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

Bo-Han Wu1,∗, Saikat Guha2,3, Quntao Zhuang2,3,† 1 Department of Physics,
University of Arizona, Tucson, Arizona 85721, USA  2 J. C. Wyant College of
Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA  3
Department of Electrical and Computer Engineering, University of Arizona,
Tucson, Arizona 85721, USA
$^*$gowubohan<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

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.

##  1 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 [1-7]. Conventional radars
use a classical coherent EM field probe. With the recent emergence of quantum
sensing technology [8-11], quantum radar has been proposed utilizing
entanglement for higher sensitivity [12-14]. The notion of quantum radar
started with the detection of absence or presence of a target in the quantum
illumination protocol [15-17], 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.

##  2 Radar Angle estimation

To precisely determine the angle relative to the vertical direction, $\phi$,
shown in Fig. 1, we consider two quantum receivers, each with an individual
aperture, separated by distance $d$. By assuming a distant target ($L\gg d$),
the return microwave wave vectors are approximated as parallel between the two
receivers. The overall return-path transmissivity (source-to-target-to-
receiver) $\kappa\ll 1$ is assumed to be small, and known a priori.

Our proposed quantum radar detects the angle $\phi$ by analyzing the
difference of signal arrival time at the two receivers. Based on any prior
knowledge of $\phi$, for example acquired using passive imaging on a
classical-radar pre-estimate, we direct the dual-receivers toward
$\phi_{c}\sim\phi$ 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
$\kappa_{\phi,\phi_{c}}=\kappa\cos\left(\phi-\phi_{c}\right)$, which decays
with the error of the prior knowledge $|\phi-\phi_{c}|$.

Figure 1: Scheme of dual-receiver bistatic radar. R1: receiver 1, R2: receiver
2. T: transmitter. QM: quantum memory.

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
$\omega_{0}/2\pi\sim 100$ 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,

$N_{B}=\frac{1}{e^{\hbar\omega/k_{B}T_{B}}-1},$ (1)

where $\hbar$ 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., $N_{B}\gg 1$). 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
$\sqrt{\text{photons}/\text{second}}$) are the same. For transmitted field
$\hat{E}\left(t\right)$, the return field at receiver 1(2)
$\hat{E}_{1(2)}\left(t\right)$ can be described as

$\hat{E}_{\text{1}(2)}\left(t\right)=\sqrt{\kappa_{\phi,\phi_{c}}}\exp{\left(i\xi\right)}\hat{E}\left[t-\left(\tau\mp
d\sin\phi/2c\right)\right]+\sqrt{1-\kappa_{\phi,\phi_{c}}}\hat{\mathcal{V}}_{1(2)}\left(t\right),$
(2)

where $\xi\in\left[0,2\pi\right)$ denotes the phase picked up from the
reflection of the target, $\tau=2L/c$ is the time of flight of the microwave
probe pulse, $c$ is the speed of light, $\hat{\mathcal{V}}_{1}\left(t\right)$
and $\hat{\mathcal{V}}_{2}\left(t\right)$ are the environmental noise field
operators that correspond to the ‘$-$’ and ‘$+$’ signs in ‘$\mp$’ of Eq. (2).
In a general phase-incoherent scenario, $\xi$ is random and unknown; to begin
with, we will consider the phase-coherent case of known $\xi$, recognizing
that the results obtained are lower bounds on the phase-incoherent
counterparts, similar to Ref. [18]. Both $\hat{E}_{1(2)}\left(t\right)$ and
$\hat{\mathcal{V}}_{1(2)}\left(t\right)$ satisfy the commutation relations,

$\left[\hat{E}_{1(2)}\left(t\right),\hat{E}^{\dagger}_{1(2)}\left(t^{\prime}\right)\right]=\left[\hat{\mathcal{V}}_{1(2)}\left(t\right),\hat{\mathcal{V}}^{\dagger}_{1(2)}\left(t^{\prime}\right)\right]=\delta\left(t-t^{\prime}\right).$
(3)

Figure 2: Mean photon number per mode in the thermal environment, for the
Planck-Law distribution.

