- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Quantum work: Reconciling quantum mechanics and thermodynamics

Thales A. B. Pinto Silva* and David Gelbwaser-Klimovsky

  • Schulich Faculty of Chemistry and Helen Diller Quantum Center, Technion–Israel Institute of Technology, Haifa 3200003, Israel

  • *pinto_silva@campus.technion.ac.il
PDF

Phys. Rev. Research 6, L022036 – Published 9 May, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L022036

Abstract

It has been claimed that no protocol for measuring quantum work can satisfy standard physical principles, casting doubts on the compatibility between quantum mechanics, thermodynamics, and the classical limit. In this Letter, we present a solution for this incompatibility. We demonstrate that the standard formulation of these principles fails to address the classical limit properly. By proposing changes in this direction, we prove that all the essential principles can be satisfied when work is defined as a quantum observable, reconciling quantum work statistics and thermodynamics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (126)

  1. H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (Wiley, New York, 1985).
  2. E. Fermi, Thermodynamics (Dover, New York, 1956).
  3. F. Reif, Fundamentals of Statistical and Thermal Physics (Waveland Press, Long Grove, IL, 2009).
  4. K. Sekimoto, Stochastic Energetics (Springer, Berlin, 2010).
  5. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  6. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  7. J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski's equality, Science 296, 1832 (2002).
  8. G. Huber, F. Schmidt-Kaler, S. Deffner, and E. Lutz, Employing trapped cold ions to verify the quantum Jarzynski equality, Phys. Rev. Lett. 101, 070403 (2008).
  9. F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource theory of quantum states out of thermal equilibrium, Phys. Rev. Lett. 111, 250404 (2013).
  10. J. Åberg, Fully quantum fluctuation theorems, Phys. Rev. X 8, 011019 (2018).
  11. A. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry, Fluctuating work: From quantum thermodynamical identities to a second law equality, Phys. Rev. X 6, 041017 (2016).
  12. C. Jarzynski, Nonequilibrium work theorem for a system strongly coupled to a thermal environment, J. Stat. Mech.: Theory Exp. (2004) P09005.
  13. M. Campisi, P. Talkner, and P. Hänggi, Fluctuation theorem for arbitrary open quantum systems, Phys. Rev. Lett. 102, 210401 (2009).
  14. K. Maruyama, F. Nori, and V. Vedral, Colloquium: The physics of Maxwell's demon and information, Rev. Mod. Phys. 81, 1 (2009).
  15. T. Sagawa and M. Ueda, Minimal energy cost for thermodynamic information processing: Measurement and information erasure, Phys. Rev. Lett. 102, 250602 (2009).
  16. M. Esposito and C. Van den Broeck, Second law and Landauer principle far from equilibrium, Europhys. Lett. 95, 40004 (2011).
  17. G. Manzano, F. Plastina, and R. Zambrini, Optimal work extraction and thermodynamics of quantum measurements and correlations, Phys. Rev. Lett. 121, 120602 (2018).
  18. S. Alipour, F. Benatti, F. Bakhshinezhad, M. Afsary, S. Marcantoni, and A. T. Rezakhani, Correlations in quantum thermodynamics: Heat, work, and entropy production, Sci. Rep. 6, 35568 (2016).
  19. J. J. Park, H. Nha, S. W. Kim, and V. Vedral, Information fluctuation theorem for an open quantum bipartite system, Phys. Rev. E 101, 052128 (2020).
  20. J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, The role of quantum information in thermodynamics—a topical review, J. Phys. A: Math. Theor. 49, 143001 (2016).
  21. M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).
