Estimation of a global parameter defined as a weighted linear combination of unknown multiple parameters can be enhanced by using quantum resources. Advantageous quantum strategies may vary depending on the weight distribution, requiring the study of an optimal scheme achieving a maximal quantum advantage for a given sensing scenario. In this work, we propose a Heisenberg-limited distributed quantum phase sensing scheme using Gaussian states for an arbitrary distribution of the weights with positive and negative signs. The proposed scheme exploits entanglement of Gaussian states only among the modes assigned with equal signs of the weights, but separates the modes with opposite weight signs. We show that the estimation precision of the scheme exhibits the Heisenberg scaling in the mean photon number and it can be achieved by injecting two single-mode squeezed states into the respective linear beam-splitter networks and performing homodyne detection on them in the absence of loss. Interestingly, the proposed scheme is proven to be optimal for Gaussian probe states with zero displacement. We also provide an intuitive understanding of our results by focusing on the two-mode case, in comparison with the cases using non-Gaussian probe states. We expect this work to motivate further studies on quantum-enhanced distributed sensing schemes considering various types of physical parameters with an arbitrary weight distribution.
Estimation of a global parameter defined as a weighted linear combination of unknown multiple parameters can be enhanced by using quantum resources. Advantageous quantum strategies may vary depending on the weight distribution, requiring the study of an optimal scheme achieving a maximal quantum advantage for a given sensing scenario. In this work, we propose a Heisenberg-limited distributed quantum phase sensing scheme using Gaussian states for an arbitrary distribution of the weights with positive and negative signs. The proposed scheme exploits entanglement of Gaussian states only among the modes assigned with equal signs of the weights, but separates the modes with opposite weight signs. We show that the estimation precision of the scheme exhibits the Heisenberg scaling in the mean photon number and it can be achieved by injecting two single-mode squeezed states into the respective linear beam-splitter networks and performing homodyne detection on them in the absence of loss. Interestingly, the proposed scheme is proven to be optimal for Gaussian probe states with zero displacement. We also provide an intuitive understanding of our results by focusing on the two-mode case, in comparison with the cases using non-Gaussian probe states. We expect this work to motivate further studies on quantum-enhanced distributed sensing schemes considering various types of physical parameters with an arbitrary weight distribution.
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.
Quantum metrology enables parameter estimation beyond classical limits by exploiting nonclassical resources such as squeezing and entanglement. In distributed quantum sensing, Heisenberg scaling has been extended from 1/N2 to 1/(NM)2 through entanglement across both particles and spatial modes, where N denotes the photon number and M the number of spatially distributed modes. However, the overall sensitivity has remained limited to linear scaling with the number of measurement repetitions R. Here, we show that exploiting entanglement across temporal modes via time-domain multiplexing enables a scaling advantage with respect to R. As a result, the sensitivity can asymptotically approach simultaneous Heisenberg scaling in photons, spatial modes, and repetitions, yielding an overall sensitivity approaching Δ2φ∝1/(NMR)2. Using the Bogoliubov transformation formalism, we prove the optimality of the protocol within the class of Gaussian states and show that the scaling is realizable via homodyne detection and maximum-likelihood estimation. We further show that the advantage persists under optical loss and propose an experimentally feasible loop-based photonic sensing scheme. Our results open a route to incorporating time-multiplexing techniques into quantum metrology.
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.
Private distributed quantum sensing aims to estimate an authorized collective parameter while preventing independent estimation of individual local parameters. Here, we analytically characterize Gaussian quantum networks satisfying this perfect local privacy condition. Using a graph representation of the Gaussian pairing matrix, we show that two-mode squeezed vacuum states constitute the essential building blocks of privacy-preserving Gaussian states. We then analytically derive the optimal sensitivity under perfect local privacy and determine a Gaussian probe that achieves Heisenberg scaling with respect to both the photon number and the number of sensing modes. Using local homodyne measurements and maximum-likelihood estimation, we numerically verify quantum-enhanced sensitivity beyond the shot-noise limit while preserving perfect local privacy. We further show that the local-privacy condition is preserved under arbitrary phase-independent quantum channels, including optical loss. Our results provide a general framework for constructing and optimizing privacy-preserving continuous-variable quantum sensing networks.
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.
Distributed quantum metrology has drawn intense interest as it outperforms the optimal classical counterparts in estimating multiple distributed parameters. However, most schemes so far have required entangled resources consisting of photon numbers equal to or more than the parameter numbers, which is a fairly demanding requirement as the number of nodes increases. Here, we present a distributed quantum sensing scenario in which quantum-enhanced sensitivity can be achieved with fewer photons than the number of parameters. As an experimental demonstration, using a two-photon entangled state, we estimate four phases distributed 3 km away from the central node, resulting in a 2.2 dB sensitivity enhancement from the standard quantum limit. Our results show that the Heisenberg scaling can be achieved even when using fewer photons than the number of parameters. We believe our scheme will open a pathway to perform large-scale distributed quantum sensing with currently available entangled sources.