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Optimal Time Estimation and the Clock Uncertainty Relation for Stochastic Processes

Kacper Prech1,*, Gabriel T. Landi2,†, Florian Meier3, Nuriya Nurgalieva4, Patrick P. Potts1, Ralph Silva5, and Mark T. Mitchison6,7,‡

  • 1Department of Physics and Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
  • 2Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
  • 3Atominstitut, TU Wien, 1020 Vienna, Austria
  • 4Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
  • 5Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
  • 6School of Physics, Trinity College Dublin, College Green, Dublin 2, D02 K8N4, Ireland
  • 7Department of Physics, King’s College London, Strand, London, WC2R 2LS, United Kingdom

  • *Contact author: kacper.prech@unibas.ch
  • †Contact author: glandi@ur.rochester.edu
  • ‡Contact author: mark.mitchison@kcl.ac.uk
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Phys. Rev. X 15, 031068 – Published 11 September, 2025

DOI: https://doi.org/10.1103/rpls-mp8z

Abstract

Time estimation is a fundamental task that underpins precision measurement, global navigation systems, financial markets, and the organization of everyday life. Many biological processes also depend on time estimation by nanoscale clocks, whose performance can be significantly impacted by random fluctuations. In this work, we formulate the problem of optimal time estimation for Markovian stochastic processes, and present its general solution in the asymptotic (longtime) limit. Specifically, we obtain a tight upper bound on the precision of any time estimate constructed from sustained observations of a classical, Markovian jump process. This bound is controlled by the mean residual time, i.e., the expected wait before the first jump is observed. As a consequence, we obtain a universal bound on the signal-to-noise ratio of arbitrary currents and counting observables in the steady state. This bound is similar in spirit to the kinetic uncertainty relation but provably tighter, and we explicitly construct the counting observables that saturate it. Our results establish ultimate precision limits for an important class of observables in nonequilibrium systems and demonstrate that the mean residual time, not the dynamical activity, is the measure of freneticity that tightly constrains fluctuations far from equilibrium.

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Popular Summary

Can a sequence of random events be used to estimate the time? This problem is faced not only by unlucky castaways who have lost their watch but also by anyone trying to construct or calibrate a clock in the unavoidable presence of noise. In our study, we tackle this question using a general mathematical model called a classical Markovian jump process, which captures a wide range of physical and biological systems that evolve through random jumps.

We derive a fundamental bound on timekeeping accuracy for such systems, showing that precision is limited by the mean residual time—the average time until the first observed event. In simple terms, the faster and more frequently a system undergoes jumps, the better it can serve as a clock. Our bound sets a clear theoretical limit on the accuracy achievable for a class of classical clocks. It also applies more broadly, constraining the fluctuations of any current in systems that are out of equilibrium.

Beyond offering a practical limit on timekeeping, our results have deeper implications. If a system appears to violate our bound, it may indicate hidden features such as unseen states, memory effects, or even quantum behavior. This opens new paths for detecting complex dynamics from limited data. Overall, our work connects ideas from stochastic physics and precision measurement, helping to chart the limits of clock performance across both classical and quantum systems.

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Phys. Rev. X 15, 031068– Published 11 September, 2025
  • Received 11 July 2024
  • Revised 15 February 2025
  • Accepted 30 June 2025
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