Subsequently, we take the Fourier transform on Eq. (2) to convert field
operators into the frequency domain:

$\hat{\tilde{E}}_{\text{1}(\text{2})}\left(\omega\right)=\sqrt{\kappa_{\phi,\phi_{c}}}\exp{\left[i\xi^{(\mp)}_{\phi,\omega}\right]}\hat{\tilde{E}}\left(\omega\right)+\sqrt{1-\kappa_{\phi,\phi_{c}}}\hat{\tilde{\mathcal{V}}}_{1(2)}\left(\omega\right),$
(4)

where $\xi^{\mp}_{\phi,\omega}=\xi-\omega\left(\tau\mp d\sin\phi/2c\right)$,

$\displaystyle\hat{E}\left(t\right)$
$\displaystyle=\frac{1}{2\pi}\int\hat{\tilde{E}}\left(\omega\right)e^{-i\omega
t}d\omega,$ (5) $\displaystyle\hat{E}_{1(2)}\left(t\right)$
$\displaystyle=\frac{1}{2\pi}\int\hat{\tilde{E}}_{1(2)}\left(\omega\right)e^{-i\omega
t}d\omega,$ $\displaystyle\hat{\mathcal{V}}_{1(2)}\left(t\right)$
$\displaystyle=\frac{1}{2\pi}\int\hat{\tilde{\mathcal{V}}}_{1(2)}\left(\omega\right)e^{-i\omega
t}d\omega.$

The noise mode $\hat{\tilde{\mathcal{V}}}_{1,2}\left(\omega\right)$ (in units
of $\sqrt{\text{photons}/\text{Hz}}$) satisfy the auto-correlation relations:

$\langle\hat{\tilde{\mathcal{V}}}^{\dagger}_{1,2}\left(\omega\right)\hat{\tilde{\mathcal{V}}}_{1,2}\left(\omega^{\prime}\right)\rangle=2\pi
N_{B}\;\delta\left(\omega-\omega^{\prime}\right).$ (6)

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 well-defined, for evaluating various quantities. Denote the number of
frequency bins as $N$. The matrix elements of the CM, $\mathbb{V}$, of an
$N$-mode Gaussian state are given by:

$\left[\mathbb{V}\right]_{j,l}\equiv\langle\hat{\mathbb{x}}_{j}\hat{\mathbb{x}}_{l}\rangle-\langle\hat{\mathbb{x}}_{j}\rangle\langle\hat{\mathbb{x}}_{l}\rangle,\;\;\forall
j,l=\left\\{1,2,\cdots,N\right\\},$ (7)

where
$\hat{\mathbb{x}}=\left\\{\hat{q}_{1},\hat{p}_{1},\hat{q}_{2},\hat{p}_{2}\cdots,\hat{q}_{N},\hat{p}_{N}\right\\}^{T}$
with quadrature operator $\hat{q}_{s}=\hat{a}_{s}+\hat{a}_{s}^{\dagger}$ and
$\hat{p}_{s}=\left(\hat{a}_{s}-\hat{a}_{s}^{\dagger}\right)/i$, $\forall
s\in\left\\{1,2,\cdots,N\right\\}$. Here $\hat{a}_{s}$’s are the annihilation
operators of the modes.

