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Operational quasiprobability in quantum thermodynamics: Work extraction by coherence and nonjoint measurability

Jeongwoo Jae1,*, Junghee Ryu2,3,†, and Hoon Ryu4,‡

  • *Contact author: jeongwoo.jae@samsung.com
  • †Contact author: junghee@kisti.re.kr
  • ‡Contact author: elec1020@kumoh.ac.kr
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Phys. Rev. Research 8, 013232 – Published 3 March, 2026

DOI: https://doi.org/10.1103/qbs6-nj3l

Abstract

We employ the operational quasiprobability (OQ) as a work distribution, which reproduces the Jarzynski equality and yields the average work consistent with the classical definition. The OQ distribution can be implemented through two simple experimental settings: the end-point measurement and the two-point measurement. Using this framework, we demonstrate the explicit contribution of coherence to the work, which can be formulated by the second-law-type relation. In a two-level system, we show that nonjoint measurability, a generalized notion of measurement incompatibility, can increase the amount of extractable work beyond the classical bound imposed by jointly measurable measurements. We further prove that the real part of Kirkwood-Dirac quasiprobability (KDQ) and the OQ are equivalent in two-level systems, and they are non-negative for binary unbiased measurements if and only if the measurements are jointly measurable. In a three-level nitrogen-vacancy center system, the OQ and the KDQ exhibit different amounts of negativities while enabling the same work extraction, implying that the magnitude of negativity is not a faithful indicator of nonclassical work. These results highlight that coherence and nonjoint measurability play fundamental roles in the enhancement of work.

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References (40)

