- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

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
  • Open Access

Complete Characterization of State Conversions by Work Extraction

Chung-Yun Hsieh1,* and Manuel Gessner2,†

  • *Contact author: chung-yun.hsieh@bristol.ac.uk
  • †Contact author: manuel.gessner@uv.es
PDF

Phys. Rev. Lett. 137, 070403 – Published 12 August, 2026

DOI: https://doi.org/10.1103/4f1s-5q8b

Abstract

We introduce a thermodynamic work-extraction task that describes the energy storage enhancement of quantum systems. This task induces majorizationlike conditions that provide a necessary and sufficient characterization of state conversions in general quantum resource theories. When applied to specific resources, these conditions reduce to the majorization conditions under unital channels and provide a thermodynamic version of Nielsen’s theorem in entanglement theory. We show how this result establishes the first universal resource certification class based on thermodynamics, and how it can be employed to quantify general quantum resources based on work extraction.

View figure in article

Physics Subject Headings (PhySH)

See Also

General quantum resources providing advantages in work-extraction tasks

Chung-Yun Hsieh and Manuel Gessner
Phys. Rev. A 114, 022456 (2026)

Article Text

References (111)

  1. K. Kuroiwa, R. Takagi, G. Adesso, and H. Yamasaki, Every quantum helps: Operational advantage of quantum resources beyond convexity, Phys. Rev. Lett. 132, 150201 (2024).
  2. K. Kuroiwa, R. Takagi, G. Adesso, and H. Yamasaki, Robustness- and weight-based resource measures without convexity restriction: Multicopy witness and operational advantage in static and dynamical quantum resource theories, Phys. Rev. A 109, 042403 (2024).
  3. F. Meier and H. Yamasaki, Energy-consumption advantage of quantum computation, PRX Energy 4, 023008 (2025).
  4. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
  5. C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett. 69, 2881 (1992).
  6. L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  7. Z. Ren, W. Li, A. Smerzi, and M. Gessner, Metrological detection of multipartite entanglement from young diagrams, Phys. Rev. Lett. 126, 080502 (2021).
  8. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  9. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  10. A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-independent security of quantum cryptography against collective attacks, Phys. Rev. Lett. 98, 230501 (2007).
  11. R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, Quantum steering, Rev. Mod. Phys. 92, 015001 (2020).
  12. D. Cavalcanti and P. Skrzypczyk, Quantum steering: A review with focus on semidefinite programming, Rep. Prog. Phys. 80, 024001 (2016).
  13. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301(R) (2012).
  14. O. Gühne, E. Haapasalo, T. Kraft, J.-P. Pellonpää, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys. 95, 011003 (2023).
  15. C.-Y. Hsieh, R. Uola, and P. Skrzypczyk, Quantum complementarity: A novel resource for unambiguous exclusion and encryption, arXiv:2309.11968.
  16. D. Rosset, F. Buscemi, and Y.-C. Liang, Resource theory of quantum memories and their faithful verification with minimal assumptions, Phys. Rev. X 8, 021033 (2018).
  17. X. Yuan, Y. Liu, Q. Zhao, B. Regula, J. Thompson, and M. Gu, Universal and operational benchmarking of quantum memories, npj Quantum Inf. 7, 108 (2021).
  18. H.-Y. Ku, J. Kadlec, A. Černoch, M. T. Quintino, W. Zhou, K. Lemr, N. Lambert, A. Miranowicz, S.-L. Chen, F. Nori, and Y.-N. Chen, Quantifying quantumness of channels without entanglement, PRX Quantum 3, 020338 (2022).
  19. L. B. Vieira, H.-Y. Ku, and C. Budroni, Entanglement-breaking channels are a quantum memory resource, Phys. Rev. Res. 7, 043281 (2025).
  20. P. Abiuso, Verification of continuous-variable quantum memories, Quantum Sci. Technol. 9, 01LT02 (2023).
