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arXiv:2509.08230

Scalable Network of Mach-Zehnder Interferometers with a Single Entangled Resource

Zhihui Yan, Yanni Feng, Luca Pezzè, Zhaoqing Zeng, Jingxu Ma, Xiaoyu Zhou, Augusto Smerzi, Xiaojun Jia, Kunchi Peng

Published Sep 10, 2025• 0 citations
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Abstract

Distributed quantum sensing exploits entanglement to enhance the estimation of multiple parameters across a network of spatially-separated sensors, achieving sensitivities beyond the classical limit. Potential applications cover a plethora of technologies, from precision navigation to biomedical imaging and environmental monitoring. However, practical implementations are challenged by the complex optimal distribution of entanglement throughout the sensing nodes, which affects scalability and robustness. Here we demonstrate a reconfigurable network of Mach-Zehnder interferometers entangled via a single shared squeezed-vacuum resource. We achieve joint noise suppression of $4.36 \pm 0.35$ dB below the standard quantum limit at the phase-uncertainty level of $10^{-9}$ . Furthermore, after full optimization in the low-intensity regime, we demonstrate a crossover from the standard quantum limit to the Heisenberg limit. The network estimates arbitrary linear combinations of phases, saturates the quantum Cramer-Rao bound in the ideal case, remains robust under realistic photon losses, and scales favorably with the number of sensors.