  22. J. Roßnagel, S. T. Dawkins, K. N. Tolazzi, O. Abah, E. Lutz, F. Schmidt-Kaler, and K. Singer, A single-atom heat engine, Science 352, 325 (2016).
  23. G. N. Bochkov and Y. E. Kuzovlev, General theory of thermal fluctuations in nonlinear systems, Zh. Eksp. Teor. Fiz. 72, 238 (1977) [Sov. Phys. JETP 45, 125 (1977)].
  24. R. Alicki, The quantum open system as a model of the heat engine, J. Phys. A: Math. Gen. 12, L103 (1979).
  25. J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics (Springer, Berlin, 2005).
  26. P. Hänggi and P. Talkner, The other QFT, Nat. Phys. 11, 108 (2015).
  27. J. Millen and A. Xuereb, Perspective on quantum thermodynamics, New J. Phys. 18, 011002 (2016).
  28. P. Talkner and P. Hänggi, Aspects of quantum work, Phys. Rev. E 93, 022131 (2016).
  29. S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information (Morgan & Claypool, San Rafael, CA, 2019).
  30. F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso, Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Fundamental Theories of Physics Vol. 195 (Springer International Publishing, Cham, 2018).
  31. P. Strasberg and A. Winter, First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy, PRX Quantum 2, 030202 (2021).
  32. D. Gelbwaser-Klimovsky, R. Alicki, and G. Kurizki, Work and energy gain of heat-pumped quantized amplifiers, Europhys. Lett. 103, 60005 (2013).
  33. D. Gelbwaser-Klimovsky and G. Kurizki, Work extraction from heat-powered quantized optomechanical setups, Sci. Rep. 5, 7809 (2015).
  34. D. Gelbwaser-Klimovsky and G. Kurizki, Heat-machine control by quantum-state preparation: From quantum engines to refrigerators, Phys. Rev. E 90, 022102 (2014).
  35. R. Gallego, J. Eisert, and H. Wilming, Thermodynamic work from operational principles, New J. Phys. 18, 103017 (2016).
  36. R. Dann and R. Kosloff, Unification of the first law of quantum thermodynamics, New J. Phys. 25, 043019 (2023).
  37. M. Esposito, K. Lindenberg, and C. Van den Broeck, Entropy production as correlation between system and reservoir, New J. Phys. 12, 013013 (2010).
  38. F. Barra, The thermodynamic cost of driving quantum systems by their boundaries, Sci. Rep. 5, 14873 (2015).
  39. M. J. Kewming and S. Shrapnel, Entropy production and fluctuation theorems in a continuously monitored optical cavity at zero temperature, Quantum 6, 685 (2022).
  40. N. V. Gnezdilov, A. I. Pavlov, V. Ohanesjan, Y. Cheipesh, and K. Schalm, Ultrafast dynamics of cold Fermi gas after a local quench, Phys. Rev. A 107, L031301 (2023).
  41. D. Das, G. Thomas, and A. N. Jordan, Quantum Stirling heat engine operating in finite time, Phys. Rev. A 108, 012220 (2023).
  42. A. M. Lacerda, A. Purkayastha, M. Kewming, G. T. Landi, and J. Goold, Quantum thermodynamics with fast driving and strong coupling via the mesoscopic leads approach, Phys. Rev. B 107, 195117 (2023).
  43. V. Ohanesjan, Y. Cheipesh, N. V. Gnezdilov, A. I. Pavlov, and K. Schalm, Energy dynamics, information and heat flow in quenched cooling and the crossover from quantum to classical thermodynamics, J. High Energy Phys. 05 (2023) 237.
  44. D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett. 130, 210401 (2023).
  45. N. Lörch, C. Bruder, N. Brunner, and P. P. Hofer, Optimal work extraction from quantum states by photo-assisted Cooper pair tunneling, Quantum Sci. Technol. 3, 035014 (2018).
  46. A. Levy, M. Göb, B. Deng, K. Singer, E. Torrontegui, and D. Wang, Single-atom heat engine as a sensitive thermal probe, New J. Phys. 22, 093020 (2020).
  47. A. Auffèves, Quantum technologies need a quantum energy initiative, PRX Quantum 3, 020101 (2022).
  48. S. L. Jacob, G. T. Landi, M. Esposito, and F. Barra, Two-point measurement energy statistics from particle scattering, Phys. Rev. Res. 5, 043160 (2023).