  21. V. Narasimhachar, J. Thompson, J. Ma, G. Gour, and M. Gu, Quantifying memory capacity as a quantum thermodynamic resource, Phys. Rev. Lett. 122, 060601 (2019).
  22. C.-Y. Hsieh, B. Stratton, C.-H. Wu, and H.-Y. Ku, Dynamical resource theory of incompatibility preservability, Phys. Rev. A 111, 022422 (2025).
  23. R. Takagi, K. Wang, and M. Hayashi, Application of the resource theory of channels to communication scenarios, Phys. Rev. Lett. 124, 120502 (2020).
  24. C.-Y. Hsieh, Communication, dynamical resource theory, and thermodynamics, PRX Quantum 2, 020318 (2021).
  25. C.-Y. Hsieh, Dynamical Landauer principle: Quantifying information transmission by thermodynamics, Phys. Rev. Lett. 134, 050404 (2025).
  26. C.-Y. Hsieh, Dynamical Landauer principle: Thermodynamic criteria of transmitting classical information, Phys. Rev. A 111, 022207 (2025).
  27. C.-Y. Hsieh, Resource preservability, Quantum 4, 244 (2020).
  28. Z.-W. Liu and A. Winter, Resource theories of quantum channels and the universal role of resource erasure, arXiv:1904.04201.
  29. Y. Liu and X. Yuan, Operational resource theory of quantum channels, Phys. Rev. Res. 2, 012035 (2020).
  30. C.-Y. Hsieh, M. Lostaglio, and A. Acín, Entanglement preserving local thermalization, Phys. Rev. Res. 2, 013379 (2020).
  31. A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Measuring quantum coherence with entanglement, Phys. Rev. Lett. 115, 020403 (2015).
  32. B. Stratton, C.-Y. Hsieh, and P. Skrzypczyk, Dynamical resource theory of informational nonequilibrium preservability, Phys. Rev. Lett. 132, 110202 (2024).
  33. M. Lostaglio, An introductory review of the resource theory approach to thermodynamics, Rep. Prog. Phys. 82, 114001 (2019).
  34. P. Faist, J. Oppenheim, and R. Renner, Gibbs-preserving maps outperform thermal operations in the quantum regime, New J. Phys. 17, 043003 (2015).
  35. M. Horodecki, K. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen(De), and U. Sen, Local information as a resource in distributed quantum systems, Phys. Rev. Lett. 90, 100402 (2003).
  36. M. Horodecki, P. Horodecki, and J. Oppenheim, Reversible transformations from pure to mixed states and the unique measure of information, Phys. Rev. A 67, 062104 (2003).
  37. E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019).
  38. M. Piani and J. Watrous, All entangled states are useful for channel discrimination, Phys. Rev. Lett. 102, 250501 (2009).
  39. R. Takagi, B. Regula, K. Bu, Z.-W. Liu, and G. Adesso, Operational advantage of quantum resources in subchannel discrimination, Phys. Rev. Lett. 122, 140402 (2019).
  40. P. Skrzypczyk and N. Linden, Robustness of measurement, discrimination games, and accessible information, Phys. Rev. Lett. 122, 140403 (2019).
  41. J. Bae, D. Chruściński, and M. Piani, More entanglement implies higher performance in channel discrimination tasks, Phys. Rev. Lett. 122, 140404 (2019).
  42. R. Takagi and B. Regula, General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks, Phys. Rev. X 9, 031053 (2019).
  43. P. Skrzypczyk, I. Šupić, and D. Cavalcanti, All sets of incompatible measurements give an advantage in quantum state discrimination, Phys. Rev. Lett. 122, 130403 (2019).
  44. C.-Y. Hsieh, G. N. M. Tabia, Y.-C. Yin, and Y.-C. Liang, Resource marginal problems, Quantum 8, 1353 (2024).
  45. C.-Y. Hsieh, M. Lostaglio, and A. Acín, Quantum channel marginal problem, Phys. Rev. Res. 4, 013249 (2022).
  46. A. F. Ducuara and P. Skrzypczyk, Operational interpretation of weight-based resource quantifiers in convex quantum resource theories, Phys. Rev. Lett. 125, 110401 (2020).
  47. R. Uola, T. Bullock, T. Kraft, J.-P. Pellonpää, and N. Brunner, All quantum resources provide an advantage in exclusion tasks, Phys. Rev. Lett. 125, 110402 (2020).
  48. K. C. Tan, V. Narasimhachar, and B. Regula, Fisher information universally identifies quantum resources, Phys. Rev. Lett. 127, 200402 (2021).
  49. R. Uola, T. Kraft, and A. A. Abbott, Quantification of quantum dynamics with input-output games, Phys. Rev. A 101, 052306 (2020).

