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Entanglement-assisted multi-aperture pulse-compression radar for angle resolving detection

Entanglement has been known to boost target detection, despite it being destroyed by lossy-noisy propagation. Recently, [Phys. Rev. Lett. 128, 010501 (2022)] proposed a quantum pulse-compression radar to extend entanglement's benefit to target range estimation. In a radar application, many other aspects of the target are of interest, including angle, velocity and cross section. In this study, we propose a dual-receiver radar scheme that employs a high time-bandwidth product microwave pulse entangled with a pre-shared reference signal available at the receiver, to investigate the direction of a distant object and show that the direction-resolving capability is significantly improved by entanglement, compared to its classical counterpart under the same parameter settings. We identify the applicable scenario of this quantum radar to be short-range and high-frequency, which enables entanglement's benefit in a reasonable integration time.


Introduction
Radio Detection and Ranging (radar) system exploits the techniques of transmitting and receiving electromagnetic (EM) field for detecting properties of distant objects. In a radar system, the ranging measurement of an object can be inferred by the time-of-flight of a pulse [1][2][3][4][5][6][7]. Conventional radars use a classical coherent EM field probe. With the recent emergence of quantum sensing technology [8][9][10][11], quantum radar has been proposed utilizing entanglement for higher sensitivity [12][13][14]. The notion of quantum radar started with the detection of absence or presence of a target in the quantum illumination protocol [15][16][17], where a six-decibel advantage was found in the error exponent. Recent works have also extended the applicability of quantum radar to ranging [18,19], showing a huge entanglement advantage in range accuracy due to the nonlinear nature of the range estimation problem. However, radar detection is a complex task aiming at estimating various properties of the target, such as range, angle, velocity and cross section. The benefit of quantum radar has not been fully explored in estimating these different properties. In this paper, we consider a bistatic dual-receiver quantum radar scheme to resolve the angular elevation of a distant target. In the large signal-to-noise ratio (SNR) limit, we identify a factor of two angle resolution advantage from entanglement. Furthermore, in the intermediate SNR region, by evaluating the quantum Ziv-Zakai bound [20], we identify a huge angle resolution advantage from entanglement at the SNR threshold, similar to Ref. [18]. To connect to practical scenarios, we analyze entanglement's advantage in angle detection of an unmanned aerial system (UAS) versus its range and integration time, where we identify short-range of a hundred meters to be the parameter region where entanglement's benefit over a classical radar is applicable.

