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Optimal work extraction from finite-time closed quantum dynamics

Shoki Sugimoto1,2,*, Takahiro Sagawa1,3, and Ryusuke Hamazaki2,4

  • *Contact author: shoki.sugimoto@ap.t.u-tokyo.ac.jp
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Phys. Rev. Research 8, 023321 – Published 22 June, 2026

DOI: https://doi.org/10.1103/5cy2-3xjk

Abstract

Extracting useful work from quantum systems is a fundamental problem in quantum thermodynamics. In scenarios where rapid protocols are desired—whether due to practical constraints or deliberate design choices—a fundamental trade-off between power and efficiency is yet to be established. Here, we investigate the problem of finite-time optimal work extraction from closed quantum systems, subject to a constraint on the magnitude of the control Hamiltonian. We first reveal the trade-off relation between power and work under a general setup, showing that these fundamental performance metrics cannot be maximized simultaneously. We then identify a solvable class of finite-time optimal work-extraction problems. This class includes nontrivial many-body models such as the Heisenberg model and the SU(n)-Hubbard model. The key assumption is that the control Hamiltonian is optimized over a Lie algebra preserved by the uncontrolled dynamics. Within this class, the optimal work-extraction problem admits an exact reduction to a nonlinear self-consistent equation, circumventing extensive search over time-dependent control paths. The resulting optimal protocol turns out to be particularly simple: It suffices to use a time-independent control Hamiltonian in the interaction picture, determined by that equation. By exploiting the Lie-algebraic structure of the controllable terms, our approach is applicable to quantum many-body systems through efficient numerical computation. Our results highlight the necessity of rapid protocols to achieve the maximum power and provide an exact route to finite-time optimal work extraction in many-body quantum systems.

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References (74)

