# Autonomous Ticking Clocks from Axiomatic Principles

Mischa P. Woods

Institute for Theoretical Physics, ETH Zurich, Switzerland

### Abstract

There are many different types of time keeping devices. We use the phrase $\textit{ticking clock}$ to describe those which – simply put – tick'' at approximately regular intervals. Various important results have been derived for ticking clocks, and more are in the pipeline. It is thus important to understand the underlying models on which these results are founded. The aim of this paper is to introduce a new ticking clock model from axiomatic principles that overcomes concerns in the community about the physicality of the assumptions made in previous models. The ticking clock model in [1] achieves high accuracy, yet lacks the autonomy of the less accurate model in [2]. Importantly, the model we introduce here achieves the best of both models: it retains the autonomy of [2] while allowing for the high accuracies of [1]. What is more, [2] is revealed to be a special case of the new ticking clock model.

### ► References

[1] M. P. Woods, R. Silva, G. Pütz, S. Stupar, and R. Renner, Quantum clocks are more accurate than classical ones,'' (2018a), arXiv:1806.00491v2 [quant-ph].
arXiv:1806.00491v2

[2] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Phys. Rev. X 7, 031022 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.031022

[3] H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958).
https:/​/​doi.org/​10.1103/​PhysRev.109.571

[4] A. Peres, Am. J. Phys 48, 552 (1980).
https:/​/​doi.org/​10.1119/​1.12061

[5] V. Bužek, R. Derka, and S. Massar, Phys. Rev. Lett. 82, 2207 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.82.2207

[6] P. Erker, The Quantum Hourglass,'' (2014), ETH Zürich.
https:/​/​doi.org/​10.3929/​ethz-a-010514644

[7] S. Ranković, Y.-C. Liang, and R. Renner, Quantum clocks and their synchronisation | the Alternate Ticks Game,'' (2015), arXiv:1506.01373v1 [quant-ph].
arXiv:1506.01373v1

[8] M. P. Woods, R. Silva, and J. Oppenheim, Ann. Henri Poincaré (2018b), 10.1007/​s00023-018-0736-9.
https:/​/​doi.org/​10.1007/​s00023-018-0736-9

[9] S. Khandelwal, M. P. Lock, and M. P. Woods, Quantum 4, 309 (2020).
https:/​/​doi.org/​10.22331/​q-2020-08-14-309

[10] Y. Yang, L. Baumgärtner, R. Silva, and R. Renner, Accuracy enhancing protocols for quantum clocks,'' (2019), arXiv:1905.09707 [quant-ph].
arXiv:1905.09707

[11] N. Yunger Halpern and D. T. Limmer, Phys. Rev. A 101, 042116 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.042116

[12] P. A. Hoehn, A. R. H. Smith, and M. P. E. Lock, The Trinity of Relational Quantum Dynamics,'' (2019), arXiv:1912.00033 [quant-ph].
arXiv:1912.00033

[13] W. Pauli, in Handbuch der Physik, Vol. 24 (Springer, 1933) pp. 83–272.
https:/​/​doi.org/​10.1007/​978-3-642-52619-0_2

[14] W. Pauli, Encyclopedia of Physics, Vol. 1 (Springer, Berlin, 1958) p. 60.

[15] Á. Rivas, S. F. Huelga, and M. B. Plenio, Reports on Progress in Physics 77, 094001 (2014).
https:/​/​doi.org/​10.1088/​0034-4885/​77/​9/​094001

[16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44 (Springer New York, New York, NY, 1983).
https:/​/​doi.org/​10.1007/​978-1-4612-5561-1

[17] R. Gandy and C. Yates, Mathematical Logic, Vol. 4 (Elsevier, 2001) (see page 267).
https:/​/​www.sciencedirect.com/​book/​9780444504234/​mathematical-logic

[18] A. Degasperis, L. Fonda, and G. C. Ghirardi, Il Nuovo Cimento A (1965-1970) 21, 471 (1974).
https:/​/​doi.org/​10.1007/​BF02731351

[19] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977).
https:/​/​doi.org/​10.1063/​1.523304

[20] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2007) . See section 3.3 Microscopic derivations. In particular, 3.3.1 Weak-coupling limit and 3.3.3 Singular-coupling limit.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199213900.001.0001

[21] P. F. Palmer, J. Math. Phys. 18, 527 (1977).
https:/​/​doi.org/​10.1063/​1.523296

[22] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. Sudarshan, Rep. Math. Phys. 13, 149 (1978).
https:/​/​doi.org/​10.1016/​0034-4877(78)90050-2

[23] S. Stupar, C. Klumpp, N. Gisin, and R. Renner, Performance of Stochastic Clocks in the Alternate Ticks Game,'' (2018), arXiv:1806.08812 [quant-ph].
arXiv:1806.08812

[24] Y. Yang and R. Renner, Ultimate limit on time signal generation,'' (2020), arXiv:2004.07857 [quant-ph].
arXiv:2004.07857

[25] I. Pikovski, M. Zych, F. Costa, and Č. Brukner, Nat. Phys. 11, 668 (2015).
https:/​/​doi.org/​10.1038/​nphys3366

[26] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).
https:/​/​doi.org/​10.1007/​BF01608499

[27] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976).
https:/​/​doi.org/​10.1063/​1.522979

[28] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed. (Cambridge University Press, USA, 2011).
https:/​/​doi.org/​10.5555/​1972505

[29] K. Kraus, A. Böhm, J. D. Dollard, and W. H. Wootters, eds., States, Effects, and Operations Fundamental Notions of Quantum Theory, Lecture Notes in Physics, Vol. 190 (Springer Berlin Heidelberg, Berlin, Heidelberg, 1983).
https:/​/​doi.org/​10.1007/​3-540-12732-1

[30] M.-D. Choi, Linear Algebra and its Applications 10, 285 (1975).
https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0

### Cited by

[1] Antoine Rignon-Bret, Giacomo Guarnieri, John Goold, and Mark T. Mitchison, "Thermodynamics of precision in quantum nanomachines", Physical Review E 103 1, 012133 (2021).

[2] Emanuel Schwarzhans, Maximilian P. E. Lock, Paul Erker, Nicolai Friis, and Marcus Huber, "Autonomous Temporal Probability Concentration: Clockworks and the Second Law of Thermodynamics", Physical Review X 11 1, 011046 (2021).

[3] G. J. Milburn, "The thermodynamics of clocks", Contemporary Physics 61 2, 69 (2020).

[4] Knud Thomsen, "Timelessness strictly inside the Quantum Realm", arXiv:2009.07087.

The above citations are from Crossref's cited-by service (last updated successfully 2021-05-06 17:15:17) and SAO/NASA ADS (last updated successfully 2021-05-06 17:15:19). The list may be incomplete as not all publishers provide suitable and complete citation data.