Radar Angle estimation
To precisely determine the angle relative to the vertical direction, φ, shown in Fig. 1, we consider two quantum receivers, each with an individual aperture, separated by distance d. By assuming a distant target (L d), the return microwave wave vectors are approximated as parallel between the two receivers. The overall return-path transmissivity (source-to-target-to-receiver) κ 1 is assumed to be small, and known a priori.
Our proposed quantum radar detects the angle φ by analyzing the difference of signal arrival time at the two receivers. Based on any prior knowledge of φ, for example acquired using passive imaging on a classical-radar pre-estimate, we direct the dual-receivers toward φ c ∼ φ to maximize the transmissivity of the return field. (Here 'c' denotes compensation.) By doing so, the effective transmissivity, projected on the plane of the receiver, is κ φ,φ c = κ cos (φ − φ c ), which decays with the error of the prior knowledge |φ − φ c |.
Radar, in general, is operated at microwave frequency for both quantum [21,22] and classical [5,6]. In our case, we choose the microwave frequency at ω 0 /2π ∼ 100 GHz in W-band, since that this frequency band is robust to degraded visual environments [23] with high precision and especially suitable for the applications of UAS localization [24]. The thermal bath has mean photon number per mode following the Planck-law distribution, plotted in Fig. 2, where is the reduced Planck constant, k B is the Boltzmann constant and T B is the temperature of the thermal bath. From this, we can see that the W-band domain is especially noisy (i.e., N B 1). Thereby, the return-path propagation in W-band can be modeled as a very noisy and lossy channel.
Regardless of the radar scheme being classical or quantum, the input-output relation for the field operators (in units photons/second) are the same. For transmitted fieldÊ (t), the return field at receiver 1(2)Ê 1(2) (t) can be described aŝ where ξ ∈ [0, 2π) denotes the phase picked up from the reflection of the target, τ = 2L/c is the time of flight of the microwave probe pulse, c is the speed of light,V 1 (t) andV 2 (t) are the environmental noise field operators that correspond to the '−' and '+' signs in '∓' of Eq. (2). In a general phase-incoherent scenario, ξ is random and unknown; to begin with, we will consider the phase-coherent case of known ξ, recognizing that the results obtained are lower bounds on the phase-incoherent counterparts, similar to Ref. [18]. BothÊ 1(2) (t) andV 1(2) (t) satisfy the commutation relations, Subsequently, we take the Fourier transform on Eq.
(2) to convert field operators into the frequency domain:Ê The noise modeV 1,2 (ω) (in units of photons/Hz) satisfy the auto-correlation relations: For simplicity, in the following section, we model each transmitted temporal mode of the microwave probe for the classical and the quantum radars respectively, as a coherent state and a two-mode squeezed vacuum (TMSV) state. Since the aforementioned two states are both Gaussian and can be fully characterized by their quadrature mean and covariance matrices (CMs), we analyze the two radars with their corresponding mean and CMs. Note that although we have introduced the continuous-time field operators to describe the radar signals, in a finite-time analysis one can always discretize the field into the orthogonal frequency modes, each with a finite frequency bin size [18]. We will adopt this discrete-mode approach, where finite dimensional CMs are welldefined, for evaluating various quantities. Denote the number of frequency bins as N. The matrix elements of the CM, V, of an N-mode Gaussian state are given by: wherex = {q 1 ,p 1 ,q 2 ,p 2 · · · ,q N ,p N } T with quadrature operatorq s =â s +â † s andp s = â s −â † s /i, ∀s ∈ {1, 2, · · · , N}. Hereâ s 's are the annihilation operators of the modes.

Classical radar
Classical radar transmits a coherent state | √ Es (t) e −iω 0 t in a compressed chirped pulse, where E is the total mean photon number, T d is the pulse duration (i.e., 2π/T d ∆ω ω 0 ) and ∆ω is the bandwidth. The pulse has the spectrum Considering the received fields at the two receiversÊ 1 andÊ 2 , after discretizing each field into N frequency modes, the global system has 2N modes. The overall quantum stateρ φ C of the system is specified by its mean and CM, with the CM given by a direct sum of each 4by-4 CM of two received modes at the same frequency, namely V C = Ω (2N B + 1) I 4 , and quadrature mean q φ C = Ω q φ,Ω C . Here ' Ω ' is the direct sum of over all frequencies and I 4 is the 4 × 4 identity matrix. The quadrature mean can be obtained as q φ,Ω is the mean photon number per mode of the flat-top spectral mode with width 1/T d centered at ω 0 + Ω. The total mean photon number is

Quantum Radar
In a quantum radar, the transmitted microwave pulse is entangled with an idler pulse, which is stored in a quantum memory at the location of the two receivers. As the signal pulse is returned from the target and to receiver 1 and 2, we perform a joint measurement on the quantum state of the idler and return from both receivers. The signal (idler) field of the microwave pulse has the field operator centered at the frequency ω 0 as,Ê where the '−' in the '±' corresponds to the idler and we have denotedÂ S (I) (Ω) ≡Ê S (I) (ω 0 + Ω) for simplicity. The field operators in Eq. (13) have spectral auto-correlations and cross-correlations, where S (p) (Ω) = S (n) (Ω) S (n) (Ω) + 1 . The total mean photon number of the signal (idler) mode is, and its average bandwidth is Eq. (16) and Eq. (17) coincide with the mean photon number in Eq. (12) and the bandwidth in Eq. (10) of the classical framework, showing that the power and the bandwidth of our quantumradar transmitter are identical with the classical case. After the discretization, the overall received field in the quantum radar case can be described by a collection of N mode triplets, each triplet consisting of one idler mode and two received modes at receiver 1 and receiver 2. The global stateρ φ Q is zero-mean Gaussian state characterized by the