  1. A. Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J. Stat. Phys. 19, 575 (1978).
  2. W. Pusz and S. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58, 273 (1978).
  3. P. Skrzypczyk, R. Silva, and N. Brunner, Passivity, complete passivity, and virtual temperatures, Phys. Rev. E 91, 052133 (2015).
  4. Y. Mitsuhashi, K. Kaneko, and T. Sagawa, Characterizing symmetry-protected thermal equilibrium by work extraction, Phys. Rev. X 12, 021013 (2022).
  5. A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett. 67, 565 (2004).
  6. R. Alicki and M. Fannes, Entanglement boost for extractable work from ensembles of quantum batteries, Phys. Rev. E 87, 042123 (2013).
  7. M. Perarnau-Llobet, K. V. Hovhannisyan, M. Huber, P. Skrzypczyk, N. Brunner, and A. Acín, Extractable work from correlations, Phys. Rev. X 5, 041011 (2015).
  8. G. Francica, F. C. Binder, G. Guarnieri, M. T. Mitchison, J. Goold, and F. Plastina, Quantum coherence and ergotropy, Phys. Rev. Lett. 125, 180603 (2020).
  9. A. Touil, B. Çakmak, and S. Deffner, Ergotropy from quantum and classical correlations, J. Phys. A 55, 025301 (2022).
  10. A. Hokkyo and M. Ueda, Universal upper bound on ergotropy and no-go theorem by the eigenstate thermalization hypothesis, Phys. Rev. Lett. 134, 010406 (2025).
  11. K. G. Paulson, H. Terletska, and H. F. Fotso, Work extraction from a controlled quantum emitter, J. Phys. Photonics 7, 025023 (2025).
  12. C.-K. Hu, J. Qiu, P. J. P. Souza, J. Yuan, Y. Zhou, L. Zhang, J. Chu, X. Pan, L. Hu, J. Li, Y. Xu, Y. Zhong, S. Liu, F. Yan, D. Tan, R. Bachelard, C. J. Villas-Boas, A. C. Santos, and D. Yu, Optimal charging of a superconducting quantum battery, Quantum Sci. Technol. 7, 045018 (2022).
  13. J. Q. Quach, K. E. McGhee, L. Ganzer, D. M. Rouse, B. W. Lovett, E. M. Gauger, J. Keeling, G. Cerullo, D. G. Lidzey, and T. Virgili, Superabsorption in an organic microcavity: Toward a quantum battery, Sci. Adv. 8, eabk3160 (2022).
  14. J. Joshi and T. S. Mahesh, Experimental investigation of a quantum battery using star-topology NMR spin systems, Phys. Rev. A 106, 042601 (2022).
  15. G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys. 95, 025003 (2023).
  16. F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries, Rev. Mod. Phys. 96, 031001 (2024).
  17. A. Sone and S. Deffner, Quantum and classical ergotropy from relative entropies, Entropy 23, 1107 (2021).
  18. M. Campisi, Unified theory of classical and quantum ergotropy, Europhys. Lett. 154, 50002 (2026).
  19. L. P. García-Pintos, A. Hamma, and A. del Campo, Fluctuations in extractable work bound the charging power of quantum batteries, Phys. Rev. Lett. 125, 040601 (2020).
  20. B. Mohan and A. K. Pati, Quantum speed limits for observables, Phys. Rev. A 106, 042436 (2022).
  21. D. Shrimali, B. Panda, and A. K. Pati, Stronger speed limit for observables: Tighter bound for the capacity of entanglement, the modular Hamiltonian, and the charging of a quantum battery, Phys. Rev. A 110, 022425 (2024).
  22. R. Hamazaki, Speed limits to fluctuation dynamics, Commun. Phys. 7, 361 (2024).
  23. L. Mandelstam and I. Tamm, The uncertainty relation between energy and time in non-relativistic quantum mechanics, in Selected Papers, edited by B. M. Bolotovskii, V. Y. Frenkel, and R. Peierls (Springer, Berlin, 1991), pp. 115–123, originally published in J. Phys. (USSR) 9, 249 (1945).
  24. N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D 120, 188 (1998).
  25. S. Deffner and S. Campbell, Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control, J. Phys. A: Math. Theor. 50, 453001 (2017).
  26. Z. Gong and R. Hamazaki, Bounds in nonequilibrium quantum dynamics, Int. J. Mod. Phys. B 36, 2230007 (2022).
  27. J.-Y. Gyhm, D. Rosa, and D. Šafránek, Minimal time required to charge a quantum system, Phys. Rev. A 109, 022607 (2024).
  28. Y. M. Bershchanskii, An iterative method for solving optimal problems with phase constraints, USSR Comput. Math. Math. Phys. 10, 98 (1970).
  29. J. Somlói, V. A. Kazakov, and D. J. Tannor, Controlled dissociation of I2 via optical transitions between the X and B electronic states, Chem. Phys. 172, 85 (1993).
  30. S. E. Sklarz and D. J. Tannor, Loading a Bose-Einstein condensate onto an optical lattice: An application of optimal control theory to the nonlinear Schrödinger equation, Phys. Rev. A 66, 053619 (2002).
  31. S. G. Schirmer and P. D. Fouquieres, Efficient algorithms for optimal control of quantum dynamics: The Krotov method unencumbered, New J. Phys. 13, 073029 (2011).
  32. O. V. Morzhin and A. N. Pechen, Krotov method for optimal control of closed quantum systems, Russ. Math. Surv. 74, 851 (2019).
  33. N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: Design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson. 172, 296 (2005).
  34. J. Dominy and H. Rabitz, Exploring families of quantum controls for generating unitary transformations, J. Phys. A 41, 205305 (2008).
  35. T. Caneva, T. Calarco, and S. Montangero, Chopped random-basis quantum optimization, Phys. Rev. A 84, 022326 (2011).
  36. E. Zahedinejad, S. Schirmer, and B. C. Sanders, Evolutionary algorithms for hard quantum control, Phys. Rev. A 90, 032310 (2014).
  37. S. Machnes, E. Assémat, D. Tannor, and F. K. Wilhelm, Tunable, flexible, and efficient optimization of control pulses for practical qubits, Phys. Rev. Lett. 120, 150401 (2018).
  38. M. Bukov, A. G. R. Day, D. Sels, P. Weinberg, A. Polkovnikov, and P. Mehta, Reinforcement learning in different phases of quantum control, Phys. Rev. X 8, 031086 (2018).
