A Realistic Framework for Quantum Sensing under Finite Resources

Abstract

Quantum-enhanced sensing is commonly benchmarked using the quantum Fisher information (QFI), often interpreted as a direct indicator of achievable precision. However, this quantity acquires operational meaning only within a fully specified inference framework that consistently incorporates state preparation, measurement design, resource accounting, estimator construction, prior information, and finite data effects. Here we establish a realistic end-to-end framework for quantum sensing under finite resources and identify general principles required for operationally meaningful performance assessment. A central conceptual point is that the relevant unit of estimation is not a single detection event but the inference data set required to construct a consistent estimator. We apply this approach to several paradigmatic sensing strategies frequently cited in the literature. Revisiting phase estimation with NOON states within a Bayesian framework under equal total photon resources, we explicitly construct optimal estimators and show that such schemes offer no performance advantage over repeated classical interferometry for global phase estimation with finite prior width. The apparent Heisenberg-like scaling arises predominantly from prior constraints rather than from information gained in the measurement, which is operationally negligible in the resource-normalized sense considered here. We further analyse Holland-Burnett interferometry and homodyne detection with squeezed states, demonstrating how estimator construction and repetition number determine the attainable precision and when QFI provides a reliable diagnostic. Our results clarify the conditions under which nonclassical resources lead to genuine metrological advantages and provide a practical methodology for designing and evaluating quantum sensing protocols under realistic experimental constraints.

Figures (3)

• Figure 1: Schematic of an interferometric phase-estimation protocol using an $N$-photon NOON state (upper panel) and an $N$-times repeated standard Mach--Zehnder interferometer with a single photon (lower panel). For phase estimation around zero phase shift, the posterior distributions scale as $P_{\rm NOON} \propto \cos^2\!\left(\tfrac{N\phi}{2}\right)$ and $P_{\rm MZ} \propto \cos^{2N}\!\left(\tfrac{\phi}{2}\right)$, as illustrated in the insets. The apparent "Heisenberg-like" scaling does not originate from the detection scheme but is imposed by prior information.
• Figure 2: Schematic of an interferometric phase--estimation protocol based on the Holland--Burnett scheme. The phase is estimated at the operating point $\phi=0$ using $n$ repeated detections for fixed resources $N= 2n j .$ The upper inset compares the oscillating posterior phase distribution obtained from a single detection ($n=1$, solid blue line) with a conventional strategy in which all resources are injected into a single input port of a Mach--Zehnder interferometer (black dashed line). The lower inset plots the variance of posterior distribution as a function of the number of repetitions $n$ (blue points). The optimal resolution is achieved for $n=4.$ The classical variance (black dashed line) and the hypothetical variance for QFI (red points) are shown for comparison.
• Figure 3: Schematic of homodyne detection scheme for bright and vacuum squeezed field. Notation of phase distinguishes between phase of coherent amplitude $\phi$ and phase of the squeezing $\theta.$ Whereas the the former one is linked with Gaussian distribution, the latter one is related to $\chi^2$ distribution.