Variance bound of estimator
In this section, we evaluate lower bounds of the mean squared error (MSE) in angle estimation. The Cramér-Rao bound (CRB) provides an asymptotically tight lower bound on the minimum possible MSE among all unbiased estimators. Whereas CRB is well known to be tight in the limit of infinite time-bandwidth product (i.e., ∆ωT d → ∞), it underestimates the achievable error as the time-bandwidth product is finite (i.e., ∆ωT d < ∞). The Ziv-Zakai bound (ZZB) is another lower bound, obtained by analyzing the error probability in a binary hypothesis problem, and is proved to be a tighter bound than CRB in multiple cases of the non-asymptotic region [18,20,[25][26][27][28][29][30]. In the following, we compare the estimation of φ by CRBs and ZZBs in the quantum and classical cases.

Cramér-Rao bound
In radar detection, the target angle φ is encoded into the output statê ρ φ . CRB indicates the minimum variance lower bound of the unbias estimator for estimating φ fromρ φ , and is the Uhlmann fidelity [31] between statesρ φ andρ φ+ . For our evaluation, in the classical scenario, we assign ρ φ ,ρ φ+ = ρ φ C ,ρ φ+ C , whereas in quantum scenario, ρ φ Q ,ρ φ+ Q . Under the 6 approximation N S , κ 1 and N B 1, we have the fidelities of classical and quantum cases as Followed by the definition of QFI in Eq. (19), we derive the QFI for classical and quantum radar as, Since QFI is additive across all the frequency modes Ω, we integrate Eq. (22) over the whole fluorescence spectrum. This can be justified by taking a continuous limit of a discrete set of frequency modes [18]; in other words, we calculate It is interesting to note that the QFI of quantum radar is twice of that of the classical radar, similar to the ranging case [18]. Owing to the fact that CRB is only tight in large SNR limit, we take φ c = φ for evaluating the CRB. Ultimately, we derive the quantum CRB (QCRB) and classical CRB (CCRB) as δφ 2 CCRB = 1/F C,φ and δφ 2 QCRB = 1/F Q,φ , and plot them as the red and blue dotted lines in Fig. 3. 2.3.2. Ziv-Zakai bound ZZB is a Bayesian bound, calculated by averaging the MSE over a priori probability density function of the estimating parameter so as to incorporate knowledge of a priori parameter space [26,27]. Specifically, consider a random variable X with the prior distribution P X (·), and then the ZZB of the MSE can be evaluated as where P X (x) denotes the probability density of prior knowledge at angle x, V denotes the valleyfilling operation, i.e., V f (τ) = max η≥0 f (τ + η), and Pr(x; x + ζ) denotes the minimum error probability to distinguish two hypotheses, Considering uniform prior-knowledge in the range [φ − ∆φ/2, φ + ∆φ/2], where ∆φ denotes the uncertainty range, we rewrite Eq. (24) as Note that as Pr (x; x + ζ) decreases with ζ increasing, and the distribution is uniform, so the principle value is always achieved at ζ. Considering the limit of ∆φ 1, we can approximate the above results as We will justify this assumption later. The error probability Pr (·) in Eq. (26) or Eq. (27) is obtained from the maximum likelyhood-ratio test in classical scenario or from the Helstrom limit in quantum. While classical ZZB (CZZB) is fairly straightforward to calculate, quantum ZZB (QZZB) is challenging due to the integration of the Helstrom limit. To enable efficient evaluation, we approximate the Helstrom limit with the quantum Chernoff bound (QCB) (i.e., Pr → P (QCB) ). As QCB is exponentially tight, we expect the results to reveal the advantage of entanglement, similar to previous works [16,18]. Similar to Refs. [16,32,33], we can derive the QCBs for classical and quantum as (see more details in Appendix Appendix A) under the approximation of N S , κ 1 and N B 1. In angle detection, the error exponent of QCB achieves 6-dB advantage over the classical one, same conclusion as QI. Akin to CRB evaluation, we set φ c = φ and plug the upper bounds of P (QCB) C and P (QCB) Q from Eq. (28) into Eq. (27) to evaluate the ZZBs normalized to δφ 2 ref = ∆φ 2 /12 (i.e., δφ 2 ref is the ZZB by assuming Pr = 1/2) in Fig. 3.
Under appropriate parameter settings, Fig. 3 shows a huge quantum advantage (∼ 30 dB) at the SNR threshold of the quantum radar, where the precision improves drastically with SNR increasing (will be later quantified in Section 2.4). In the high SNR regime, QZZB coincides with the QCRB while CZZB has an 3 dB offset higher than CCRB. Moreover, the approximation at the ∆φ 1 limit used in Eq. (27) can be justified by the concurrence of the blue solid curve (evaluated via Eq. (27)) and green dashed curve (evaluated via Eq. (26)) in Fig. 3 when setting a small ∆φ (e.g., ∆φ = π/100).

