Energy conservation and fluctuation theorem are incompatible for quantum work

Karen V. Hovhannisyan1 and Alberto Imparato2

1University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-Str. 24-25, 14476 Potsdam, Germany
2Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus, Denmark

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


Characterizing fluctuations of work in coherent quantum systems is notoriously problematic. Here we reveal the ultimate source of the problem by proving that ($\mathfrak{A}$) energy conservation and ($\mathfrak{B}$) the Jarzynski fluctuation theorem cannot be observed at the same time. Condition $\mathfrak{A}$ stipulates that, for any initial state of the system, the measured average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange $deterministically$ zero work. Condition $\mathfrak{B}$ is only for thermal initial states and encapsulates the second law of thermodynamics. We prove that $\mathfrak{A}$ and $\mathfrak{B}$ are incompatible for work measurement schemes that are differentiable functions of the state and satisfy two mild structural constraints. This covers all existing schemes and leaves the theoretical possibility of jointly observing $\mathfrak{A}$ and $\mathfrak{B}$ open only for a narrow class of exotic schemes. For the special but important case of state-independent schemes, the situation is much more rigid: we prove that, essentially, only the two-point measurement scheme is compatible with $\mathfrak{B}$.

Work is one of the most basic notions in physics — it quantifies the transactions of “ordered” mechanical energy. At macroscopic scale, work is justifiably perceived as deterministic. However, at the microscale, especially in the quantum regime, significant fluctuations do occur, and they matter. Therefore, no characterization of energy consumption or storage capability of a quantum device will be complete without specifying the full statistics of consumed or delivered work.

Classically, the statistics of work are defined unambiguously: a deterministic value of work corresponds to each phase-space trajectory, and those trajectories have a well-defined probability distribution. In quantum mechanics, there are famously no phase-space trajectories, and therefore a qualitatively new philosophy is required for accessing the statistics of work. Many attempts have been made in the literature in that direction, but all of them eventually run into problems. Most commonly, issues arise whenever quantum coherence is to be accounted for. It has remained unclear whether a satisfactory approach to measuring work would ultimately be found.

We give a compelling resolution to this long-standing problem by showing that the two most fundamental laws associated with work — energy conservation and fluctuation theorem — cannot be observed simultaneously. Namely, no quantum measurement can yield results that would satisfy both laws. This means that any measurement of work inevitably comes with a trade-off, and therefore, each time a statement is made about work in the quantum regime, it must come with a disclaimer stipulating for which measurement scheme it is expected to be true.

► BibTeX data

► References

[1] G. N. Bochkov and Y. B. Kuzovlev, General theory of thermal fluctuations in nonlinear systems, Sov. Phys. JETP 45, 125 (1977).

[2] C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).

[3] J. Kurchan, A quantum fluctuation theorem, arXiv:cond-mat/​0007360.

[4] H. Tasaki, Jarzynski relations for quantum systems and some applications, arXiv:cond-mat/​0009244.

[5] A. E. Allahverdyan and T. M. Nieuwenhuizen, Fluctuations of work from quantum subensembles: The case against quantum work-fluctuation theorems, Phys. Rev. E 71, 066102 (2005).

[6] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009).

[7] A. E. Allahverdyan, Nonequilibrium quantum fluctuations of work, Phys. Rev. E 90, 032137 (2014).

[8] P. Talkner and P. Hänggi, Aspects of quantum work, Phys. Rev. E 93, 022131 (2016).

[9] M. Perarnau-Llobet, E. Bäumer, K. V. Hovhannisyan, M. Huber, and A. Acín, No-go theorem for the characterization of work fluctuations in coherent quantum systems, Phys. Rev. Lett. 118, 070601 (2017).

[10] R. Sampaio, S. Suomela, T. Ala-Nissila, J. Anders, and T. G. Philbin, Quantum work in the Bohmian framework, Phys. Rev. A 97, 012131 (2018).

[11] E. Bäumer, M. Lostaglio, M. Perarnau-Llobet, and R. Sampaio, Fluctuating work in coherent quantum systems: Proposals and limitations, 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 Publishing, Cham, 2018) pp. 275–300.

[12] O. Brodier, K. Mallick, and A. M. Ozorio de Almeida, Semiclassical work and quantum work identities in Weyl representation, J. Phys. A 53, 325001 (2020).

[13] S. Yukawa, A quantum analogue of the Jarzynski equality, J. Phys. Soc. Jpn. 69, 2367 (2000).

[14] P. Talkner, E. Lutz, and P. Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102(R) (2007).

[15] P. Solinas and S. Gasparinetti, Full distribution of work done on a quantum system for arbitrary initial states, Phys. Rev. E 92, 042150 (2015).

[16] 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).

[17] J. Åberg, Fully quantum fluctuation theorems, Phys. Rev. X 8, 011019 (2018).

[18] 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).

[19] 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).

[20] B.-M. Xu, J. Zou, L.-S. Guo, and X.-M. Kong, Effects of quantum coherence on work statistics, Phys. Rev. A 97, 052122 (2018).

