. T.B. Batalhão, A.M. Souza, L. Mazzola, R. Auccaise, R.S. Sarthour, I.S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, R.M. Serra, Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system. Phys. Rev. Lett. 113, 140601 (2014)

    Article  ADS  Google Scholar 

  56. S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H.T. Quan, K. Kim, Experimental test of the quantum Jarzynski equality with a trapped-ion system. Nat. Phys. 11, 193–199 (2015)

    Article  Google Scholar 

  57. I. Bentsoon, K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (University Press, Cambridge, 2006)

    Book  Google Scholar 

  58. I. Callens, W. De Roeck, T. Jacobs, C. Maes, K. Netočný, Quantum entropy production as a measure of irreversibility. Phys. D 187, 383–391 (2004)

    Article  MathSciNet  Google Scholar 

  59. W. De Roeck, C. Maes, Quantum version of free-energy-irreversiblework relations. Phys. Rev. E 69, 026115 (2004)

    Article  ADS  Google Scholar 

  60. C. Jarzynski, Equalities and inequalities: irreversibility and the second law of thermodynamics at the nanoscale. Ann. Rev. Condens. Matter Phys. 2, 329–351 (2011)

    Article  ADS  Google Scholar 

  61. S. Deffner, E. Lutz, Nonequilibrium entropy production for open quantum systems. Phys. Rev. Lett. 107, 140404 (2011)

    Article  ADS  Google Scholar 

  62. R. Landauer, Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5, 183–191 (1961)

    Article  MathSciNet  Google Scholar 

  63. C.H. Bennett, The thermodynamics of computation-a review. Int. J. Theor. Phys. 21, 905–940 (1982)

    Article  Google Scholar 

  64. D. Reeb, M.M. Wolf, An improved Landauer principle with finite-size corrections. New J. Phys. 16, 103011 (2014)

    Article  ADS  Google Scholar 

  65. M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii, Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007)

    Article  ADS  Google Scholar 

  66. M.A. Cazalilla, Effect of suddenly turning on interactions in the Luttinger model. Phys. Rev. Lett. 97, 156403 (2006)

    Article  ADS  Google Scholar 

  67. M. Rigol, V. Dunjko, M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854–858 (2008)

    Article  ADS  Google Scholar 

  68. M. Cramer, C.M. Dawson, J. Eisert, T.J. Osborne, Exact relaxation in a class of nonequilibrium quantum lattice systems. Phys. Rev. Lett. 100, 030602 (2008)

    Article  ADS  Google Scholar 

  69. P. Calabrese, F.H.L. Essler, M. Fagotti, Quantum quench in the transverse-field ising chain. Phys. Rev. Lett. 106, 227203 (2011)

    Article  ADS  Google Scholar 

  70. A.C. Cassidy, C.W. Clark, M. Rigol, Generalized thermalization in an integrable lattice system. Phys. Rev. Lett. 106, 140405 (2011)

    Article  ADS  Google Scholar 

  71. J.-S. Caux, R.M. Konik, Constructing the generalized Gibbs ensemble after a quantum quench. Phys. Rev. Lett. 109, 175301 (2012)

    Article  ADS  Google Scholar 

  72. M. Fagotti, F.H.L. Essler, Reduced density matrix after a quantum quench. Phys. Rev. B 87, 245107 (2013)

    Article  ADS  Google Scholar 

  73. T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I.E. Mazets, T. Gasenzer, J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble. Science 348, 207–211 (2015)

    Article  MathSciNet  ADS  Google Scholar 

  74. R. Hamazaki, T.N. Ikeda, M. Ueda, Generalized Gibbs ensemble in a nonintegrable system with an extensive number of local symmetries. Phys. Rev. E 93, 032116 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  75. J.M. Hickey, S. Genway, Fluctuation theorems and the generalized Gibbs ensemble in integrable systems. Phys. Rev. E 90, 022107 (2014)

    Article  ADS  Google Scholar 

  76. N.Y. Halpern, J.M. Renes, Beyond heat baths: generalized resource theories for small-scale thermodynamics. Phys. Rev. E 93, 022126 (2016)

    Article  ADS  Google Scholar 

  77. Y. Guryanova, S. Popescu, A.J. Short, R. Silva, P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities. Nat. Commun. 7, 12049 (2016)

    Article  ADS  Google Scholar 

  78. E.T. Jaynes, Information theory and statistical mechanics. II. Phys. Rev. 108, 171–190 (1957)

    Article  MathSciNet  ADS  Google Scholar 

  79. J.A. Vaccaro, S.M. Barnett, Information erasure without an energy cost. Proc. Roy. Soc. Lond. A 467, 1770–1778 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  80. S.M. Barnett, J.A. Vaccaro, Beyond Landauer erasure. Entropy 15, 4956–4968 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  81. R. Alicki, The quantum open system as a model of a heat engine. J. Phys. A 12, L103 (1979)

    Article  ADS  Google Scholar 

  82. H. Spohn, J.L. Lebowitz, Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs, in Advances in Chemical Physics: For Ilya Prigogine, vol. 38, ed. by S.A. Rice (Wiley, Hoboken, USA, 1978)

    Google Scholar 

  83. F.W.J. Hekking, J.P. Pekola, Quantum jump approach for work and dissipation in a two-level system. Phys. Rev. Lett. 111, 093602 (2013)

    Article  ADS  Google Scholar 

  84. J. Dereziński, W. De Roeck, C. Maes, Fluctuations of quantum currents and unravelings of master equations. J. Stat. Phys. 131, 341–356 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  85. G.E. Crooks, On the Jarzynski relation for dissipative quantum dynamics, J. Stat. Mech.: Theor. Exp. 10, P10023 (2008)

    Article  MathSciNet  Google Scholar 

  86. T. Albash, S. Boixo, D.A. Lidar, P. Zanardi, Quantum adiabatic Markovian master equations. New J. Phys. 14, 123016 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  87. S. Suomela, J. Salmilehto, I.G. Savenko, T. Ala-Nissila, M. Möttönen, Fluctuations of work in nearly adiabatically driven open quantum systems. Phys. Rev. E 91, 022126 (2015)

    Article  ADS  Google Scholar 

  88. G.B. Cuetara, A. Engel, M. Esposito, Stochastic thermodynamics of rapidly driven systems. New J. Phys. 17, 055002 (2015)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Scuola Normale Superiore, Pisa, Italy

    Dr. Gonzalo Manzano Paule

  2. International Center for Theoretical Physics, Trieste, Italy

    Dr. Gonzalo Manzano Paule

Authors
  1. Dr. Gonzalo Manzano Paule
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gonzalo Manzano Paule .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Manzano Paule, G. (2018). Fluctuation Theorems for Quantum Maps. In: Thermodynamics and Synchronization in Open Quantum Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93964-3_7

Download citation

Publish with us

Policies and ethics

Search

Navigation