Quantum advantage versus pulse duration and range
To understand the practical use scenario of quantum radar, it is necessary to evaluate the tradeoff between the quantum advantage with the pulse duration at a given set of physical parameter (e.g., κ and N B ). To calculate the ZZB without considering the long pulse approximation, we have to calculate the QCB numerically rather than adopting the asymptotic formula in Eq. (28). At the same time we will adopt the full numerical approach in Eq. (26), instead of the approximated result in Eq. (27).
We will focus on the SNR threshold of the quantum radar to evaluate the quantum advantage. Before proceeding to the evaluation, we make our definition of SNR threshold precise. The intermediate SNR that results in a significant drop of QZZB is defined as the SNR threshold (SNR th ) of quantum radar, manifested in Fig. 3. In the high SNR regime (SNR SNR th ), the major contribution of the integration over ζ in Eq. (27) comes from the values near the origin and, thereafter, QZZB can be asymptotically derived as where Υ φ was defined in Eq. (23). On the other hand, the QZZB in the low SNR regime (SNR SNR th ) follows the inequality, i.e., 0 ≤Θ φ,ζ ≤ 2. SNR th is defined as the particular SNR that matches the asymptotic limit in Eq. (30) and the asymptotic lower bound limit in Eq. (31) (i.e., δφ 2 ref exp [−4 (SNR)]), and formulated as where g y is the inverse function of ye −y , ∀y > 1. To understand the quantum advantage trade-off, we specify our radar system to the application of UAS. In UAS detection, when the target is at distance L, the transmissivity of the interrogation channel where G T = A R / (2πc/ω 0 ) 2 is radar's antenna gain, A R is the antenna's area, σ is the target's cross section area. Plugging these parameters into our simulation model and fixing SNR = SNR th by tuning N S , we numerically calculate the quantum advantage trade off in Fig. 4. Under this parameter setting, the quantum advantage is appreciable ( 15 dB) for distance L ∼ 500 m by setting a practical pulse duration of T d = 0.1 second.

Conclusion and discussion
In this work, we propose a two-receiver bistatic radar framework to employ a microwave probe entangled with a reference pre-shared with the receivers, in detecting the direction of the target and prove that quantum radar outperforms the classical competitor. The proposed quantum radar has the SNR threshold (i.e., SNR th ) 6 dB lower than the classical one's and, as a result, the quantum advantage is significant when we enact the radar at the SNR th of quantum radar at the long integration time limit. When the integration time is finite, the quantum advantage applies to short range precise ranging of small targets such as UAS. Our proposed quantum radar generalizes the previous results in quantum radar ranging [18] towards a general quantum radar detection system capable of detecting various properties of targets. One might argue that our dual-receiver system does not make use of the spatial field distribution on the imaging plane; indeed, concatenating our analysis with spatial mode sorter and photodetection could potentially improve the estimation of φ; however, we show that such a mode sorting approach only brings marginal improvement even in the best-case scenario (see more details in Appendix Appendix B), as the usual size of aperture is small compared with the separation of the apertures. Hence, considering the level of complexity in the receiver design, it is not necessary to incorporate such processing into our design.