###  2.1 Classical radar

Classical radar transmits a coherent state $|\sqrt{{\cal
E}}s\left(t\right)e^{-i\omega_{0}t}\rangle$ in a compressed chirped pulse,

$s\left(t\right)=\left(2\pi/T_{d}^{2}\right)^{-1/4}\exp{\left[i\Delta\omega^{2}/2T_{d}-t^{2}/4T_{d}^{2}\right]},$
(8)

where ${\cal E}$ is the total mean photon number, $T_{d}$ is the pulse
duration (i.e., $2\pi/T_{d}\ll\Delta\omega\ll\omega_{0}$) and $\Delta\omega$
is the bandwidth. The pulse has the spectrum

$\displaystyle
S\left(\Omega\right)=\int_{-\infty}^{\infty}s\left(t\right)e^{-i\Omega
t}dt\simeq\left(\Delta\omega^{2}/2\pi\right)^{-1/4}\exp{\left[-\frac{\Omega^{2}}{4\Delta\omega^{2}}\right]}$
(9)

with

$\frac{1}{2\pi}\int_{-\infty}^{\infty}\Omega^{2}|S\left(\Omega\right)|^{2}d\Omega\simeq\Delta\omega^{2}.$
(10)

Considering the received fields at the two receivers $\hat{E}_{1}$ and
$\hat{E}_{2}$, after discretizing each field into $N$ frequency modes, the
global system has $2N$ modes. The overall quantum state
$\hat{\rho}_{\text{C}}^{\phi}$ of the system is specified by its mean and CM,
with the CM given by a direct sum of each $4$-by-$4$ CM of two received modes
at the same frequency, namely
$\mathbb{V}_{\text{C}}=\bigoplus_{\Omega}\left(2N_{B}+1\right)\mathbb{I}_{4}$,
and quadrature mean
$\langle\hat{\mathbb{q}}\rangle^{\phi}_{\text{C}}=\bigoplus_{\Omega}\langle\hat{\mathbb{q}}\rangle^{\phi,\Omega}_{\text{C}}$.
Here ‘$\bigoplus_{\Omega}$’ is the direct sum of over all frequencies and
$\mathbb{I}_{4}$ is the $4\times 4$ identity matrix. The quadrature mean can
be obtained as
$\langle\hat{\mathbb{q}}\rangle^{\phi,\Omega}_{\text{C}}=\sqrt{2S^{(n)}\left(\Omega\right)\kappa_{\phi,\phi_{c}}}\left(\cos\xi^{(-)}_{\phi},\sin\xi^{(-)}_{\phi},\cos\xi^{(+)}_{\phi},\sin\xi^{(+)}_{\phi}\right)^{T}$,
where

$S^{(n)}\left(\Omega\right)=\sqrt{2\pi}N_{S}\exp{\left[-\Omega^{2}/2\Delta\omega^{2}\right]}$
(11)

is the mean photon number per mode of the flat-top spectral mode with width
$1/T_{d}$ centered at $\omega_{0}+\Omega$. The total mean photon number is

$\displaystyle\int^{T_{d}}_{0}dt\langle\hat{E}^{\dagger}\left(t\right)\hat{E}\left(t\right)\rangle$
$\displaystyle\simeq\frac{T_{d}}{2\pi}\int^{\infty}_{-\infty}d\Omega\,S^{(n)}\left(\Omega\right)=N_{S}\Delta\omega
T_{d}\equiv{\cal E}.$ (12)

###  2.2 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 $\omega_{0}$ as,

$\hat{E}_{S(I)}\left(t\right)=\frac{1}{2\pi}\int\hat{\tilde{A}}_{S(I)}\left(\Omega\right)e^{-i\left(\omega_{0}\pm\Omega\right)t}d\Omega,$
(13)

where the ‘$-$’ in the ‘$\pm$’ corresponds to the idler and we have denoted
$\hat{\tilde{A}}_{S(I)}\left(\Omega\right)\equiv\hat{\tilde{E}}_{S(I)}\left(\omega_{0}+\Omega\right)$
for simplicity. The field operators in Eq. (13) have spectral auto-
correlations