  50. See also Ref. [51] for spacelike separated resources.

  51. D. Rosset, D. Schmid, and F. Buscemi, Type-independent characterization of spacelike separated resources, Phys. Rev. Lett. 125, 210402 (2020).
  52. R. Kosloff and T. Feldmann, Discrete four-stroke quantum heat engine exploring the origin of friction, Phys. Rev. E 65, 055102 (2002).
  53. T. Feldmann and R. Kosloff, Quantum lubrication: Suppression of friction in a first-principles four-stroke heat engine, Phys. Rev. E 73, 025107 (2006).
  54. W. Ji, Z. Chai, M. Wang, Y. Guo, X. Rong, F. Shi, C. Ren, Y. Wang, and J. Du, Spin quantum heat engine quantified by quantum steering, Phys. Rev. Lett. 128, 090602 (2022).
  55. K. Beyer, K. Luoma, and W. T. Strunz, Steering heat engines: A truly quantum Maxwell demon, Phys. Rev. Lett. 123, 250606 (2019).
  56. F.-J. Chan, Y.-T. Huang, J.-D. Lin, H.-Y. Ku, J.-S. Chen, H.-B. Chen, and Y.-N. Chen, Maxwell’s two-demon engine under pure dephasing noise, Phys. Rev. A 106, 052201 (2022).
  57. T. Biswas, C. Datta, and L. P. García-Pintos, Quantum thermodynamic advantage in work extraction from steerable quantum correlations, Phys. Rev. Lett. 135, 110402 (2025).
  58. D. Jennings and T. Rudolph, Entanglement and the thermodynamic arrow of time, Phys. Rev. E 81, 061130 (2010).
  59. M. Lostaglio, D. Jennings, and T. Rudolph, Thermodynamic resource theories, non-commutativity and maximum entropy principles, New J. Phys. 19, 043008 (2017).
  60. S. Majidy, W. F. Braasch, A. Lasek, T. Upadhyaya, A. Kalev, and N. Yunger Halpern, Noncommuting conserved charges in quantum thermodynamics and beyond, Nat. Rev. Phys. 5, 689 (2023).
  61. N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges, Nat. Commun. 7, 12051 (2016).
  62. Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Thermodynamics of quantum systems with multiple conserved quantities, Nat. Commun. 7, 12049 (2016).
  63. M. Lostaglio, Certifying quantum signatures in thermodynamics and metrology via contextuality of quantum linear response, Phys. Rev. Lett. 125, 230603 (2020).
  64. A. Levy and M. Lostaglio, Quasiprobability distribution for heat fluctuations in the quantum regime, PRX Quantum 1, 010309 (2020).
  65. S. Puliyil, M. Banik, and M. Alimuddin, Thermodynamic signatures of genuinely multipartite entanglement, Phys. Rev. Lett. 129, 070601 (2022).
  66. T. Upadhyaya, W. F. Braasch, G. T. Landi, and N. Y. Halpern, Non-Abelian transport distinguishes three usually equivalent notions of entropy production, PRX Quantum 5, 030355 (2024).
  67. F. Centrone and M. Gessner, Breaking local quantum speed limits with steering, Phys. Rev. Res. 6, L042067 (2024).
  68. P. Lipka-Bartosik, G. F. Diotallevi, and P. Bakhshinezhad, Fundamental limits on anomalous energy flows in correlated quantum systems, Phys. Rev. Lett. 132, 140402 (2024).
  69. 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).
  70. L. d. Rio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, The thermodynamic meaning of negative entropy, Nature (London) 474, 61 (2011).
  71. P. Skrzypczyk, A. J. Short, and S. Popescu, Work extraction and thermodynamics for individual quantum systems, Nat. Commun. 5, 4185 (2014).
  72. N. Shiraishi and R. Takagi, Arbitrary amplification of quantum coherence in asymptotic and catalytic transformation, Phys. Rev. Lett. 132, 180202 (2024).
  73. K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, The extraction of work from quantum coherence, New J. Phys. 18, 023045 (2016).
  74. C.-Y. Hsieh and S.-L. Chen, Thermodynamic approach to quantifying incompatible instruments, Phys. Rev. Lett. 133, 170401 (2024).
  75. A. de Oliveira Junior, J. B. Brask, and P. Lipka-Bartosik, Heat as a witness of quantum properties, Phys. Rev. Lett. 134, 050401 (2025).
  76. R. A. Macêdo, A. de Oliveira Junior, N. E. Comar, L. L. Keller, J. B. Brask, L. C. Céleri, and R. Chaves, Every little thing heat does is magic, arXiv:2604.08663.
  77. F. Buscemi, All entangled quantum states are nonlocal, Phys. Rev. Lett. 108, 200401 (2012).
  78. G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics, Phys. Rep. 583, 1 (2015).
  79. C.-Y. Hsieh and M. Gessner, companion paper, General quantum resources provide advantages in work extraction tasks, arXiv:2403.18753 [Phys. Rev. A (to be published)].
  80. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).

  81. We always assume OR is compact in the topology induced by the diamond distance, i.e., the distance measure induced by diamond norm ∥·∥⋄ [82] (see also, e.g., Ref. [83]).