Acknowledgments
The project was supported by Raytheon Missiles and Defense under a corporate IRAD project led by Mark J. Meisner. QZ also acknowledges support from NSF CAREER Award CCF-2142882.

Appendix A. Quantum Chernoff bound
The error probability of distinguishing two hypotheses of quantum state (i.e., H 1 and H 2 ) is upper bounded by the QCB, Tr ρ s whereρ 1 andρ 2 are the density operators corresponding to H 1 and H 2 . In radar scenario, the two hypotheses refer to parameters specified in Eq. (25), and the QCB defined in Eq. (A.1) [32] under the approximation ∆φ 1 can be derived as with each frequency contributing and Here N denotes the number of involved mode (i.e., N = 2 in classical and N = 3 in quantum), δv φ,ζ Ω = q φ+ζ Ω − q φ Ω denotes the quadrature mean difference, G (±) λ φ j is the j-th symplectic eigenvalue associated with the symplectic matrix S x Ω in parameter φ = {φ, φ + ζ}.
Subsequently, we claim that the infimum in Eq. (A.3) occurs at s = 1/2 and support it with numerical justifications and perturbation theory. Fig. A1 shows the plot of Eq. (A.4) in quantum case with Ω = 0 as a function of s (i.e., P 0,s Q ). In Fig. A1(a), we fix κ while changing N S ; in Fig. A1(b), N S is fixed while κ changes. Obviously, in these parameter settings, all minimal values, consistently, occur at the choices of s = 1/2. To be more strict on justifying s = 1/2, we employ the perturbation theory, introduced by Ref. [34], on our angle-resolving radar model with the approximation N S , κ 1 and N B 1, and easily conclude that infinimum does occur at s = 1/2 by the Theorem 4 of Ref. [34].
Therefore, we can calculate Eq. (A.3) in both cases as P Ω

Appendix B. Mode sorter
In this section, we apply mode sorter on the proposed radar and study the potential improvement. Instead of dual-receiver, let us underpin the scheme of single-receiver, shown in Fig. B1, to simplify the calculation, and we anticipate that mode sorter brings the same or, at least, similar improvement in single-and dual-receiver schemes. In a single receiver scenario, we set the target at the angle φ, same as in dual-receiver radar, but set the compensation angle to be zero φ c = 0 (i.e., the face of radar is vertically directed, shown in Fig. B1) and consider a soft-aperture located at the focal plane of the paraboloid antenna in Fig. B2(a). The received pulse is collected by the paraboloid antenna as an elliptical Gaussian beam w (t; whose overall phase is set zero, r is the half length of the minor axis of ellipse (i.e., cross section of paraboloid), and the field has the spectrum W (Ω; x, y) = S (Ω) ε (x, y), where s (t) and S (Ω) were denoted in Eq. (8) and Eq. (9). The asymmetricity of the two axes comes from the angle deviation  φ between the incident plane and the imaging plane, yielding the y-axis, shown in Fig. B2(a), being elongated whereas x-axis being the same. Hermite Gaussian (HG) mode is treated as a discriminator to sort the spatial mode of the field on the imaging plane [35,36], shown in Fig. B2(a). The proposed HG mode has the eigenfunction indexed by non-negative integers n, m ∈ N 0 , where β = the Fresnel number, A δ = πδ 2 /4 is the aperture area (i.e., δ is its diameter), ν = 1 + iω 0 /cβ 2 L /2 1/2, and H n (·) denotes the n-th order Hermite polynomial. The received signal can be regarded as a far field if D f 1 and is decomposed by k HG modes and an additional mode that covers the residual higher order ones. The field distribution projected on the imaging plane depends on the parameter φ, as demonstrated in the inset of Fig. B2(a).
To begin our analyses, we evaluate the overlap of a Gaussian function with each basis ψ n,m (x, y) at the image plane, leading to the associate occupation probability where χ = 2β 2 r 2 and this decomposition applies to our Gaussian beam imaging system [36]. For smooth communication, we relabel the indices of probability at each mode as , · · · and plot the lowest three order modes in Fig. B3. This mathematical decomposition process can be visualized by passing the incident Gaussian beam through a k-series beamsplitters, shown in Fig. B2(b). The j-th mode (i.e., 1 ≤ j ≤ k) beamsplitter has the matrix form where These k beamsplitters entangle the k vacuum modesV at the input and result in k+1 output modes. In the following, we apply the mode sorter on the analysis of classical radar and quantum radar. For simplicity, we consider the mode sorter approach that involves only one beamsplitter, k = 1 and set δ/r = 2 such that the fundamental mode (ψ 0,0 ) dominates all other HG modes when the concerning angle is small φ 1, shown in Fig. B3.