[21] S. Gherardini, A. Belenchia, M. Paternostro, and A. Trombettoni, End-point measurement approach to assess quantum coherence in energy fluctuations, Phys. Rev. A 104, L050203 (2021).

[22] K. Beyer, K. Luoma, and W. T. Strunz, Work as an external quantum observable and an operational quantum work fluctuation theorem, Phys. Rev. Research 2, 033508 (2020).

[23] K. Micadei, G. T. Landi, and E. Lutz, Extracting Bayesian networks from multiple copies of a quantum system, arXiv:2103.14570.

[24] 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).

[25] M. Janovitch and G. T. Landi, Quantum mean-square predictors and thermodynamics, Phys. Rev. A 105, 022217 (2022).

[26] K. Beyer, R. Uola, K. Luoma, and W. T. Strunz, Joint measurability in nonequilibrium quantum thermodynamics, Phys. Rev. E 106, L022101 (2022).

[27] 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).

[28] M. Lostaglio, Quantum fluctuation theorems, contextuality, and work quasiprobabilities, Phys. Rev. Lett. 120, 040602 (2018).

[29] U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).

[30] M. Campisi, P. Hänggi, and P. Talkner, Colloquium: Quantum fluctuation relations: Foundations and applications, Rev. Mod. Phys. 83, 771 (2011).

[31] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part I (Pergamon, New York, 1980).

[32] G. Lindblad, Non-Equilibrium Entropy and Irreversibility (Reidel, Dordrecht, 1983).

[33] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Understanding quantum measurement from the solution of dynamical models, Phys. Reps. 525, 1 (2013).

[34] L. Masanes, T. D. Galley, and M. P. Müller, The measurement postulates of quantum mechanics are operationally redundant, Nat. Commun. 10, 1361 (2019).

[35] K. Abdelkhalek, Y. Nakata, and D. Reeb, Fundamental energy cost for quantum measurement, arXiv:1609.06981.

[36] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, Cambridge, England, 2010).

[37] C. Jarzynski, H. T. Quan, and S. Rahav, Quantum-classical correspondence principle for work distributions, Phys. Rev. X 5, 031038 (2015).

[38] I. García-Mata, A. J. Roncaglia, and D. A. Wisniacki, Semiclassical approach to the work distribution, Europhys. Lett. 120, 30002 (2017).

[39] K. Funo and H. T. Quan, Path integral approach to quantum thermodynamics, Phys. Rev. Lett. 121, 040602 (2018).

[40] Z. Fei, H. T. Quan, and F. Liu, Quantum corrections of work statistics in closed quantum systems, Phys. Rev. E 98, 012132 (2018).

[41] D. Petz, A survey of certain trace inequalities, Banach Cent. Publ. 30, 287 (1994).

[42] R. Pan, Z. Fei, T. Qiu, J.-N. Zhang, and H. T. Quan, Quantum-classical correspondence of work distributions for initial states with quantum coherence, arXiv:1904.05378.

[43] 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).

[44] T. B. Batalhão, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Experimental reconstruction of work distribution and study of fluctuation relations in a closed quantum system, Phys. Rev. Lett. 113, 140601 (2014).

[45] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q. Yin, H. T. Quan, and K. Kim, Experimental test of the quantum Jarzynski equality with a trapped-ion system, Nat. Phys. 11, 193 (2015).

[46] A. E. Allahverdyan, R. Balian, and T. M. Nieuwenhuizen, Maximal work extraction from finite quantum systems, Europhys. Lett. 67, 565 (2004).

[47] D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett. 130, 210401 (2023).

[48] H. M. Wiseman, Adaptive phase measurements of optical modes: Going beyond the marginal $Q$ distribution, Phys. Rev. Lett. 75, 4587 (1995).

[49] M. A. Armen, J. K. Au, J. K. Stockton, A. C. Doherty, and H. Mabuchi, Adaptive homodyne measurement of optical phase, Phys. Rev. Lett. 89, 133602 (2002).

[50] J. L. O’Brien, A. Furusawa, and J. Vučković, Photonic quantum technologies, Nature Photonics 3, 687 (2009).

[51] A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, Ab initio quantum-enhanced optical phase estimation using real-time feedback control, Nature Photonics 9, 577 (2015).

[52] 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, 4944 (2019).

[53] K.-D. Wu, E. Bäumer, J.-F. Tang, K. V. Hovhannisyan, M. Perarnau-Llobet, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Minimizing backaction through entangled measurements, Phys. Rev. Lett. 125, 210401 (2020).

[54] R. A. Horn and C. R. Johnson, Matrix analysis, 2nd ed. (Cambridge University Press, New York, 2013).

[55] A. Levy and M. Lostaglio, Quasiprobability distribution for heat fluctuations in the quantum regime, PRX Quantum 1, 010309 (2020).

[56] M. H. Mohammady, A. Auffèves, and J. Anders, Energetic footprints of irreversibility in the quantum regime, Commun. Phys. 3, 89 (2020).

[57] Y. V. Nazarov and M. Kindermann, Full counting statistics of a general quantum mechanical variable, Eur. Phys. J. B 35, 413 (2003).

