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Autonomous Quantum Machines and Finite-Sized Clocks


Processes such as quantum computation, or the evolution of quantum cellular automata, are typically described by a unitary operation implemented by an external observer. In particular, an interaction is generally turned on for a precise amount of time, using a classical clock. A fully quantum mechanical description of such a device would include a quantum description of the clock whose state is generally disturbed because of the back-reaction on it. Such a description is needed if we wish to consider finite-sized autonomous quantum machines requiring no external control. The extent of the back-reaction has implications on how small the device can be, on the length of time the device can run, and is required if we want to understand what a fully quantum mechanical treatment of an observer would look like. Here, we consider the implementation of a unitary by a finite-sized device and show that the back-reaction on it can be made exponentially small in the device’s dimension while its energy only increases linearly with dimension. As a result, an autonomous quantum machine need only be of modest size and energy. We are also able to solve a long-standing open problem by using a finite-sized quantum clock to approximate the continuous evolution of an idealised clock. The result has implications for how well quantum devices can be controlled and on the equivalence of different paradigms of control.


  1. 1.

    Howard, J.: Molecular motors: structural adaptations to cellular functions. Nature 389(6651), 561–567 (1997)

    ADS  Article  Google Scholar 

  2. 2.

    Frank, J. (ed.): Molecular Machines in Biology. Cambridge University Press, Cambridge (2011). Cambridge Books Online

    Google Scholar 

  3. 3.

    Douglas, S.M., Bachelet, I., Church, G.M.: A logic-gated nanorobot for targeted transport of molecular payloads. Science 335(6070), 831–834 (2012)

    ADS  Article  Google Scholar 

  4. 4.

    Scovil, H.E.D., Schulz-DuBois, E.O.: Three-level masers as heat engines. Phys. Rev. Lett. 2, 262–263 (1959)

    ADS  Article  Google Scholar 

  5. 5.

    Geusic, J.E., Schulz-DuBois, E.O., Scovil, H.E.D.: Quantum equivalent of the carnot cycle. Phys. Rev. 156, 343–351 (1967)

    ADS  Article  Google Scholar 

  6. 6.

    Linden, N., Popescu, S., Skrzypczyk, P.: How small can thermal machines be? The smallest possible refrigerator. Phys. Rev. Lett. 105(13), 130401 (2010)

    ADS  Article  Google Scholar 

  7. 7.

    Brask, J.B., Haack, G., Brunner, N., Huber, M.: Autonomous quantum thermal machine for generating steady-state entanglement. New J. Phys. 17(11), 113029 (2015)

    ADS  Article  Google Scholar 

  8. 8.

    Brandão, F.G.S.L., Horodecki, M., Oppenheim, J., Renes, J.M., Spekkens, R.W.: Resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett. 111(25), 250404 (2013)

    ADS  Article  Google Scholar 

  9. 9.

    Malabarba, A.S.L., Short, A.J., Kammerlander, P.: Clock-driven quantum thermal engines. New J. Phys. 17(4), 045027 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Tonner, F., Mahler, G.: Autonomous quantum thermodynamic machines. Phys. Rev. E 72(6), 066118 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Gelbwaser-Klimovsky, D., Kurizki, G.: Heat-machine control by quantum-state preparation: from quantum engines to refrigerators. Phys. Rev. E 90(2), 022102 (2014)

    ADS  Article  Google Scholar 

  12. 12.

    Correa, L.A., Palao, J.P., Alonso, D., Adesso, G.: Quantum-enhanced absorption refrigerators. Sci. Rep. 4, 3949 (2014)

    Google Scholar 

  13. 13.

    Tonner, F., Mahler, G.: Quantum Limit of the Carnot Engine. 1807–2007 Knowledge for Generations (2007)

  14. 14.

    Feynman, R.P.: The Feynman Lectures on Physics, vol. 2. Addison-Wesley, Boston (1963)

    Google Scholar 

  15. 15.

    Linden, N., Popescu, S., Skrzypczyk, P.: How small can thermal machines be? The smallest possible refrigerator. Phys. Rev. Lett. 105, 130401 (2010)

    ADS  Article  Google Scholar 

  16. 16.

    Erker, P., Mitchison, M.T., Silva, R., Woods, M.P., Brunner, N., Huber, M.: Autonomous quantum clocks: Does thermodynamics limit our ability to measure time? Phys. Rev. X 7, 031022 (2017)

    Google Scholar 

  17. 17.

    Geusic, J.E., Schulz-DuBios, E.O., Scovil, H.E.D.: Quantum equivalent of the Carnot cycle. Phys. Rev. 156, 343–351 (1967)

    ADS  Article  Google Scholar 

  18. 18.