  82. J. Watrous, The Theory of Quantum Information (Cambridge University Press, Cambridge, England, 2018).
  83. B. Regula, R. Takagi, and M. Gu, Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps, Quantum 5, 522 (2021).
  84. F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. U.S.A. 112, 3275 (2015).
  85. P. Ćwikliński, M. Studziński, M. Horodecki, and J. Oppenheim, Limitations on the evolution of quantum coherences: Towards fully quantum second laws of thermodynamics, Phys. Rev. Lett. 115, 210403 (2015).
  86. G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian, Quantum majorization and a complete set of entropic conditions for quantum thermodynamics, Nat. Commun. 9, 5352 (2018).
  87. T. Theurer, E. Zanoni, C. Maria Scandolo, and G. Gour, Thermodynamic state convertibility is determined by qubit cooling and heating, New J. Phys. 25, 123017 (2023).
  88. G. Gour, Role of quantum coherence in thermodynamics, PRX Quantum 3, 040323 (2022).
  89. 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).

  90. Note that W(ρ,H) is the average work that can be extracted in the limit of infinitely many copies of ρ [89], or in the limit of infinitely many steps in the protocol [71].

  91. H. Umegaki, Conditional expectation in an operator algebra, IV (Entropy And Information), Kodai Math. Sem. Rep. 14, 59 (1962).
  92. J. Oppenheim, M. Horodecki, P. Horodecki, and R. Horodecki, Thermodynamical approach to quantifying quantum correlations, Phys. Rev. Lett. 89, 180402 (2002).
  93. M. Alimuddin, T. Guha, and P. Parashar, Independence of work and entropy for equal-energetic finite quantum systems: Passive-state energy as an entanglement quantifier, Phys. Rev. E 102, 012145 (2020).
  94. M. Alimuddin, T. Guha, and P. Parashar, Bound on ergotropic gap for bipartite separable states, Phys. Rev. A 99, 052320 (2019).
  95. J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs, Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules, Phys. Rev. A 77, 032114 (2008).
  96. H. Kwon, H. Jeong, D. Jennings, B. Yadin, and M. S. Kim, Clock–work trade-off relation for coherence in quantum thermodynamics, Phys. Rev. Lett. 120, 150602 (2018).
  97. S. Vinjanampathy and J. Anders, Quantum thermodynamics, Contemp. Phys. 57, 545 (2016).
  98. M. A. Ciampini, L. Mancino, A. Orieux, C. Vigliar, P. Mataloni, M. Paternostro, and M. Barbieri, Experimental extractable work-based multipartite separability criteria, npj Quantum Inf. 3, 10 (2017).
  99. G. Francica, J. Goold, F. Plastina, and M. Paternostro, Daemonic ergotropy: Enhanced work extraction from quantum correlations, npj Quantum Inf. 3, 12 (2017).
  100. 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).
  101. J. Monsel, M. Fellous-Asiani, B. Huard, and A. Auffèves, The energetic cost of work extraction, Phys. Rev. Lett. 124, 130601 (2020).
  102. T. Opatrný, A. Misra, and G. Kurizki, Work generation from thermal noise by quantum phase-sensitive observation, Phys. Rev. Lett. 127, 040602 (2021).
  103. X. Yang, Y.-H. Yang, M. Alimuddin, R. Salvia, S.-M. Fei, L.-M. Zhao, S. Nimmrichter, and M.-X. Luo, Battery capacity of energy-storing quantum systems, Phys. Rev. Lett. 131, 030402 (2023).
  104. C. Datta, R. Ganardi, T. V. Kondra, and A. Streltsov, Is there a finite complete set of monotones in any quantum resource theory?, Phys. Rev. Lett. 130, 240204 (2023).
  105. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed. (Cambridge University Press, Cambridge, England, 1988).

  106. To see this, it suffices to combine Lemmas 6, 10, 16, and Definition 15 in Ref. [78].

  107. M. Sion, On general minimax theorems, Pac. J. Math. 8, 171 (1958).
  108. M. Navascués and S. Popescu, How energy conservation limits our measurements, Phys. Rev. Lett. 112, 140502 (2014).
  109. M. A. Nielsen, Conditions for a class of entanglement transformations, Phys. Rev. Lett. 83, 436 (1999).
  110. A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett. 67, 565 (2004).
  111. S. Tirone, R. Salvia, S. Chessa, and V. Giovannetti, Quantum work capacitances: Ultimate limits for energy extraction on noisy quantum batteries, SciPost Phys. 17, 041 (2024).

Outline

Information

Phys. Rev. Lett. 137, 070403– Published 12 August, 2026
  • Received 5 April 2024
  • Revised 9 March 2026
  • Accepted 7 July 2026
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