$\displaystyle\langle\hat{\tilde{A}}_{S}^{\dagger}\left(\Omega\right)\hat{\tilde{A}}_{S}\left(\Omega^{\prime}\right)\rangle$
$\displaystyle=\langle\hat{\tilde{A}}_{I}^{\dagger}\left(\Omega\right)\hat{\tilde{A}}_{I}\left(\Omega^{\prime}\right)\rangle=2\pi
S^{(n)}\left(\Omega\right)\delta\left(\Omega-\Omega^{\prime}\right),$ (14)

and cross-correlations,

$\langle\hat{\tilde{A}}_{S}^{\dagger}\left(\Omega\right)\hat{\tilde{A}}_{I}\left(\Omega^{\prime}\right)\rangle=\langle\hat{\tilde{A}}_{I}^{\dagger}\left(\Omega\right)\hat{\tilde{A}}_{S}\left(\Omega^{\prime}\right)\rangle=2\pi
S^{(p)}\left(\Omega\right)\delta\left(\Omega-\Omega^{\prime}\right),$ (15)

where
$S^{(p)}\left(\Omega\right)=\sqrt{S^{(n)}\left(\Omega\right)\left(S^{(n)}\left(\Omega\right)+1\right)}$.
The total mean photon number of the signal (idler) mode is,

$\int^{T_{d}}_{0}dt\,\langle\hat{E}^{\dagger}_{S\left(I\right)}\left(t\right)\hat{E}_{S\left(I\right)}\left(t\right)\rangle=\frac{T_{d}}{2\pi}\int^{\infty}_{-\infty}d\Omega\,S^{(n)}\left(\Omega\right)=N_{S}\Delta\omega
T_{d}\equiv{\cal E},$ (16)

and its average bandwidth is

$\int_{-\infty}^{\infty}S^{(n)}\left(\Omega\right)\Omega^{2}d\Omega/\int_{-\infty}^{\infty}S^{(n)}\left(\Omega\right)d\Omega=\Delta\omega^{2}.$
(17)

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 quantum-radar 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 state $\hat{\rho}_{\text{Q}}^{\phi}$ is zero-mean Gaussian state
characterized by the CM
$\mathbb{V}^{\phi}_{\text{Q}}=\bigoplus_{\Omega}\mathbb{V}_{\text{Q}}^{\phi,\Omega}$,
where

$\displaystyle\mathbb{V}^{\phi,\Omega}_{\text{Q}}=\begin{pmatrix}A_{\Omega}\mathbb{I}_{2}&C^{\Omega}_{\phi}\mathbb{R}_{\phi^{(-)}_{\Omega}}&C^{\Omega}_{\phi}\mathbb{R}_{\phi_{\Omega}^{(+)}}\\\
C^{\Omega}_{\phi}\mathbb{R}_{\phi^{(-)}_{\Omega}}&B^{\Omega}_{\phi}\mathbb{I}_{2}&D^{\Omega}_{\phi}\mathbb{W}_{\phi_{\Omega}^{(+)}-\phi_{\Omega}^{(-)}}\\\
C^{\Omega}_{\phi}\mathbb{R}_{\phi^{(+)}_{\Omega}}&D^{\Omega}_{\phi}\mathbb{W}_{\phi_{\Omega}^{(-)}-\phi_{\Omega}^{(+)}}&B^{\Omega}_{\phi}\mathbb{I}_{2}\end{pmatrix}$
(18)