Appendix B.1. Classical radar
In classical single-receiver scheme, the global state, at the angle φ, has the CM V C = (2N B + 1) Ω I 4 and quadrature mean q φ C = Ω B φ 1 q φ,Ω C , where q φ,Ω C = 2S (n) (Ω) κ φ,0 (cos Ξ, sin Ξ, 0, 0) T is the quadrature at frequency ω 0 + Ω in the basis of q S ,p S ,q ⊥ S ,p ⊥ S , where Ξ ∈ (0, 2π] is the overall phase of the return signal, {q S ,p S } and q ⊥ S ,p ⊥ S are the quadrature pairs of signal, projected on ψ 0,0 (i.e., occupation probability P φ 1 ), and the residual HG modes ψ ⊥ (i.e., occupation probability 1 − P φ 1 ). Akin to calculating CRB and ZZB of dualreceiver radar in the main text, under the approximation N S , κ 1, N B 1, we asymptotically and analytically derive the CCRB (red dotted line in Fig. B4), Similarly, the QCB for distinguishing the hypotheses in Eq. (25) can be obtained as rior knowledge in φ − ∆φ/2, φ + ∆φ/2 , CZZB is numerically derived by plugging the upper bound of P (QCB) C into Eq. (27) with the approximation ∆φ 1 and is plotted as the red solid curve in Fig. B4.
In Eq. (B.6), note that the size of the receiver antenna comes in via the transmissivity κ in the SNR (see Eq. (33)). The area of aperture A δ at the focal plane needs to match the antenna's cross section area A R in a way to receive all the beams that is reflected from the surface of paraboloid antenna, so that no additional loss occurs to degrade the SNR.

Appendix B.2. Quantum radar
In a quantum single-receiver scheme, the collected microwave field is from the signal mode of a TMSV state. After HG mode decomposition, the CM of global state at angle x = {x, x + ζ} is in quadrature basis q I ,p I ,q S ,p S ,q ⊥ S ,p ⊥ S , where {q I ,p I } denotes the quadrature pair of idler mode, B Ω x = 2N B + 2κ x ,0 S (n) (Ω) + 1, C Ω x = 2 √ κ x ,0 S (p) (Ω). The QCRB (blue dotted line in Fig. B4) is asymptotically and analytically derived as, and QCB, for distinguishing H 1 and H 2 in Eq. (25), P (QCB) Q (x; x + ζ) ≤ exp − (SNR) Γ x,ζ /2 under the approximation N S , κ 1 and N B 1. With uniform prior knowledge in φ − ∆φ/2, φ + ∆φ/2 , QZZBs can be numerically calculated with or without the approximation ∆φ 1 by Eq. (27), which are plotted as the blue solid and green dashed curves in Fig. B4. Same conclusion as dualreceiver, the concurrence of two curves justifies the formula of Eq. (27) and they both coincide with the QCRB in high SNR regime. In Fig. B4, we plot ZZBs in both classical and quantum scenarios by fixing the uncertainty tolerance ∆φ = π/100 and setting φ = π/2 − ∆φ/2. The choice of φ = π/2 − ∆φ/2 has the maximal distinguishability between H 1 and H 2 , because the occupation probability in HG fundamental mode varies dramatically as target angle approaches π/2. However, in Fig. B4, despite the optimal choice of φ, the noticeable reduction of δφ 2 /δφ 2 ref can only be observed when SNR goes to very high (i.e., > 30 dB); conversely, the dual-receiver radar mode sorter has significant reduction outcome even at low SNR regime (i.e., ∼ 1 dB).