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arXiv:2602.03727

Distributed Phase-Insensitive Displacement Sensing

Piotr T. Grochowski, Matteo Fadel, Radim Filip

quant-phSubmitted Feb 3, 2026• Announced Feb 4, 2026• 0 citations
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Abstract

Distributed quantum sensing leverages quantum correlations among multiple sensors to enhance the precision of parameter estimation beyond classical limits. Most existing approaches target phase estimation and rely on a shared phase reference between the signal and the probe, yet many relevant scenarios deal with regimes where such a reference is absent, making the estimation of force or field amplitudes the main task. We study this phase-insensitive regime for bosonic sensors that undergo identical displacements with common phases randomly varying between experimental runs. We derive analytical bounds on the achievable precision and show that it is determined by first-order normal correlations between modes in the probe state, constrained by their average excitations. These correlations yield a collective sensitivity enhancement over the standard quantum limit, with a gain that grows linearly in the total excitation number, revealing a distributed quantum advantage even without a global phase reference. We identify families of multimode states with definite joint parity that saturate this limit and can be probed efficiently via local parity measurements already demonstrated or emerging in several quantum platforms. We further demonstrate that experimentally relevant decoherence channels favor two distinct sensing strategies: splitting of a single-mode nonclassical state among the modes, which is robust to loss and heating, and separable probes, which are instead resilient to dephasing and phase jitter. Our results are relevant to multimode continuous platforms, including trapped-ion, solid-state mechanical, optomechanical, superconducting, and photonic systems.