    Brandão, F., Horodecki, M., Ng, N., Oppenheim, J., Wehner, S.: The second laws of quantum thermodynamics. Proc. Natl. Acad. Sci. 112(11), 3275–3279 (2015)

    ADS  Article  Google Scholar 

  19. 19.

    van Dam, W., Hayden, P.: Universal entanglement transformations without communication. Phys. Rev. A 67(6), 060302 (2003)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011)

    ADS  Article  Google Scholar 

  21. 21.

    Ranković, S., Liang, Y.C., Renner, R.: Quantum clocks and their synchronisation—the alternate ticks game (2015). arXiv:1506.01373v1

  22. 22.

    Peres, A.: Measurement of time by quantum clocks. Am. J. Phys. 48(7), 552 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Bužek, V., Derka, R., Massar, S.: Optimal quantum clocks. Phys. Rev. Lett. 82, 2207–2210 (1999)

    ADS  Article  Google Scholar 

  24. 24.

    Allcock, G.R.: The time of arrival in quantum mechanics i. Formal considerations. Ann. Phys. 53(2), 253–285 (1969)

    ADS  Article  Google Scholar 

  25. 25.

    Salecker, H., Wigner, E.P.: Quantum limitations of the measurement of space-time distances. Phys. Rev. 109, 571–577 (1958)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Frenzel, M.F., Jennings, D., Rudolph, T.: Quasi-autonomous quantum thermal machines and quantum to classical energy flow. New J. Phys. 18(2), 023037 (2016)

    ADS  Article  Google Scholar 

  27. 27.

    Pauli, W.: Handbuch der Physik, vol. 24, pp. 83–272. Springer, Berlin (1933)

    Google Scholar 

  28. 28.

    Pauli, W.: Encyclopedia of Physics, vol. 1, p. 60. Springer, Berlin (1958)

    Google Scholar 

  29. 29.

    Gross, D., Nesme, V., Vogts, H., Werner, R.F.: Index theory of one dimensional quantum walks and cellular automata. Commun. Math. Phys. 310(2), 419–454 (2012)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Massar, S., Spindel, P.: Uncertainty relation for the discrete fourier transform. Phys. Rev. Lett. 100, 190401 (2008)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Woods, M.P., Silva, R., Pütz, G., Stupar, S.R., Renner, R.: Quantum clocks are more accurate than classical ones. ArXiv:1806.00491

  32. 32.

    Pegg, D.T., Barnett, S.M.: Phase properties of the quantized single-mode electromagnetic field. Phys. Rev. A 39, 1665–1675 (1989)

    ADS  Article  Google Scholar 

  33. 33.

    Busch, P.: No information without disturbance: quantum limitations of measurement. In: Christian, W., Myrvold, J. (eds.) Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: An International Conference in Honour of Abner Shimony. Springer (2006). arXiv:0706.3526v1

  34. 34.

    Busch, P.: The time–energy uncertainty relation. In: Muga, J.G., Sala Mayato, R., Egusquiza, I.L. (eds.) Time in Quantum Mechanics, 2nd edn, pp. 69–98. Springer, Berlin (2002)

    MATH  Chapter  Google Scholar 

  35. 35.

    Åberg, J.: Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014)

    ADS  Article  Google Scholar 

  36. 36.

    Abramowitz, M., Stegun, I.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. Am. J. Phys. 56, 958 (1988)

    ADS  MATH  Article  Google Scholar 

  37. 37.

    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions, 2 edn. Birkhäuser, Basel (2002).

  38. 38.

    Berend, D., Tassa, T.: Improved bounds on bell numbers and on moments of sums of random variables. Probab. Math. Stat. 30, 185–205 (2010)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Sophus, L., Friedrich, E.: Theorie der Transformationsgruppen. 1st edition, Leipzig; 2nd edition, AMS Chelsea Publishing, 1970 (1888)

  40. 40.

    Garrison, J.C., Wong, J.: Canonically conjugate pairs, uncertainty relations, and phase operators. J. Math. Phys. 11, 2242–2249 (1970)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Fourier Analysis, self-Adjointness. Number v. 2 in Methods of Modern Mathematical Physics. Academic Press, London (1975)

    MATH  Google Scholar 

  42. 42.

    Weyl, H.: Quantenmechanik und gruppentheorie. Z. Phys. 46(1), 1–46 (1927)

    ADS  MATH  Article  Google Scholar 

  43. 43.

    Grafakos, L.: Classical Fourier Analysis. Springer, Berlin (2014)

    MATH  Google Scholar 

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Correspondence to Mischa P. Woods.

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Communicated by David Pérez-García.

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Woods, M.P., Silva, R. & Oppenheim, J. Autonomous Quantum Machines and Finite-Sized Clocks. Ann. Henri Poincaré 20, 125–218 (2019).

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