DOI
10.48550/arxiv.2509.08230

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Distributed quantum sensing exploits entanglement to enhance the estimation of multiple parameters across a network of spatially-separated sensors, achieving sensitivities beyond the classical limit. Potential applications cover a plethora of technologies, from precision navigation to biomedical imaging and environmental monitoring. However, practical implementations are challenged by the complex optimal distribution of entanglement throughout the sensing nodes, which affects scalability and robustness. Here we demonstrate a reconfigurable network of Mach-Zehnder interferometers entangled via a single shared squeezed-vacuum resource. We achieve joint noise suppression of 4.36±0.354.36 \pm 0.354.36±0.35 dB below the standard quantum limit at the phase-uncertainty level of 10−910^{-9}10−9 . Furthermore, after full optimization in the low-intensity regime, we demonstrate a crossover from the standard quantum limit to the Heisenberg limit. The network estimates arbitrary linear combinations of phases, saturates the quantum Cramer-Rao bound in the ideal case, remains robust under realistic photon losses, and scales favorably with the number of sensors.
We study an entangled distributed quantum sensing scheme based on an array of ddd Mach-Zehnder interferometers (MZIs) for the estimation of relative phase shifts. The scheme uses ddd coherent states and a single squeezed-vacuum state that is distributed among the MZIs by a quantum circuit (QC). The protocol can be optimized analytically: it overcomes the shot-noise limit and reaches the Heisenberg limit with respect to the average total number of probe particles, n‾T{\overline{n}}_{T}nT​, for the estimation of arbitrary linear combinations of the ddd phases. We compare the entangled strategy with a separable one that uses ddd coherent and ddd squeezed-vacuum states and the same n‾T{\overline{n}}_{T}nT​. The entangled strategy benefits for a substantial reduction of resource overhead and can achieve a maximum gain equal to ddd when using the same total squeezed-light intensity as the separable strategy. Interestingly, the entangled strategy using a single squeezed-vacuum state can reach the same sensitivity as the separable strategy that uses ddd copies of the same state. Finally, given a random choices of the QC, we identify the optimal linear combination of the phases that can be estimated with maximum sensitivity. Our scheme paves the ways for a variety of applications in distributed quantum sensing with photonic and atomic interferometers.
We present a theoretical study of distributed quantum sensing using a reconfigurable network of NNN Mach--Zehnder interferometers to simultaneously estimate weighted linear combinations of relative phase shifts. The proposed scheme relies on injecting a Perelomov coherent state, generated from the SU(1,1) group representation, into one port of a passive quantum circuit while the remaining modes are initialized in the vacuum state, thereby establishing non-trivial quantum correlations among the sensors. Based on multi-parameter quantum estimation theory, our analysis shows that this entangled strategy outperforms the standard shot-noise limit and asymptotically achieves the Heisenberg limit. Furthermore, we highlight the crucial role of the first beam splitter's asymmetry; specifically, the optimal transmission coefficient that maximizes sensitivity corresponds to its maximum imbalance. Compared to a separable approach utilizing the same total average photon number, the entangled strategy yields a more efficient use of quantum resources, guaranteeing an accuracy gain between 111 and NNN. This demonstrates that the entangled strategy is systematically superior to its separable counterpart. Ultimately, these results illustrate the potential of entangled distributed quantum sensor networks for high-precision measurements and provide clear guidelines for optimizing interferometric architectures in quantum metrology.
We study a sensor network of distributed Mach-Zehnder interferometers (MZIs) for the parallel (simultaneous) estimation of an arbitrary number d≥1d \geq 1d≥1 of phase shifts. The scheme uses a squeezed-vacuum state that is split between ddd modes by a quantum circuit (QC). Each output mode of the QC is the input of one of ddd MZIs, the other input of each MZI being a coherent state. In particular, i{\it i}i) we identify the optimal configuration of the sensor network that allows the estimation of any linear combination of the ddd phases with maximal sensitivity. The protocol overcomes the shot-noise limit and reaches Heisenberg scalings with respect to the total average number of particles in the overall probe state, the multiphase estimation only requiring local photocounting. Furthermore, the parallel multiphase estimation overcomes optimal separable strategies for the estimation of any linear combination of the phases: the sensitivity gain being a factor ddd, at most. Viceversa, ii{\it ii}ii) given a specific QC, we identify the optimal linear combination of the phases that maximizes the sensitivity and show that results are robust against random choices of the QC. Our scheme paves the ways to a variety of applications in distributed quantum sensing.
We study a phase estimation protocol based on the distribution of a single squeezed state among an array of Mach-Zehnder interferometers (MZIs). The fundamental component of our scheme is the quantum circuit (QC), a linear network that optimally distributes the squeezing generated at one of its inputs among the d MZIs, where d unknown parameters θ1, . . . , θd are then imprinted and the number of photons at the outputs finally measured. For any given linear combination of the parameters, we can optimize the QC and achieve sub-shot-noise sensitivity. Our parallel strategy, based on the mode-entanglement created by the QC, outperforms the rival and more common sequential strategy, in which the same unknown parameters are estimated independently. It also saturates the ultimate sensitivity bound, the quantum Cramer-Rao bound, in a relevant regime of parameters, thus constituting an optimal estimation method in that regime.
We study the multiparameter sensitivity bounds of a network of d spatially distributed Mach-Zehnder interferometers (MZIs). A generic single nonclassical state is mixed with d-1 vacuums to create a d-mode entangled state, each mode entering one input port of a MZI, while a coherent state enters its second port. We show that local measurements, independently performed on each MZI, are sufficient to provide a sensitivity saturating the quantum Cramér-Rao bound. The sensor network can overcome the shot noise limit for the estimation of arbitrary linear combinations of the d phase shifts, provided that the nonclassical probe state has an antisqueezed quadrature variance. We compare the sensitivity bounds of this sensor with that achievable with d independent MZIs, each probed with a nonclassical state and a coherent state. We find that the d independent interferometers can achieve the same sensitivity of the entangled protocol, but at the cost of using d nonclassical states rather than a single one. Considering the same average number of particles per shot in the two protocols, n[over ¯]_{T}, we find analytically a sensitivity scaling 1/n[over ¯]_{T}^{2} for the entangled case that provides a gain factor d with respect to the separable case where the sensitivity scales as d/n[over ¯]_{T}^{2}. We have numerical evidence that the gain factor d is also found when fixing the total average number of particles, namely when optimizing with respect to the number of repeated measurements.
In recent years, distributed quantum sensing has gained interest for a range of applications requiring networks of sensors, from global-scale clock synchronization to high energy physics. In particular, a network of entangled sensors can improve not only the sensitivity beyond the shot noise limit, but also enable a Heisenberg scaling with the number of sensors. Here, using bright entangled twin beams, we theoretically and experimentally demonstrate the detection of a linear combination of two distributed phases beyond the shot noise limit with a truncated SU(1,1) interferometer. Specifically, we show a quantum noise reduction of 1.7 ± 0.3 dB below what is possible with the corresponding classical configuration. Additionally, we theoretically extend the use of a truncated SU(1,1) interferometer to a multi-phase-distributed sensing scheme that leverages entanglement as a resource to achieve a quantum improvement in the scaling with the number of sensors in the network. Our results pave the way for developing quantum-enhanced sensor networks that can achieve an entanglement-enhanced sensitivity.