Distributed quantum sensing network has the potential of enhancing the precision in estimating a global function of local parameters by utilizing an entangled probe, compared with that achieved with separable probes. This advantage is often characterized as a quadratic improvement of the quantum Cramér-Rao bound (QCRB). This argument is incomplete in that QCRB assumes a team of all-powerful sensors that can perform arbitrary joint measurements allowed by quantum mechanics. An immediate question arises as to whether such an advantage persists for isolated sensors with physically motivated constraints in their measurement strategies. In this paper, we first consider local operations and classical communication (LOCC) strategies and prove that the QCRB is indeed asymptotically attainable for arbitrary pure probe states, by extending previous work on single-parameter estimation [1]. We further numerically analyze a more restricted scenario where the sensors can only make independent local measurements, and provide evidence that the QCRB is not informative enough for comparing different probe states.
Distributed quantum sensing network has the potential of enhancing the precision in estimating a global function of local parameters by utilizing an entangled probe, compared with that achieved with separable probes. This advantage is often characterized as a quadratic improvement of the quantum Cramér-Rao bound (QCRB). This argument is incomplete in that QCRB assumes a team of all-powerful sensors that can perform arbitrary joint measurements allowed by quantum mechanics. An immediate question arises as to whether such an advantage persists for isolated sensors with physically motivated constraints in their measurement strategies. In this paper, we first consider local operations and classical communication (LOCC) strategies and prove that the QCRB is indeed asymptotically attainable for arbitrary pure probe states, by extending previous work on single-parameter estimation [1]. We further numerically analyze a more restricted scenario where the sensors can only make independent local measurements, and provide evidence that the QCRB is not informative enough for comparing different probe states.
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.
Abstract The quantum Cramér–Rao bound (QCRB) provides an ultimate precision limit allowed by quantum mechanics in parameter estimation. Given any quantum state dependent on a single parameter, there is always a positive-operator valued measurement (POVM) saturating the QCRB. However, the QCRB-saturating POVM cannot always be implemented efficiently, especially in multipartite systems. In this paper, we show that the POVM based on local operations and classical communication is QCRB-saturating for arbitrary pure states or rank-two mixed states with varying probability distributions over fixed eigenbasis. Local measurements without classical communication, however, is not QCRB-saturating in general.
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 investigate strategies for reaching the ultimate limit on the precision of frequency estimation when the number of probes used in each run of the experiment is fixed. That limit is set by the quantum Cramér-Rao bound (QCRB), which predicts that the use of maximally entangled probes enhances the estimation precision, when compared with the use of independent probes. However, the bound is only achievable if the statistical model used in the estimation remains identifiable throughout the procedure. This in turn sets different limits on the maximal sensing time used in each run of the estimation procedure, when entangled and independent probes are used. When those constraints are taken into account, one can show that, when the total number of probes and the total duration of the estimation process are counted as fixed resources, the use of entangled probes is, in fact, disadvantageous when compared with the use of independent probes. In order to counteract the limitations imposed on the sensing time by the requirement of identifiability of the statistical model, we propose a time-adaptive strategy, in which the sensing time is adequately increased at each step of the estimation process, calculate an attainable error bound for the strategy, and discuss how to optimally choose its parameters in order to minimize that bound. We show that the proposed strategy leads to much better scaling of the estimation uncertainty with the total number of probes and the total sensing time than the traditional fixed-sensing-time strategy. We also show that, when the total number of probes and the total sensing time are counted as resources, independent probes and maximally entangled ones have now the same performance, in contrast to the nonadaptive strategy, where the use of independent is more advantageous than the use of maximally entangled ones. Published by the American Physical Society 2024
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.
Breaking the standard quantum limit in the sensing of parameters at different spatial locations, such as in a quantum network, is of great importance. Using the framework of quantum Fisher information, many strategies based on squeezed quantum probes and multipath multiphoton or multiqubit entangled states have been considered. In this context there is always the question of what the simplest measurement is that would saturate the quantum Cramér-Rao bound (QCRB). The simplest quantity to measure would be characteristics of photon flux or population distribution in the case of qubits. Previous studies have shown that the error sensitivity in SU(1,1) interferometry, also known by several other terms such as nonlinear interferometry and time-reversed measurements, does saturate the QCRB for single parameters such as phase, displacement, and loss. In this work we reveal the great utility of generalized SU(1,1) interferometry in distributed sensing. Generalized SU(1,1) interferometry is a combination of SU ( m ) and SU(1,1) elements, where m is the number of nodes in the network. The SU ( m ) element is used to produce distributed entanglement starting from a squeezed photonic or matter probe. We demonstrate how error sensitivity measurement or more precisely the method of moment sensitivity at just one output port can saturate or nearly saturate the QCRB and thus results in Heisenberg sensitivity of network sensing.