  49. T. A. B. Pinto Silva, and R. M. Angelo, Quantum mechanical work, Phys. Rev. A 104, 042215 (2021).
  50. T. A. B. Pinto Silva, and R. M. Angelo, Fluctuation theorems for genuine quantum mechanical regimes, Phys. Rev. A 107, 052211 (2023).
  51. A. Sone, Y. X. Liu, and P. Cappellaro, Quantum Jarzynski equality in open quantum systems from the one-time measurement scheme, Phys. Rev. Lett. 125, 060602 (2020).
  52. M. Janovitch and G. T. Landi, Quantum mean-square predictors and thermodynamics, Phys. Rev. A 105, 022217 (2022).
  53. M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities, Phys. Rev. Lett. 120, 040602 (2018).
  54. C. Jarzynski, Comparison of far-from-equilibrium work relations, C. R. Phys. 8, 495 (2007).
  55. J. Horowitz and C. Jarzynski, Comparison of work fluctuation relations, J. Stat. Phys.: Theory Exp. (2007) P11002.
  56. J. M. G. Vilar and J. M. Rubi, Failure of the work-Hamiltonian connection for free-energy calculations, Phys. Rev. Lett. 100, 020601 (2008).
  57. J. M. G. Vilar and J. M. Rubi, Failure of the work-Hamiltonian connection for free-energy calculations, Phys. Rev. Lett. 101, 098902 (2008).
  58. J. M. G. Vilar and J. M. Rubi, Failure of the work-Hamiltonian connection for free-energy calculations, Phys. Rev. Lett. 101, 098904 (2008).
  59. J. Horowitz and C. Jarzynski, Failure of the work-Hamiltonian connection for free-energy calculations, Phys. Rev. Lett. 101, 098901 (2008).
  60. L. Peliti, Failure of the work-Hamiltonian connection for free-energy calculations, Phys. Rev. Lett. 101, 098903 (2008).
  61. H. J. D. Miller and J. Anders, Time-reversal symmetric work distributions for closed quantum dynamics in the histories framework, New J. Phys. 19, 062001 (2017).
  62. S. Deffner, J. P. Paz, and W. H. Zurek, Quantum work and the thermodynamic cost of quantum measurements, Phys. Rev. E 94, 010103(R) (2016).
  63. K. Beyer, K. Luoma, and W. T. Strunz, Work as an external quantum observable and an operational quantum work fluctuation theorem, Phys. Rev. Res. 2, 033508 (2020).
  64. K. Beyer, R. Uola, K. Luoma, and W. T. Strunz, Joint measurability in nonequilibrium quantum thermodynamics, Phys. Rev. E 106, L022101 (2022).
  65. A. E. Allahverdyan, Nonequilibrium quantum fluctuations of work, Phys. Rev. E 90, 032137 (2014).
  66. F. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantum thermodynamics of general quantum processes, Phys. Rev. E 91, 032119 (2015).
  67. P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102(R) (2007).
  68. M. Perarnau-Llobet, E. Bäumer, K. V. Hovhannisyan, M. Huber, and A. Acin, No-go theorem for the characterization of work fluctuations in coherent quantum systems, Phys. Rev. Lett. 118, 070601 (2017).
  69. E. Bäumer, M. Lostaglio, M. Perarnau-Llobet, and R. Sampaio, in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International, Cham, 2018), pp. 275–300.
  70. G. Francica, Class of quasiprobability distributions of work with initial quantum coherence, Phys. Rev. E 105, 014101 (2022).
  71. G. Francica, Most general class of quasiprobability distributions of work, Phys. Rev. E 106, 054129 (2022).
  72. A. J. Roncaglia, F. Cerisola, and J. P. Paz, Work measurement as a generalized quantum measurement, Phys. Rev. Lett. 113, 250601 (2014).
  73. F. Campaioli, F. A. Pollock, and S. Vinjanampathy, in Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, edited by F. Binder, L. A. Correa, C. Gogolin, J. Anders, and G. Adesso (Springer International, Cham, 2018), pp. 207–225.
  74. J.-Y. Gyhm, D. Šafránek, and D. Rosa, Quantum charging advantage cannot be extensive without global operations, Phys. Rev. Lett. 128, 140501 (2022).
