The Coherent Crooks Equality

  • Zoe HolmesEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 195)


This chapter reviews an information theoretic approach (Part V) to deriving quantum fluctuation theorems (see Chap.  10) that was developed in [37, 38]. When a thermal system is driven from equilibrium, random quantities of work are required or produced: the Crooks equality is a classical fluctuation theorem that quantifies the probabilities of these work fluctuations. The framework summarised here generalises the Crooks equality to the quantum regime by modeling not only the driven system but also the control system and energy supply that enables the system to be driven. As is reasonably common within the information theoretic approach but high unusual for fluctuation theorems, this framework explicitly accounts for the energy conservation using only time independent Hamiltonians. We focus on explicating a key result of [37]: a Crooks-like equality for when the energy supply is allowed to exist in a superposition of energy eigenstates states.



The author thanks Johan Åberg, Álvaro Alhambra, Janet Anders, Florian Mintert, Erick Hinds Mingo, Tom Hebdige and Jake Lishman for commenting on drafts and David Jennings for numerous indispensable discussions. The author is supported by the Engineering and Physical Sciences Research Council Centre for Doctoral Training in Controlled Quantum Dynamics.


  1. 1.
    M. Esposito, U. Harbola, S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81(4), 1665–1702 (2009).
  2. 2.
    M. Campisi, P. Hänggi, P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys. 83(3), 771–791 (2011).
  3. 3.
    P. Hänggi, P. Talkner, The other QFT. Nat. Phys. 11(2), 108–110 (2015).
  4. 4.
    S. Vinjanampathy, J. Anders, Quantum thermodynamics. Contemp. Phys. 57(4), 545–579 (2016).
  5. 5.
    G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60(3), 2721–2726 (1999).
  6. 6.
    C. Jarzynski, How does a system respond when driven away from thermal equilibrium? Proc. Natl. Acad. Sci. 98(7), 3636–3638 (2001).
  7. 7.
    H. Tasaki, Jarzynski relations for quantum systems and some applications (2000). arXiv:cond-mat/0009244
  8. 8.
    J. Kurchan, A quantum fluctuation theorem. (2000). arXiv:cond-mat/0007360
  9. 9.
    P. Talkner, P. Hänggi, The Tasaki Crooks quantum fluctuation theorem. J. Phys. A: Math. Theor. 40(26), F569 (2007).
  10. 10.
    M. Campisi, P. Talkner, P. Hänggi, Fluctuation theorem for arbitrary open quantum systems. Phys. Rev. Lett. 102(21), 210401 (2009).
  11. 11.
    C.J. Jarzynski, Nonequilibrium work theorem for a system strongly coupled to a thermal environment. Stat. Mech. 2004(9), P09005 (2004).
  12. 12.
    G. Manzano, J.M. Horowitz, J.M.R. Parrondo, Nonequilibrium potential and fluctuation theorems for quantum maps. Phys. Rev. E 92(3), 032129 (2015).
  13. 13.
    T. Albash, D.A. Lidar, M. Marvian, P. Zanardi, Fluctuation theorems for quantum processes. Phys. Rev. E 88(3), 032146 (2013). Scholar
  14. 14.
    A.E. Rastegin, Non-equilibrium equalities with unital quantum channels. J. Stat. Mech. 2013(06), P06016 (2013).
  15. 15.
    P. Talkner, P. Hänggi, Aspects of quantum work. Phys. Rev. E 93(2), 022131 (2016). Scholar
  16. 16.
    M. Perarnau-Llobet, E. Bäumer, K.V. Hovhannisyan, M. Huber, A. Acin, No-Go Tteorem for the characterization of work fluctuations in coherent quantum systems. Phys. Rev. Lett. 118(7), 070601 (2017).
  17. 17.
    M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities. Phys. Rev. Lett. 120(4), 040602 (2018).
  18. 18.
    A. E. Allahverdyan, Nonequilibrium quantum fluctuations of work. Phys. Rev. E 90(3), 032137 (2014).
  19. 19.
    P. Solinas, H.J.D. Miller, J. Anders, Measurement-dependent corrections to work distributions arising from quantum coherences. Phys. Rev. A 96(5), 052115 (2017). Scholar
  20. 20.
    H.J.D. Miller, J. Anders, Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework. New J. Phys. 19(6), 062001 (2017).
  21. 21.
    J.M. Horowitz, Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator. Phys. Rev. E 85(3), 031110 (2012).