  1. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  2. P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102 (2007).
  3. 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).
  4. M. Lostaglio, A. Belenchia, A. Levy, S. Hernández-Gómez, N. Fabbri, and S. Gherardini, Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables, Quantum 7, 1128 (2023).
  5. M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities, Phys. Rev. Lett. 120, 040602 (2018).
  6. G. Francica, Class of quasiprobability distributions of work with initial quantum coherence, Phys. Rev. E 105, 014101 (2022).
  7. G. Francica, Most general class of quasiprobability distributions of work, Phys. Rev. E 106, 054129 (2022).
  8. J.-H. Pei, J.-F. Chen, and H. T. Quan, Exploring quasiprobability approaches to quantum work in the presence of initial coherence: Advantages of the Margenau-Hill distribution, Phys. Rev. E 108, 054109 (2023).
  9. S. Gherardini and G. De Chiara, Quasiprobabilities in quantum thermodynamics and many-body systems, PRX Quantum 5, 030201 (2024).
  10. D. R. M. Arvidsson-Shukur, N. Yunger Halpern, H. V. Lepage, A. A. Lasek, C. H. W. Barnes, and S. Lloyd, Quantum advantage in postselected metrology, Nat. Commun. 11, 3775 (2020).
  11. N. Lupu-Gladstein, Y. B. Yilmaz, D. R. M. Arvidsson-Shukur, A. Brodutch, A. O. T. Pang, A. M. Steinberg, and N. Y. Halpern, Negative quasiprobabilities enhance phase estimation in quantum-optics experiment, Phys. Rev. Lett. 128, 220504 (2022).
  12. N. Yunger Halpern, Jarzynski-like equality for the out-of-time-ordered correlator, Phys. Rev. A 95, 012120 (2017).
  13. N. Yunger Halpern, B. Swingle, and J. Dressel, Quasiprobability behind the out-of-time-ordered correlator, Phys. Rev. A 97, 042105 (2018).
  14. D. R. M. Arvidsson-Shukur, W. F. Braasch Jr., S. De Bièvre, J. Dressel, A. N. Jordan, C. Langrenez, M. Lostaglio, J. S. Lundeen, and N. Y. Halpern, Properties and applications of the Kirkwood–Dirac distribution, New J. Phys. 26, 121201 (2024).
  15. S. Hernández-Gómez, S. Gherardini, A. Belenchia, M. Lostaglio, A. Levy, and N. Fabbri, Projective measurements can probe nonclassical work extraction and time correlations, Phys. Rev. Res. 6, 023280 (2024).
  16. D. R. M. Arvidsson-Shukur, J. C. Drori, and N. Y. Halpern, Conditions tighter than noncommutation needed for nonclassicality, J. Phys. A: Math. Theor. 54, 284001 (2021).
  17. P. Busch, Unsharp reality and joint measurements for spin observables, Phys. Rev. D 33, 2253 (1986).
  18. T. Heinosaari, T. Miyadera, and M. Ziman, An invitation to quantum incompatibility, J. Phys. A: Math. Theor. 49, 123001 (2016).
  19. K. Beyer, R. Uola, K. Luoma, and W. T. Strunz, Joint measurability in nonequilibrium quantum thermodynamics, Phys. Rev. E 106, L022101 (2022).
  20. 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).
  21. M. T. Quintino, T. Vértesi, and N. Brunner, Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality, Phys. Rev. Lett. 113, 160402 (2014).
  22. R. Uola, C. Budroni, O. Gühne, and J.-P. Pellonpää, One-to-one mapping between steering and joint measurability problems, Phys. Rev. Lett. 115, 230402 (2015).
  23. N.-L. Liu, L. Li, S. Yu, and Z.-B. Chen, Duality relation and joint measurement in a Mach-Zehnder interferometer, Phys. Rev. A 79, 052108 (2009).
  24. P. Busch, P. Lahti, and R. F. Werner, Proof of Heisenberg's error-disturbance relation, Phys. Rev. Lett. 111, 160405 (2013).
  25. P. Busch, P. Lahti, and R. F. Werner, Heisenberg uncertainty for qubit measurements, Phys. Rev. A 89, 012129 (2014).
  26. J. Ryu, J. Lim, S. Hong, and J. Lee, Operational quasiprobabilities for qudits, Phys. Rev. A 88, 052123 (2013).
  27. J. Jae, J. Ryu, and J. Lee, Operational quasiprobabilities for continuous variables, Phys. Rev. A 96, 042121 (2017).
  28. J. Ryu, S. Hong, J.-S. Lee, K. H. Seol, J. Jae, J. Lim, J. Lee, K.-G. Lee, and J. Lee, Optical experiment to test negative probability in context of quantum-measurement selection, Sci. Rep. 9, 19021 (2019).
  29. J. Jae, K. Baek, J. Ryu, and J. Lee, Necessary and sufficient condition for joint measurability, Phys. Rev. A 100, 032113 (2019).
  30. J. Jae, J. Lee, M. S. Kim, K.-G. Lee, and J. Lee, Contextual quantum metrology, npj Quantum Inf. 10, 68 (2024).
  31. T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  32. T. Heinosaari, J. Kiukas, and D. Reitzner, Noise robustness of the incompatibility of quantum measurements, Phys. Rev. A 92, 022115 (2015).
  33. L. M. Johansen, Quantum theory of successive projective measurements, Phys. Rev. A 76, 012119 (2007).
  34. S. Yu, N.-l. Liu, L. Li, and C. H. Oh, Joint measurement of two unsharp observables of a qubit, Phys. Rev. A 81, 062116 (2010).
  35. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Colloquium: Understanding quantum weak values: Basics and applications, Rev. Mod. Phys. 86, 307 (2014).
  36. R. Wagner and E. F. Galvão, Simple proof that anomalous weak values require coherence, Phys. Rev. A 108, L040202 (2023).
  37. N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, Stimulated Raman adiabatic passage in physics, chemistry, and beyond, Rev. Mod. Phys. 89, 015006 (2017).
  38. K.-D. Wu, Y. Yuan, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and M. Perarnau-Llobet, Experimentally reducing the quantum measurement back action in work distributions by a collective measurement, Sci. Adv. 5, eaav4944 (2019).
  39. H. Li, J. Xie, H. Kwon, Y. Zhao, M. S. Kim, and L. Zhang, Experimental demonstration of generalized quantum fluctuation theorems in the presence of coherence, Sci. Adv. 11, eadq6014 (2025).
  40. S. Rana, P. Parashar, and M. Lewenstein, Trace-distance measure of coherence, Phys. Rev. A 93, 012110 (2016).

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Phys. Rev. Research 8, 013232– Published 3 March, 2026
  • Received 9 November 2025
  • Accepted 5 February 2026
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