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.
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.
We study a sensor network of distributed Mach-Zehnder interferometers (MZIs) for the parallel (simultaneous) estimation of an arbitrary number d≥1 of phase shifts. The scheme uses a squeezed-vacuum state that is split between d modes by a quantum circuit (QC). Each output mode of the QC is the input of one of d MZIs, the other input of each MZI being a coherent state. In particular, i) we identify the optimal configuration of the sensor network that allows the estimation of any linear combination of the d 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 d, at most. Viceversa, 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 an entangled distributed quantum sensing scheme based on an array of d Mach-Zehnder interferometers (MZIs) for the estimation of relative phase shifts. The scheme uses d 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, nT, for the estimation of arbitrary linear combinations of the d phases. We compare the entangled strategy with a separable one that uses d coherent and d squeezed-vacuum states and the same nT. The entangled strategy benefits for a substantial reduction of resource overhead and can achieve a maximum gain equal to d 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 d 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 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 present a theoretical study of distributed quantum sensing using a reconfigurable network of N 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 1 and N. 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.
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±0.35 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.
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.