  75. L. Bresque, P. A. Camati, S. Rogers, K. Murch, A. N. Jordan, and A. Auffèves, Two-qubit engine fueled by entanglement and local measurements, Phys. Rev. Lett. 126, 120605 (2021).
  76. F. L. S. Rodrigues, G. De Chiara, M. Paternostro, and G. T. Landi, Thermodynamics of weakly coherent collisional models, Phys. Rev. Lett. 123, 140601 (2019).
  77. M. Popovic, M. T. Mitchison, and J. Goold, Thermodynamics of decoherence, Proc. R. Soc. A 479, 20230040 (2023).
  78. C. Jarzynski, H. T. Quan, and S. Rahav, Quantum-classical correspondence principle for work distributions, Phys. Rev. X 5, 031038 (2015).
  79. C. Elouard, D. A. Herrera-Martí, M. Clusel, and A. Aufféves, The role of quantum measurement in stochastic thermodynamics, npj Quantum Inf. 3, 9 (2017).
  80. M. H. Mohammady and A. Romito, Conditional work statistics of quantum measurements, Quantum 3, 175 (2019).
  81. P. Faist, F. Dupuis, J. Oppenheim, and R. Renner, The minimal work cost of information processing, Nat. Commun. 6, 7669 (2015).
  82. G. M. Andolina, M. Keck, A. Mari, M. Campisi, V. Giovannetti, and M. Polini, Extractable work, the role of correlations, and asymptotic freedom in quantum batteries, Phys. Rev. Lett. 122, 047702 (2019).
  83. S. Deffner, Energetic cost of Hamiltonian quantum gates, Europhys. Lett. 134, 40002 (2021).
  84. C. H. Bennett, The thermodynamics of computation—a review, Int. J. Theor. Phys. 21, 905 (1982).
  85. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information Information (Cambridge University Press, New York, 2010).
  86. T. A. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory (Cambridge University Press, New York, 2012).
  87. A. E. Allahverdyan and Th. M. Nieuwenhuizen, Fluctuations of work from quantum subensembles: The case against quantum work-fluctuation theorems, Phys. Rev. E 71, 066102 (2005).
  88. M. F. Gelin and D. S. Kosov, Unified approach to the derivation of work theorems for equilibrium and steady-state, classical and quantum Hamiltonian systems, Phys. Rev. E 78, 011116 (2008).
  89. G. Lindblad, Non-equilibrium Entropy and Irreversibility (D. Reidel Publishing Company, Dordrecht, 1983).
  90. S. Deffner and E. Lutz, Nonequilibrium work distribution of a quantum harmonic oscillator, Phys. Rev. E 77, 021128 (2008).
  91. S. Deffner, O. Abah, and E. Lutz, Quantum work statistics of linear and nonlinear parametric oscillators, Chem. Phys. 375, 200 (2010).
  92. J. F. Chen, T. Qiu, and H. T. Quan, Quantum-classical correspondence principle for heat distribution in quantum Brownian motion, Entropy 23, 1602 (2021).
  93. L. Zhu, Z. Gong, B. Wu, and H. T. Quan, Quantum-classical correspondence principle for work distributions in a chaotic system, Phys. Rev. E 93, 062108 (2016).
  94. I. García-Mata, A. J. Roncaglia, and D. A. Wisniacki, Chaos signatures in the short and long time behavior of the out-of-time ordered correlator, Phys. Rev. E 95, 050102(R) (2017).
  95. A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935).
  96. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  97. W. H. Zurek, Quantum darwinism, Nat. Phys. 5, 181 (2009).
  98. A. L. O. Bilobran and R. M. Angelo, A measure of physical reality, Europhys. Lett. 112, 40005 (2015).
  99. P. R. Dieguez and R. M. Angelo, Information-reality complementarity: The role of measurements and quantum reference frames, Phys. Rev. A 97, 022107 (2018).
  100. P. R. Dieguez, J. R. Guimarães, J. P. S. Peterson, R. M. Angelo, and R. M. Serra, Experimental assessment of physical realism in a quantum-controlled device, Commun. Phys. 5, 82 (2022).
  101. A. Mari, K. Kieling, B. M. Nielsen, E. S. Polzik, and J. Eisert, Directly estimating nonclassicality, Phys. Rev. Lett. 106, 010403 (2011).
