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arXiv:2608.02977

Distributed Phase Sensing with Multiphoton States in Optical Interferometry

Subhrajit Modak, Danilo Triggiani, Cosmo Lupo

quant-phSubmitted Aug 4, 2026• Announced Aug 5, 2026• 0 citations
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Abstract

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 $N$-photon states distributed across 2$N$ 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 $N$ 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.

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17 pages, 4 figures

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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.
The quantum statistical fluctuations of electromagnetic fields establish a limit, known as the shot-noise limit, on the sensitivity of optical measurements performed with classical technologies. However, quantum technologies are not constrained by this shot-noise limit. In this regard, the possibility of using every photon produced by quantum sources of light to estimate small physical parameters, beyond the shot-noise limit, constitutes one of the main goals of quantum optics. Here, we experimentally demonstrate a scalable protocol for quantum-enhanced optical phase estimation across a broad range of phases, with neither pre- nor post-selected measurements. This is achieved through the efficient design of a source of spontaneous parametric downconversion in combination with photon-number-resolving detection. The robustness of two-mode squeezed vacuum states against loss allows us to outperform schemes based on N00N states, in which the loss of a single photon is enough to remove all phase information from a quantum state. In contrast to other schemes that rely on N00N states or conditional measurements, the sensitivity of our technique could be improved through the generation and detection of high-order photon pairs. This unique feature of our protocol makes it scalable. Our work is important for quantum technologies that rely on multiphoton interference such as quantum imaging, boson sampling, and quantum networks.
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.
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.
Distributed quantum metrology can improve the sensitivity in estimating arbitrary positive-weight linear combinations beyond the classical limit. It is known that multimode entangled states are crucial for distributed metrology. However, strategies to efficiently engineer multimode entangled states and harness them in distributed quantum metrology remains largely unexplored. In this work, we introduce a concept for distributed quantum sensing utilizing nonlinear optical networks. Unlike linear optical networks, nonlinear optical networks utilize optical parametric amplifiers instead of beam splitters for beam splitting. Here, we employ both simultaneous estimation (global) and separate estimation (local) methods to estimate linear combinations and derive the quantum Cram\'er-Rao bound for the two schemes, respectively. Our findings indicate that the sensitivity of the global strategy surpasses that of the local strategy and reaches the Heisenberg limit. Notably, direct intensity detection saturates the quantum Cram\'er-Rao bound. Furthermore, we characterize loss effects on measurement sensitivity, revealing the reduction in sensitivity with increasing losses and establishing the conditions for surpassing the standard quantum limit. This scheme can be directly applied to applications such as biological imaging and quantum information processing.
We derive a bound on the ability of a linear-optical network to estimate a linear combination of independent phase shifts by using an arbitrary nonclassical but unentangled input state, thereby elucidating the quantum resources required to obtain the Heisenberg limit with a multiport interferometer. Our bound reveals that while linear networks can generate highly entangled states, they cannot effectively combine quantum resources that are well distributed across multiple modes for the purposes of metrology: In this sense, linear networks endowed with well-distributed quantum resources behave classically. Conversely, our bound shows that linear networks can achieve the Heisenberg limit for distributed metrology when the input photons are concentrated in a small number of input modes, and we present an explicit scheme for doing so.
In the field of quantum precision measurement, enhancing phase sensitivity is crucial for various applications, including quantum metrology and quantum sensing technologies. We theoretically investigate the improvement in phase sensitivity and quantum Fisher information achieved through multiphoton subtraction operations at the output port of an SU(1,1) interferometer under conditions of photon loss. We use vacuum and coherent states as the inputs and detect the outputs by intensity detection. The results indicate that internal photon losses within the SU(1,1) interferometer have a more significant impact on the phase sensitivity compared to external photon losses. Moreover, increasing the number of photon subtractions m effectively enhances both the phase sensitivity and the quantum Fisher information. Notably, even under conditions of severe photon loss, the multiphoton subtraction operations can enable the phase sensitivity to surpass the standard quantum limit, approaching both the Heisenberg limit and the quantum Cramér-Rao bound. This study provides a new theoretical basis for enhancing the phase sensitivity in the SU(1,1) interferometer.