[58] P. P. Hofer, Quasi-probability distributions for observables in dynamic systems, Quantum 1, 32 (2017).

[59] K. V. Hovhannisyan and A. Imparato, Quantum current in dissipative systems, New J. Phys. 21, 052001 (2019).

[60] C. Elouard, D. A. Herrera-Martí, M. Clusel, and A. Auffèves, The role of quantum measurement in stochastic thermodynamics, npj Quantum Inf. 3, 9 (2017).

[61] G. Manzano, J. M. Horowitz, and J. M. R. Parrondo, Quantum fluctuation theorems for arbitrary environments: Adiabatic and nonadiabatic entropy production, Phys. Rev. X 8, 031037 (2018).

[62] G. De Chiara and A. Imparato, Quantum fluctuation theorem for dissipative processes, Phys. Rev. Res. 4, 023230 (2022).

[63] T. Sagawa and M. Ueda, Minimal energy cost for thermodynamic information processing: Measurement and information erasure, Phys. Rev. Lett. 102, 250602 (2009).

[64] M. H. Mohammady and A. Romito, Conditional work statistics of quantum measurements, Quantum 3, 175 (2019).

[65] Y. Guryanova, N. Friis, and M. Huber, Ideal projective measurements have infinite resource costs, Quantum 4, 222 (2020).

[66] M. H. Mohammady, Thermodynamically free quantum measurements, J. Phys. A 55, 505304 (2023).

[67] K. Ito, P. Talkner, B. P. Venkatesh, and G. Watanabe, Generalized energy measurements and quantum work compatible with fluctuation theorems, Phys. Rev. A 99, 032117 (2019).

[68] H. F. Baker, Alternants and continuous groups, Proc. London Math. Soc. s2-3, 24 (1905).

[69] E. C. Lance, Hilbert C*-Modules: A Toolkit for Operator Algebraists (Cambridge University Press, Cambridge, 1995).

[70] G. Jameson, Khinchin’s inequality for operators, Glasgow Math. J. 38, 327 (1996).

[71] W. Pusz and S. L. Woronowicz, Passive states and KMS states for general quantum systems, Commun. Math. Phys. 58, 273 (1978).

[72] A. Lenard, Thermodynamical proof of the Gibbs formula for elementary quantum systems, J. Stat. Phys. 19, 575 (1978).

[73] A. Lenard, Generalization of the Golden-Thompson inequality $\mathrm{Tr}(e^{A} e^{B}) \geqq \mathrm{Tr} e^{A+B}$, Indiana Univ. Math. J. 21, 457 (1971).

[74] C. J. Thompson, Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21, 469 (1971).

[75] G. Adesso and F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives, J. Phys. A 40, 7821 (2007).

[76] J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Am. J. Math. 58, 141 (1936).

[77] H. Scutaru, Fidelity for displaced squeezed thermal states and the oscillator semigroup, J. Phys. A 31, 3659 (1998).

Cited by

[1] Matteo Lostaglio, Alessio Belenchia, Amikam Levy, Santiago Hernández-Gómez, Nicole Fabbri, and Stefano Gherardini, "Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables", Quantum 7, 1128 (2023).

[2] Viktor Holubec and Artem Ryabov, "Fluctuations in heat engines", Journal of Physics A Mathematical General 55 1, 013001 (2022).

[3] Alessandro Santini, Andrea Solfanelli, Stefano Gherardini, and Mario Collura, "Work statistics, quantum signatures, and enhanced work extraction in quadratic fermionic models", Physical Review B 108 10, 104308 (2023).

[4] M. Hamed Mohammady and Takayuki Miyadera, "Quantum measurements constrained by the third law of thermodynamics", arXiv:2209.06024, (2022).

[5] Konstantin Beyer, Roope Uola, Kimmo Luoma, and Walter T. Strunz, "Joint measurability in nonequilibrium quantum thermodynamics", Physical Review E 106 2, L022101 (2022).

[6] M. Hamed Mohammady and Takayuki Miyadera, "Quantum measurements constrained by the third law of thermodynamics", Physical Review A 107 2, 022406 (2023).

[7] M. Hamed Mohammady, "Thermodynamically free quantum measurements", arXiv:2205.10847, (2022).

[8] M. Hamed Mohammady, "Classicality of the heat produced by quantum measurements", Physical Review A 104 6, 062202 (2021).

[9] M. Hamed Mohammady, "Thermodynamically free quantum measurements", Journal of Physics A Mathematical General 55 50, 505304 (2022).

[10] M. Hamed Mohammady, "Classicality of the heat produced by quantum measurements", arXiv:2103.15749, (2021).

[11] Santiago Hernández-Gómez, Francesco Poggiali, Paola Cappellaro, Francesco S. Cataliotti, Andrea Trombettoni, Nicole Fabbri, and Stefano Gherardini, "Energy exchange statistics and fluctuation theorem for non-thermal asymptotic states", arXiv:2404.05310, (2024).

The above citations are from SAO/NASA ADS (last updated successfully 2024-05-26 13:53:42). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-05-26 13:53:40).