  102. H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Nonclassicality as a quantifiable resource for quantum metrology, Phys. Rev. Lett. 122, 040503 (2019).
  103. A. Kenfack and K. Życzkowski, Negativity of the Wigner function as an indicator of non-classicality, J. Opt. B: Quantum Semiclassical Opt. 6, 396 (2004).
  104. U. Chabaud, P.-E. Emeriau, and F. Grosshans, Witnessing Wigner negativity, Quantum 5, 471 (2021).
  105. H. C. F. Lemos, A. C. L. Almeida, B. Amaral, and A. C. Oliveira, Roughness as classicality indicator of a quantum state, Phys. Lett. A 382, 823 (2018).
  106. M. Campisi, P. Talkner, and P. Hänggi, Quantum Bochkov–Kuzovlev work fluctuation theorems, Phil. Trans. R. Soc. A 369, 291 (2011).
  107. R. Härtle, C. Schinabeck, M. Kulkarni, D. Gelbwaser-Klimovsky, M. Thoss, and U. Peskin, Cooling by heating in nonequilibrium nanosystems, Phys. Rev. B 98, 081404(R) (2018).
  108. D. Gelbwaser-Klimovsky, A. Bylinskii, D. Gangloff, R. Islam, A. Aspuru-Guzik, and V. Vule, Single-atom heat machines enabled by energy quantization, Phys. Rev. Lett. 120, 170601 (2018).
  109. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749 (1932).
  110. T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  111. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L022036, which includes Refs. [21, 67, 110, 112, 113, 114, 116, 120, 121, 122, 123, 124, 125, 126], for detailed proofs of results 1 and 2 and an in-depth analysis of the model described in Eqs. (4) and (5).
  112. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Volume I: Basic Concepts, Tools, and Applications, 2nd ed. (Wiley-VCH Verlag GmbH & Co, Weinheim, 2020).
  113. S. K. Suslov, An analog of the Cauchy-Hadamard formula for expansions in q-polynomials, in Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman, 2nd ed. (Springer Science & Business Media, New York, 2005).
  114. L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd ed. (McGraw-Hill, New York, 1979).
  115. E. M. Fortes, I. García-Mata, R. A. Jalabert, and D. A. Wisniacki, Gauging classical and quantum integrability through out-of-time-ordered correlators, Phys. Rev. E 100, 042201 (2019).
  116. W. P. Schleich Quantum Optics in Phase Space (Wiley-VCH Verlag GmbH & Co, Berlin, 2001).
  117. T. Upadhyaya, W. F. Braasch, Jr., G. T. Landi, and N. Y. Halpern, What happens to entropy production when conserved quantities fail to commute with each other, arXiv:2305.15480.
  118. R. Alicki, M. Šindelka, and D. Gelbwaser-Klimovsky, Violation of detailed balance in quantum open systems, Phys. Rev. Lett. 131, 040401 (2023).
  119. D. Gelbwaser-Klimovsky, N. Graham, M. Kardar, and M. Krüger, Near field propulsion forces from nonreciprocal media, Phys. Rev. Lett. 126, 170401 (2021).
  120. K. B. Athreya and S. N. Lahiri, Measure Theory and Probability Theory (Springer, New York, 2006).
  121. M. K. Transtrum and J. F. S. Van Huele, Commutation relations for functions of operators, J. Math. Phys. 46, 063510 (2005).
  122. B. E. Fristedt and L. F. Gray, A Modern Approach to Probability Theory (Springer Science & Business Media, New York, 2013).
  123. Y. R. Zhang, L. H. Shao, Y. Li, and H. Fan, Quantifying coherence in infinite-dimensional systems, Phys. Rev. A 93, 012334 (2016).
  124. J. Eisert, C. Simon, and M. B. Plenio, On the quantification of entanglement in infinite-dimensional quantum systems, J. Phys. A: Math. Gen. 35, 3911 (2002).
  125. A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  126. M. J. Lighthill, Introduction to Fourier Analysis and Generalised Functions (Cambridge University Press, London, 1959).

Outline

Information

Phys. Rev. Research 6, L022036– Published 9 May, 2024
  • Received 23 October 2023
  • Accepted 25 March 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation