Quantum metrology provides a route to conquer the measurement sensitivity limitation imposed by quantum noise in sensing devices [1–3]. As a unique quantum resource, squeezed and entangled states present the feature of quantum correlation, which can suppress statistical scaling of errors n−12 in interferometric measurements, and thereby enhance measurement precision, until approaching and reaching the Heisenberg bound 1/n [4–6]. It holds particular relevance to the power constraint introduced by optical damage or quantum measurement back-action, and an ultra-weak signal that is thoroughly submerged in inherent quantum noise. Laser radar is a prime case of ultra-weak signal sensing, and has an urgent demand for measurement sensitivity beyond the shot noise limit [7,8].

Since quantum metrology was recognized in the 1980s [9], both the theoretical scheme and experimental technology of quantum metrology have advanced dramatically [10,11]. In this regard, numerous demonstrations of quantum-enhanced metrology have been reported, for example, to enhance the performance of gravitational wave detection [12–15], magnetic field detection [16,17], and biological measurement [18] and imaging [19,20], and are expected to be applied in more practical areas. Furthermore, quantum sensing has been upgraded from a single sensor to a sensor network to realize spatially distributed sensing of multi-parameters. Theoretical schemes of distributed quantum sensing have been proposed in both the discrete variable (DV) domain [21–24] and continuous variable (CV) domain [25–29]. By utilizing entangled photons [30,31] or entangled states [32], the average optical phase was estimated experimentally with precision beyond what is achievable with seperable probes. Very recently, Xia et al. extended the application of distributed quantum sensing to a radio-frequency signal based on tripartite CV entanglement [33], and subsquently demonstrated the supervised learning assisted by an entangled sensor network for quantum-enhanced data classification [34]. Taking a typical sensing mission for phased-array radar (PAR) as an example, the fundamental requirement is to be able to discover a target and acquire its position relative to the sensor, not only to discriminate whether the target is present or not. However, for space positioning of a moving target with PAR, it is necessary to estimate three angles of arrival simultaneously, which cannot be implemented by three sensor nodes [33]. Moreover, estimating the angle of arrival and its derivative with high accuracy is necessary to implement target ranging, requiring at least four sensor nodes. In addition, the power limit of laser radar that comes from the endless pursuit of the detecting range is ultimately determined by noise performance. For a classical system, poor noise performance imposes restrictions on the intensity of the return signal, and that in turn limits the detecting range, which has the potential to be improved to some extent via squeezed light. To date, quantum positioning [35,36] and quantum ranging [37] protocols based on multipartite entanglement have been proposed theoretically, but none of these protocols have been demonstrated experimentally.

In this paper, we experimentally demonstrate a quantum sensor network empowered by four-partite CV entanglement, optimizing the quantum advantage in global parameter estimation including the higher order derivative of radio-frequency signal phase. The global parameter of four sensors φ=∑j=14βjφj, where βj is the weights for different sensing problems, is estimated in our experiment, achieving quantum noise reduction of 6.0dB±0.2dB. The quantum sensor network is configured with an equally weighted distribution to maximize the quantum advantage for measuring the angle of arrival of the return signal along three independent dimensions, showing quantum positioning. The sensor network is reconfigured with a weighted distribution of −11/6:3:−3/2:1/3 (2:−5:4:−1) to maximize the quantum advantage of the first (second) derivative of PAR phase, demonstrating quantum ranging.

PAR consists of multiple sensing elements appropriately arranged in space, whose amplitude and phase can be independently controlled to provide full space scanning without the influence of receiving sensitivity [38]. Space positioning can be realized by measuring the angle of arrival of the return signal in three dimensions [39], which can be inferred from the phase difference between two sensor nodes of each dimension [33]. Thus, to achieve space positioning, it requires at least four sensor nodes. As the other key capacity, target ranging, dependent on the angle of arrival and its derivative of the return signal in one dimension, can be obtained by configuring a diverse weighted distribution among these sensor nodes. As such, a reconfigurable beam splitter network (BSN), in terms of minimum estimation variance, distributes the needed portion of a squeezed state to these sensor nodes, leading to the optimal quantum advantage in different cases, which serves as the estimation of the angle of arrival and its first (second) derivative beyond the shot noise limit. There are two cases of sensor network topology: equilateral star topology for space positioning, and linear topology for target ranging. Compared with the demonstration of only an average outcome among all of sensor nodes, the richer physics and more intriguing application scenario of a distributed quantum sensor network are unveiled by our work.

In the quantum positioning case, three sensor nodes are arranged in an equilateral star topology, while the central node is situated at the triangle center; the schematic illustration of quantum positioning is represented in Fig. 1. When the target occurs, we can acquire three independent angles of arrival in three-dimensional space, and the target is positioned at the intersectional region of three angles of arrival. Critically, the distributed quantum sensor network must be optimized to generate minimum outcome uncertainty in a given distributed sensing problem. Here, to meet the requirement for simultaneous estimation of three angles of arrival with minimum positioning uncertainty, dependent on the phase difference between two sensor nodes with the function of θ=arccosλ(φj−φ1)2πΔx(j=2,3,4), a CV quantum sensor network with an equally weighted distribution is constructed. Under the circumstances, the additional noise coupling vanishes, leading to the minimum estimation variance of the PAR phase that depends on the initial squeezing factor. The equally weighted distribution is experimentally verified when the average variance for PAR phase φ=∑j=14βjφj(β4=β3=β2=1,β1=−1) reaches minimum. At this point, the uncertainties of three phase differences φ2−φ1, φ3−φ1, and φ4−φ1 represent the best compromise, and will have a boosted performance for target positioning.

A schematic illustration of our experimental setup is represented in Fig. 2. The squeezed state of light is generated by a below-threshold optical parametric amplifier (OPA), which has been shown in our previous publication in details [40]. The only difference is that the relative phase between the seed and pump beam is controlled to zero to ensure our OPA operates in parametric amplification condition, generating the quadrature phase squeezed state [δ2(Y^)=e−2r; r is the squeezing factor]. The BSN, which consists of three pairs of half-wave plates and polarization beam splitters, can be set to any splitting ratio by tuning the angle of half-wave plates. After the BSN, four partite CV entangled states with correlation variance [δ2(β1Y^1+β2Y^2+β3Y^3+β4Y^4)=e−2r; Y^i(i=1,2,3,4) is the phase quadrature of each partite] are generated and distributed to four sensor nodes. Each sensor node entails an electro-optical modulator (EOM) to simulate the return signal from the target, and homodyne detection (HOM) to discern the return signal with sensitivity beyond the shot noise limit. In each node, the relative phase between the local oscillator and sensing beam is controlled to ±π/2 by utilizing the interference signal as the error signal. The EOM that has high radio frequency to photonics conversion efficiency, is an excellent simulator of PAR. Four EOMs are independently driven by four clock-synchronized function generators with the same modulation frequency of 3 MHz, but the amplitude and phase are independently manipulated to generate four independent displacements αj(j=1,2,3,4) on the squeezed phase quadrature [33]. The output of four homodyne detectors through the same four bandpass filters from 1.8 MHz to 4.5 MHz is demodulated with 3 MHz signals. The demodulation signals are clock synchronized with the corresponding return signals, but their phases can be independently manipulated to compensate for the delay difference of the processing circuit. The time-domain data from four mixers are collected and postprocessed to derive the estimated parameters under different weights, and demonstrate diverse quantum-enhanced sensing applications.

The noise variances of the measured PAR phase are limited by the initial squeezing factor and system loss. It is worth noting that the parametric amplification process (for the generation of a quadrature phase squeezed state) intensifies the noise coupling between the seed and pump field, which usually results in the degradation of the squeezing factor [41]. As the pump factor is increased, the noise coupling is enhanced. Thus, we need to make a trade-off between the pump factor and noise coupling, both of which are key factors affecting the squeezing factor. Here, the OPA operates at the pump factor of 2/2, with quantum noise reduction of 8.0±0.2dB generated, providing a superior quantum resource for the downstream experiment.

During the experiment, delicate mode-matching with interference efficiency of 99.8% is accomplished. The optimal weights are scheduled in terms of the different mission requirements, and the number of optical elements is as small as possible, reducing the additional noise coupling to the maximum extent and minimizing the estimation variance of diverse physical scenarios. On the other hand, the phase of the demodulation signal is carefully optimized to suppress the noise coupling of anti-squeezing quadrature into squeezing quadrature as much as possible.

Suppose that the PAR operates in single-transmitter and multi-receiver mode; we show the experimental results of quantum-enhanced PAR in two sensing tasks. PAR1 serves as a transmitter and receiver, and all other nodes act as receivers.

Under ideal conditions, three angles of arrival define a point in space, which is the principle of target positioning. More details can be found in Appendix A. However, the quantum noise of an optical field imposes a limitation on positioning precision, defining a space region instead of a point. Positioning precision can be improved by reducing the estimation variance of each angle of arrival. According to the above analysis, a quantum sensor network with an equally weighted distribution can offer a precision advantage in the angle of arrival estimation over the other weighted one. We estimate the average displacement for radio-frequency phase (φ=φ4+φ3+φ2−φ1) by measuring the phase quadrature of each node and summing the photocurrents of four homodyne detectors. Figure 3(a) represents the estimation of average field amplitude and estimation variance for φ as a function of the applied PAR phases for all sensors synchronously. We repeatedly measured the dependence of estimated amplitude (average displacement) on applied PAR phase from zero to 2π with phase interval of 20 deg. To reach optimum estimation precision, the applied PAR phase is manipulated to generate identical displacements on the phase quadrature in each sensor node simultaneously. By minimizing the estimation variance, a 6.0dB±0.2dB quantum noise reduction is achieved, confirming the entanglement properties of the sensor network with equally weighted distribution. At this point, the splitting ratios of three variable beam splitters (VBSs) are 25:75, 33:67, and 50:50, and the three angles of arrival are quantified. Figure 3(b) shows the measured noise powers for three phase differences φj−φ1(j=2,3,4) for estimating the angle of arrival at different applied PAR phases. The measurement noise variances of three phase differences, represented by purple, green, and orange, are plotted for both quantum (square) and classical (circle) cases. The estimation variances for several cases are normalized to the shot noise limit. It is evident that the quantum-enhanced sensor network leads to a reduction in estimation variance, whereas such a noise reduction mechanism is absent without the quantum resource. The reduced noise powers of −2.1±0.1dB for φ2−φ1, −2.2±0.1dB for φ3−φ1, and −2.1±0.1dB for φ4−φ1 are obtained, which means the detection range is expanded by at least 12.8%. The noise variances are independent of the applied PAR phases, allowing the realization of 360-deg arbitrary scanning. The inset of Fig. 3(b) shows the principle of the improved positioning precision. It can be inferred that the uncertainty region of our protocol with multipartite entanglement is reduced to 0.4837 in three-dimensional space, in contrast with the classical protocol, corresponding to positioning precision improvement of 51.6%. Suppose we individually estimate each phase difference φ2−φ1 (φ3−φ1, φ4−φ1) with the optimal weights 1/2:1/2:0:0 (1/2:0:1/2:0, 1/2:0:0:1/2); then we can obtain higher sensitivity improvement (6.8 dB) than that of simultaneous estimation using similar quantum states. However, the individual estimation will spend more time performing the measurement task (three times), which is inappropriate for the positioning protocol of a moving target.