Comments
8+16 pages, 1+2 figures

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Distributed quantum sensing leverages quantum correlations among multiple sensors to enhance the precision of parameter estimation beyond classical limits. Most existing approaches target phase estimation and rely on a shared phase reference between the signal and the probe, yet many relevant scenarios deal with regimes where such a reference is absent, making the estimation of force or field amplitudes the main task. We study this phase-insensitive regime for bosonic sensors that undergo identical displacements with common phases randomly varying between experimental runs. We derive analytical bounds on the achievable precision and show that it is determined by first-order normal correlations between modes in the probe state, constrained by their average excitations. These correlations yield a collective sensitivity enhancement over the standard quantum limit, with a gain that grows linearly in the total excitation number, revealing a distributed quantum advantage even without a global phase reference. We identify families of multimode states with definite joint parity that saturate this limit and can be probed efficiently via local parity measurements already demonstrated or emerging in several quantum platforms. We further demonstrate that experimentally relevant decoherence channels favor two distinct sensing strategies: splitting of a single-mode nonclassical state among the modes, which is robust to loss and heating, and separable probes, which are instead resilient to dephasing and phase jitter. Our results are relevant to multimode continuous platforms, including trapped-ion, solid-state mechanical, optomechanical, superconducting, and photonic systems.
A naturally multiparameter sensing scenario arises in spatially distributed networks of quantum sensors, where each node probes a distinct parameter of interest. In such settings, the central challenges are to achieve quantum-enhanced sensitivity under well-defined resource constraints and to explicitly identify measurement strategies that attain the ultimate precision limits. Here, we analytically demonstrate quantum-enhanced sensitivity for a broad class of distributed quantum probes designed to simultaneously estimate a complete set of unknown reference-relative phase differences encoded locally. Although such locally encoded parameters can, in principle, be optimally estimated using separable strategies, our distributed protocol offers distinct advantages. In particular, for fixed network size and identical bare probe Hamiltonians, it is energetically more efficient and exhibits stroboscopic disentanglement at specific evolution times, enabling the accumulated pairwise information to be coherently mapped onto a specific subsystem. Consequently, all set of pairwise reference-relative phase differences can be accessed through a single collective measurement without requiring independent local estimation. We further construct the corresponding optimal measurement strategies and show that, within this class of probes, the ultimate multiparameter precision limits—the Holevo and quantum Cramér-Rao bounds—are saturable. We illustrate the framework with two concrete applications: simultaneous gravimetry across spatially separated locations and estimation of distributed coupling strengths. Feasibility analyses indicate that the proposed distributed quantum-enhanced sensing protocols operate within experimentally accessible regimes.
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Quantum metrology enables sensitivity to approach the limits set by fundamental physical laws. Even a single continuous mode offers enhanced precision, with the improvement scaling with its occupation number. Due to their high information capacity, continuous modes allow for the engineering of quantum non-Gaussian states, which not only improve metrological performance but can also be tailored to specific experimental platforms and conditions. Recent advancements in control over continuous platforms operating in the quantum regime have renewed interest in sensing weak forces, also coupling to massive macroscopic objects. In this work, we investigate a force-sensing scheme where a physical process completely randomizes the direction of the induced phase-space displacement, and the unknown force strength is inferred through excitation-number-resolving measurements. We find that N-spaced states, where only every N^{th} Fock state occupation is nonzero, approach the achievable sensing bound. Additionally, non-Gaussian states are shown to be more resilient against decoherence than their Gaussian counterparts with the same occupation number. While Fock states typically offer the best protection against decoherence, we uncover a transition in the metrological landscape-revealed through a tailored decoherence-aware Fisher-information-based reward functional-where experimental constraints favor a family of number-squeezed Schrödinger cat states. Specifically, by implementing quantum optimal control in a minimal spin-boson system, we identify these states as maximizing force sensitivity under lossy dynamics and finite system controllability. Our results provide a pathway for enhancing force sensing in a variety of continuous quantum systems, ranging from massive systems like mechanical oscillators to massless systems such as quantum light and microwave resonators.
We investigate interferometric phase-estimation using separable photon inputs that evolve into number-path entangled states through linear optical networks, followed by photon-number-resolving detection. A simple analytical expression for the classical Fisher information at zero phase is derived for arbitrary NNN-photon states distributed across 2NNN optical modes, partitioned into phase-encoding and reference blocks. Among all possible photon distributions between these blocks, the balanced configuration maximizes the phase sensitivity for every photon number NNN and uniquely exhibits a phase-independent response. The achievable sensitivity degrades monotonically with increasing asymmetry in the photon distribution. We further investigate the robustness of the protocol in the presence of realistic photon loss and extend the analysis to distributed architectures with multiple receivers. In the low photon-flux regime, vacuum fluctuations fundamentally limit local quadrature measurements, whereas nonlocal photon-number-resolving measurements exploit multiphoton interference to mitigate loss-induced sensitivity degradation. Together, these results establish a scalable framework for quantum-enhanced distributed multimode metrology.
Quantum sensing has emerged as a promising solution for precision measurement, and distributed quantum sensing enables multiple parameters to be measured across a network. Pushing distributed quantum sensing from the single to multimodal parameters is crucial for a complete and deep understanding of a complex system. To address this challenge, we have devised a high-sensitivity distributed quantum sensing of the multimodal parameters with a continuous-variable multipartite Greenberger–Horne–Zeilinger entangled state. All the sensors are equipped with Mach–Zehnder interferometers to couple the lasers with entangled modes. In such a quantum network, not only joint but also individual measurements of radio frequency phases and amplitudes as well as ultrasonic vibration signals are achieved beyond the standard quantum limit, showcasing the quantum Fisher information of the order of 10 16 . These results render multimodal parameters a promising contender in quantum metrology.
Phase-insensitive optical amplifiers uniformly amplify each quadrature of an input field and are of both fundamental and technological importance. We find the quantum limit on the precision of estimating the gain of a quantum-limited phase-insensitive amplifier using a multimode probe that may also be entangled with an ancilla system. In stark contrast to the sensing of loss parameters, the average photon number N and number of input modes M of the probe are found to be equivalent and interchangeable resources for optimal gain sensing. All pure-state probes whose reduced state on the input modes to the amplifier is diagonal in the multimode number basis are proven to be quantum optimal under the same gain-independent measurement. We compare the best precision achievable using classical probes to the performance of an explicit photon-counting-based estimator on quantum probes and show that an advantage exists even for single-photon probes and inefficient photodetection. A closed-form expression for the energy-constrained Bures distance between two product amplifier channels is also derived.