  22. 22.
    C. Elouard, D.A. Herrera-Martí, M. Clusel, A. Auffeves, The role of quantum measurement in stochastic thermodynamics. NPJ Quantum Inf. 3(1), 9 (2017).
  23. 23.
    F.G.S.L. Brandao, M.B. Plenio, Entanglement theory and the second law of thermodynamics. Nat. Phys. 4(11), 873 (2008).
  24. 24.
    L. Del Rio, J. Åberg, R. Renner, O. Dahlsten, V. Vedral, The thermodynamic meaning of negative entropy. Nature 474(7349), 61 (2011).
  25. 25.
    D. Jennings, T. Rudolph, Entanglement and the thermodynamic arrow of time. Phys. Rev. E 81(6), 061130 (2010).
  26. 26.
    S. Popescu, A.J. Short, A. Winter, Entanglement and the foundations of statistical mechanics. Nat. Phys. 2(11), 754–758 (2006).
  27. 27.
    J. Åberg, Truly work-like work extraction via a single-shot analysis. Nat. Commun. 4, 1925 (2013). Scholar
  28. 28.
    J. Åberg, Catalytic coherence. Phys. Rev. Lett. 113(15), 150402 (2014).
  29. 29.
    K. Korzekwa, M. Lostaglio, J. Oppenheim, D. Jennings, The extraction of work from quantum coherence. New J. Phys. 18(2), 023045 (2016).
  30. 30.
    F.G.S.L. Brandao, M. Horodecki, N. Ng, J. Oppenheim, S. Wehner, The second laws of quantum thermodynamics. PNAS 112(11), 3275–3279 (2015).
  31. 31.
    M. Lostaglio, D. Jennings, T. Rudolph, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun. 6, 6383 (2015). Scholar
  32. 32.
    M. Horodecki, J. Oppenheim, Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 2059 (2013),
  33. 33.
    G. Gour, D. Jennings, F. Buscemi, R. Duan, I. Marvian, Quantum majorization and a complete set of entropic conditions for quantum thermodynamics (2017).
  34. 34.
    D. Janzing, P. Wocjan, R. Zeier, R. Geiss, T. Beth, Thermodynamic cost of reliability and low temperatures: Tightening Landauer’s principle and the Second law. Int. J. Theor. Phys. 39(12), 2717–2753 (2000).
  35. 35.
    F.G.S.L. Brandao, M. Horodecki, J. Oppenheim, J.M. Renes, R.W. Spekkens, Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett. 111(25), 250404 (2013).
  36. 36.
    M. Horodecki, Fundamental limitations for quantum and nanoscale thermodynamics. J. Oppenheim, Nat. Commun. 4, 2059 (2013).
  37. 37.
    J. Åberg, Fully quantum fluctuation theorems. Phys. Rev. X 8, 011019 (2018).
  38. 38.
    Á.M. Alhambra, L. Masanes, Fluctuating work: From quantum thermodynamical identities to a second law equality. J. Oppenheim, C. Perry, Phys. Rev. X 6(4), 041017 (2016).
  39. 39.
    L.E. Ballentine, Quantum Mechanics: A Modern Development (World scientific, 1998).
  40. 40.
    C. Gerry, P. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005)Google Scholar
  41. 41.
    Z. Holmes, S. Weidt, D. Jennings, J. Anders, F. Mintert, Coherent fluctuation relations: from the abstract to the concrete (2018), arXiv:1806.11256
  42. 42.
    M. Frenzel, D. Jennings, T. Rudolph, Quasi-autonomous quantum thermal machines and quantum to classical energy flow. New J. Phys. 18(2), 023037 (2016).
  43. 43.
    M.P. Woods, R. Silva, Autonomous quantum machines and finite sized clocks. J. Oppenheim (2016).
  44. 44.
    A.S.L. Malabarba, A.J. Short, P. Kammerlander, Clock-driven quantum thermal engines. New J. Phys. 17(4), 045027 (2015).
  45. 45.
    E. Hinds Mingo, D. Jennings, Superpositions of mechanical processes, decomposable coherence and fluctuation relations (2018). arXiv:1812.08159
  46. 46.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81(2), 865–942 (2009).
  47. 47.
    A. Streltsov, G. Adesso, M.B. Plenio, Colloquium: Quantum coherence as a resource. Rev. Mod. Phys. 89(4), 041003 (2017).
  48. 48.
    G. Gour, M.P. Müller, V. Narasimhachar, R.W. Spekkens, N.Y. Halpern, The resource theory of informational nonequilibrium in thermodynamics. Phys. Rep. 583, 1–58 (2015).
  49. 49.
    D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105(1), 123–131 (1986).
  50. 50.
    S. D. Bartlett, T. Rudolph, R. W. Spekkens, Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79(2), 555–609 (2007).

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of PhysicsImperial College LondonLondonUK

Personalised recommendations