In the quantum ranging case, except for target positioning, target ranging is another critical specification of a radar system, which can be expressed as (see Appendix B for more details) r=(φ1′)2φ1′′λ36πΔx(−sinζx′−cosζy′+iz′),(1)where φ1′ and φ1′′ are the first and second derivatives of edge node PAR1; λ is the wavelength of the PAR field; Δx indicates the distance between each two sensors; x′, y′, and z′ show the movement speed of a target; ζ represents the azimuth angle of a target in terms of the PAR plane. The azimuth angle ζ can be obtained from the positioning case demonstrated in the previous section. According to Eq. (1), target ranging can be transferred to the estimation of the first and second derivatives of edge node PAR1. Exploiting the Taylor series expansion of the phase estimator with the estimated phase φ1 of edge node PAR1 [33], the optimum weights for the first and second derivatives are expressed as β1:β2:β3:β4=−11/6:3:−3/2:1/3 and 2:−5:4:−1, respectively. Details of the calculation of optimum weights can be found in Appendix C. As the number of nodes increases, higher order derivatives for edge node PAR1 can be obtained, standing for a richer physical scenario. Here, as shown in the inset of Fig. 4, four sensor nodes are arranged along a straight line, and edge node PAR1 serves as a reference.

To achieve optimum performance in quantum ranging, weighted distributions need to be configured on the grounds of the physical quantity measured. At the same time, the PAR network is arranged in a linear topology. The optimal weight −11/6:3:−3/2:1/3 (2:−5:4:−1) is arranged to implement the minimum estimation variance of the first (second) derivative of edge node PAR1; thereby, the splitting ratios of the VBSs correspond to 27.5:72.5, 62:38, and 82:18 (17:83, 50:50, and 80:20, respectively). The negative signs in the weights are introduced by adding a π-phase delay in the corresponding optical path. Figure 5 shows the estimated average amplitude of the first and second derivatives of PAR1 as a function of applied PAR phase, with the shaded area representing the estimation uncertainty of the first (second) derivative of the node, dark orange (green) for the quantum sensing protocol, and light orange (green) for the classical protocol. To satisfy 360-deg fast scanning, a precision advantage of quantum ranging should be offered at an arbitrary azimuth angle, irrespective of the applied PAR phase. Compared with the classical protocol, the almost invariable noise reductions of −5.0±0.1dB for the first derivative and −5.4±0.2dB for the second derivative are implemented in the experiment, demonstrating the full scanning range with enhanced precision. The inset of Fig. 5 represents the topology structure of quantum ranging. In addition, for the noise suppression in three independent relative phase differences, the first and second derivatives of the edge node are still maintained when the applied PAR amplitude varies, which confirms that the PAR network can be compatible with the return signal with different intensities without sacrificing precision. It can be inferred from the results of Fig. 5 that the pitch angle with enhanced precision of 68.4% is achieved on all-around scanning. Furthermore, combined with noise suppression of 5.4 dB for φ1′′, target ranging with enhanced precision of 69.3% can be realized. In other words, with the same signal to noise ratio (SNR) of the return signal, the detection range of radar is increased by 34.3% [42]. It is worth noting that quantum ranging can be realized only when both the first and second derivatives are measured simultaneously, meaning that at least three sensor nodes in line are required. In combination with quantum ranging, the space positioning of the target becomes more accurate.

In conclusion, we have implemented the demonstration of a quantum-enhanced PAR network exploiting a CV multipartite entanglement state, verifying the feasibility of quantum positioning and quantum ranging. By utilizing a sensor network with an average weighted distribution, quantum positioning offers a precision advantage of 51.6% in three-dimensional space over the classical scheme, increasing the detection range by 12.8%. Moreover, in virtue of diverse weighted distributions, quantum ranging with enhanced precision of 69.3% can be realized; thus, the detection range of radar is increased by 34.3% with the same SNR of the return signal. We expect to construct a distributed sensing network via a multipartite entanglement state, which can be applied to track moving targets in PAR. However, for realistic PAR, the frequency of a return signal from a target that is not constant depends on the speed of the moving target. Therefore, to satisfy diverse target tracking (different speeds of movement), we need the generation of a broad-bandwidth multipartite entanglement state and balanced homodyne detectors.