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  • Open Access

Quantum Metrology in the Finite-Sample Regime

Johannes Jakob Meyer1,*, Sumeet Khatri1, Daniel Stilck França2,3, Jens Eisert1,4,5, and Philippe Faist1

  • *Contact author: johannes.meyer@fu-berlin.de
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PRX Quantum 6, 030336 – Published 27 August, 2025

DOI: https://doi.org/10.1103/qbn1-p6bq

Abstract

In quantum metrology, a major application of quantum technologies, the ultimate precision of estimating an unknown parameter is often stated in terms of the Cramér-Rao bound. Yet, the latter fails to completely characterize the distribution of estimates in the nonasymptotic regime. Optimizing metrology protocols with respect to the Cramér-Rao bound can therefore lead to surprisingly poor finite-sample performance. This can be avoided by quantifying the quality of a metrology protocol by the probability of obtaining an estimate with a given accuracy. Using this intrinsically single-shot quantity naturally accommodates the finite-sample regime. We show that the fundamental limits of this figure of merit can be quantified through a multihypothesis testing problem between quantum states. Building on this connection, we derive an analogue of the Cramér-Rao bound that contains explicit corrections relevant to the finite-sample regime. We apply the finite-sample metrology framework to the example task of phase estimation with an ensemble of spin-1/2 particles, giving unambiguous evidence that large quantum Fisher information does not guarantee that a metrology protocol has a good finite-sample performance. Overall, analyzing the probability of success allows the reliable study of quantum metrology in the finite-sample regime and opens up a plethora of new avenues for research at the interface of quantum information theory and quantum metrology.

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Popular Summary

Quantum sensors are expected to revolutionize precision measurements in fields ranging from medical diagnostics to gravitational wave detection by exploiting quantum-mechanical effects to increase sensitivity. Today’s quantum sensors, however, face practical limitations: only a small number of quantum systems can be simultaneously controlled, they operate comparably slowly, and they require frequent recalibration. This means that real-world quantum sensors operate in the “finite-sample regime” where only a rather limited number of measurements can be performed in one experimental run.

The conventional approach to understanding quantum sensors theoretically relies on mathematical tools designed for situations where many measurements can be averaged together. This approach may be misleading when applied to finite-sample scenarios, potentially leading to sensors that perform poorly in practice despite looking promising in theory. Our work addresses this challenge by focussing on the probability of obtaining a sufficiently accurate measurement in a single attempt rather than averaging many measurements. We establish fundamental limits on what is achievable in this realistic setting and demonstrate our approach using the example of phase estimation with quantum spins. Surprisingly, we find that some quantum states that appear optimal according to traditional measures actually perform poorly when only a few measurements are available.

The finite-sample framework offers a more realistic way to evaluate quantum sensing protocols. While many challenges remain in translating quantum advantages to practical devices, this work provides clearer criteria for identifying promising approaches and understanding their limitations in resource-constrained settings.

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PRX Quantum 6, 030336– Published 27 August, 2025
  • Received 29 August 2023
  • Revised 22 May 2025
  • Accepted 8 July 2025
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