  39. J. J. W. H. Sørensen, M. O. Aranburu, T. Heinzel, and J. F. Sherson, Quantum optimal control in a chopped basis: Applications in control of Bose-Einstein condensates, Phys. Rev. A 98, 022119 (2018).
  40. T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A. Pollock, Spin-chain model of a many-body quantum battery, Phys. Rev. A 97, 022106 (2018).
  41. H. Katsura and A. Tanaka, Nagaoka states in the SU(n) Hubbard model, Phys. Rev. A 87, 013617 (2013).
  42. S. Deffner, Optimal control of a qubit in an optical cavity, J. Phys. B 47, 145502 (2014).
  43. G. Dirr and U. Helmke, Lie theory for quantum control, GAMM Mitt. 31, 59 (2008).
  44. D. D’Alessandro, Introduction to Quantum Control and Dynamics (Chapman and Hall/CRC, Boca Raton, 2021).
  45. N. Andrei, Modern Numerical Nonlinear Optimization, Springer Optimization and Its Applications, 2022nd ed. (Springer International Publishing, Cham, Switzerland, 2022).
  46. A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Time-optimal quantum evolution, Phys. Rev. Lett. 96, 060503 (2006).
  47. D. Allan, N. Hörnedal, and O. Andersson, Time-optimal quantum transformations with bounded bandwidth, Quantum 5, 462 (2021).
  48. F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, Quantacell: Powerful charging of quantum batteries, New J. Phys. 17, 075015 (2015).
  49. N. Shiraishi, K. Saito, and H. Tasaki, Universal trade-off relation between power and efficiency for heat engines, Phys. Rev. Lett. 117, 190601 (2016).
  50. J. M. Lee, Introduction to Riemannian Manifolds, Graduate Texts in Mathematics, 2nd ed. (Springer International Publishing, Cham, Switzerland, 2019).
  51. J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Graduate Texts in Mathematics, 1997th ed. (Springer, New York, NY, 1997).
  52. K. W. Moore Tibbetts, C. Brif, M. D. Grace, A. Donovan, D. L. Hocker, T.-S. Ho, R.-B. Wu, and H. Rabitz, Exploring the tradeoff between fidelity and time optimal control of quantum unitary transformations, Phys. Rev. A 86, 062309 (2012).
  53. M. Bukov, A. G. R. Day, P. Weinberg, A. Polkovnikov, P. Mehta, and D. Sels, Broken symmetry in a two-qubit quantum control landscape, Phys. Rev. A 97, 052114 (2018).
  54. L. T. Brady, C. L. Baldwin, A. Bapat, Y. Kharkov, and A. V. Gorshkov, Optimal protocols in quantum annealing and quantum approximate optimization algorithm problems, Phys. Rev. Lett. 126, 070505 (2021).
  55. E. Dionis and D. Sugny, Time-optimal control of two-level quantum systems by piecewise constant pulses, Phys. Rev. A 107, 032613 (2023).
  56. F. Mazzoncini, V. Cavina, G. M. Andolina, P. A. Erdman, and V. Giovannetti, Optimal control methods for quantum batteries, Phys. Rev. A 107, 032218 (2023).
  57. X. Li, Optimal control of quantum state preparation and entanglement creation in two-qubit quantum system with bounded amplitude, Sci. Rep. 13, 14734 (2023).
  58. P. Chen, D.-W. Luo, and T. Yu, Optimal entanglement generation in optomechanical systems via Krotov control of covariance matrix dynamics, Phys. Rev. Res. 7, 013161 (2025).
  59. P. Milgrom and I. Segal, Envelope theorems for arbitrary choice sets, Econometrica 70, 583 (2002).
  60. S. Taie, E. Ibarra-García-Padilla, N. Nishizawa, Y. Takasu, Y. Kuno, H.-T. Wei, R. T. Scalettar, K. R. A. Hazzard, and Y. Takahashi, Observation of antiferromagnetic correlations in an ultracold SU(N) Hubbard model, Nat. Phys. 18, 1356 (2022).
  61. A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics (Birkhäuser, Cambridge, MA, 1996).
  62. M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, 2007th ed. (Springer, New York, NY, 2006).
  63. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics Vol. 9 (Springer, New York, NY, 1972).
  64. W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, 1st ed. (Springer, New York, NY, 2022).
  65. D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M. Polini, High-power collective charging of a solid-state quantum battery, Phys. Rev. Lett. 120, 117702 (2018).
  66. R. R. Rodríguez, B. Ahmadi, G. Suárez, P. Mazurek, S. Barzanjeh, and P. Horodecki, Optimal quantum control of charging quantum batteries, New J. Phys. 26, 043004 (2024).
  67. L. P. García-Pintos, S. B. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, Unifying quantum and classical speed limits on observables, Phys. Rev. X 12, 011038 (2022).
  68. B. Yadin, S. Imai, and O. Gühne, Quantum speed limit for states and observables of perturbed open systems, Phys. Rev. Lett. 132, 230404 (2024).
  69. R. Hamazaki, Speed limits for macroscopic transitions, PRX Quantum 3, 020319 (2022).
  70. R. Hamazaki, Quantum velocity limits for multiple observables: Conservation laws, correlations, and macroscopic systems, Phys. Rev. Res. 6, 013018 (2024).
  71. C. Villani, Optimal Transport, Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, Germany, 2015).
  72. N. Beato, P. Patil, and M. Bukov, Topological phase transitions in a constrained two-qubit quantum control landscape, Phys. Rev. Lett. 135, 110803 (2025).
  73. N. Beato, P. Patil, and M. Bukov, Towards a theory of phase transitions in quantum control landscapes, Phys. Rev. X 15, 041014 (2025).
  74. B. C. Hall, Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics, 2nd ed. (Springer International Publishing, Cham, Switzerland, 2015).

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Phys. Rev. Research 8, 023321– Published 22 June, 2026
  • Received 14 October 2025
  • Accepted 26 May 2026
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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.

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