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arXiv:2609.28066

Quantum Metrology for Signals with an Unknown Causal Structure

Gwon Ryul Han, Hyukgun Kwon, Seok Hyung Lie

quant-phSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations
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Abstract

Quantum sensing protocols usually presuppose where and when each signal acts, yet this information is unavailable beforehand in experiments such as scattering and radar. We show that causal agnosticity forces a measurement scheme to be interferometric even when only discrete interactions are allowed. Describing the sensing events in the recently developed quantum state over spacetime formalism, we then prove that the Fisher information of an interferometer is bounded by the squared sum of the local generator norms, independently of how the events are ordered. Nonunitary and spatiotemporally correlated signals can be explained as well when their physical dilation has a spacetime state on the enlarged events. Notably, the bound holds without presupposing a background spacetime that orders the events whenever the full collection of target and auxiliary events has a spacetime state. We show that this bound yields the spacetime Heisenberg limit and characterize its saturation through marginals of the spacetime state `dressed' by the signal. The $N^2$ scaling of the Fisher information with $N$ sensing events arises from the coherence of a single reference system shared across spacetime, whereas independently repeated interferometry obeys the standard quantum limit.

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Quantum sensing protocols usually presuppose where and when each signal acts, yet this information is unavailable beforehand in experiments such as scattering and radar. We show that causal agnosticity forces a measurement scheme to be interferometric even when only discrete interactions are allowed. Describing the sensing events in the recently developed quantum state over spacetime formalism, we then prove that the Fisher information of an interferometer is bounded by the squared sum of the local generator norms, independently of how the events are ordered. Nonunitary and spatiotemporally correlated signals can be explained as well when their physical dilation has a spacetime state on the enlarged events. Notably, the bound holds without presupposing a background spacetime that orders the events whenever the full collection of target and auxiliary events has a spacetime state. We show that this bound yields the spacetime Heisenberg limit and characterize its saturation through marginals of the spacetime state `dressed' by the signal. The N2N^2N2 scaling of the Fisher information with NNN sensing events arises from the coherence of a single reference system shared across spacetime, whereas independently repeated interferometry obeys the standard quantum limit.
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Quantum-enhanced interferometers utilize non-classical states of light in order to surpass the limitations imposed by classical physics on phase measurement sensitivity and resolution. We review key scientific developments in this growing field of research and derive fundamental metrology concepts, namely the Cram\'{e}r-Rao bound and the Fisher information, which are used to analyze the metrological performance of quantum states of light. We propose a supersensitive quantum interferometry protocol that can be amplified to the macroscopic realm (i.e. the mean number of photons of the probe state can reach approximately 100,000 photons) using contemporary techniques of spontaneous parametric down conversion. The parity measurement on the output field asymptotically saturates the quantum Cram\'{e}r-Rao bound, which scales like the Heisenberg limit. We uncover the role of photon statistics on phase sensitivity and rule out the necessity of mode entanglement for quantum-enhanced interferometry. Based on these insights we outline a practical metrology technique (independent of mode entanglement) that measures a phase delay of a single-mode anti-squeezing operation. This scheme can also be scaled macroscopically using contemporary techniques and is shown to significantly surpass the shot-noise limit (SNL) in the presence of realistic losses. Finally, the physical resources that enable quantum-enhanced interferometry is studied by analyzing the quantum Fisher information (QFI) using the first quantization and the second quantization formalisms of quantum mechanics. It is shown that increasing the intra-mode correlations of an interferometer (as quantified by the second order Glauber coherence function) can be conducive to attaining a quantum advantage in phase estimation; whereas, introducing mode entanglement can reduce phase sensitivity. Using the first quantization description, we derive a formula for the QFI that shows explicitly how the Heisenberg scaling term depends on particle entanglement.
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