with $A_{\Omega}=2S^{(n)}\left(\Omega\right)+1$,
$B^{\Omega}_{\phi}=2N_{B}+2\kappa_{\phi,\phi_{c}}S^{(n)}\left(\Omega\right)+1$,
$C^{\Omega}_{\phi}=2\sqrt{\kappa_{\phi,\phi_{c}}}S^{(p)}\left(\Omega\right)$,
$D^{\Omega}_{\phi}=A_{\Omega}\kappa_{\phi,\phi_{c}}$,
$\mathbb{R}_{\theta}=\text{Re}\left[\exp{\left(i\theta\right)}\left(\mathbb{Z}_{2}-i\mathbb{X}_{2}\right)\right]$,
$\mathbb{W}_{\theta}=\text{Re}\left[\exp{\left(i\theta\right)}\left(\mathbb{I}_{2}+\mathbb{Y}_{2}\right)\right]$.
Here $\mathbb{I}_{2}$, $\mathbb{X}_{2}$, $\mathbb{Y}_{2}$ and $\mathbb{Z}_{2}$
are the $2\times 2$ identity, Pauli X, Y, and Z matrices,
$\phi_{\Omega}^{(\pm)}=\xi+\left(\omega_{0}+\Omega\right)\left(\tau\pm
d\sin\phi/2c\right)$,
$N_{B}=\langle\hat{\mathcal{V}}_{1(2)}^{\dagger}\hat{\mathcal{V}}_{1(2)}\rangle/\left(1-\kappa\cos\phi\right)\simeq\langle\hat{\mathcal{V}}_{1(2)}^{\dagger}\hat{\mathcal{V}}_{1(2)}\rangle$.

###  2.3 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., $\Delta\omega T_{d}\rightarrow\infty$), it underestimates the
achievable error as the time-bandwidth product is finite (i.e., $\Delta\omega
T_{d}<\infty$). 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
[18,20,25-30]. In the following, we compare the estimation of $\phi$ by CRBs
and ZZBs in the quantum and classical cases.

####  2.3.1 Cramér-Rao bound

In radar detection, the target angle $\phi$ is encoded into the output state
$\hat{\rho}_{\phi}$. CRB indicates the minimum variance lower bound of the
unbias estimator for estimating $\phi$ from $\hat{\rho}_{\phi}$,
$\delta\phi_{\text{CRB}}^{2}\equiv 1/\mathcal{F}_{\phi}$, where
$\mathcal{F}_{\phi}$ is the quantum fisher information (QFI),

$\mathcal{F}_{\phi}\equiv\lim_{\epsilon\rightarrow
0}\;8\frac{1-\sqrt{F\left(\hat{\rho}_{\phi},\hat{\rho}_{\phi+\epsilon}\right)}}{\epsilon^{2}},$
(19)

and

$F\left(\hat{\rho}_{\phi},\hat{\rho}_{\phi+\epsilon}\right)=\left[\text{Tr}\left(\sqrt{\sqrt{\hat{\rho}_{\phi}}\hat{\rho}_{\phi+\epsilon}\sqrt{\hat{\rho}_{\phi}}}\right)\right]^{2}.$
(20)

is the Uhlmann fidelity [31] between states $\hat{\rho}_{\phi}$ and
$\hat{\rho}_{\phi+\epsilon}$. For our evaluation, in the classical scenario,
we assign
$\left\\{\hat{\rho}_{\phi},\hat{\rho}_{\phi+\epsilon}\right\\}=\left\\{\hat{\rho}^{\phi}_{\text{C}},\hat{\rho}^{\phi+\epsilon}_{\text{C}}\right\\}$,
whereas in quantum scenario,
$\left\\{\hat{\rho}^{\phi}_{\text{Q}},\hat{\rho}^{\phi+\epsilon}_{\text{Q}}\right\\}$.
Under the approximation $N_{S},\kappa\ll 1$ and $N_{B}\gg 1$, we have the
fidelities of classical and quantum cases as
$F^{\Omega}_{\text{C},\phi,\epsilon}\simeq 1-\kappa
S^{(n)}\left(\Omega\right)\Theta^{\Omega}_{\phi,\epsilon}/N_{B}$ and
$F^{\Omega}_{\text{Q},\phi,\epsilon}\simeq 1-2\kappa
S^{(n)}\left(\Omega\right)\Theta^{\Omega}_{\phi,\epsilon}/N_{B}$ at frequency
$\omega_{0}+\Omega$, where

$\displaystyle\Theta^{\Omega}_{\phi,\epsilon}\equiv$
$\displaystyle\cos\left(\phi-\phi_{c}\right)+\cos\left(\phi-\phi_{c}+\epsilon\right)$
(21)
$\displaystyle-2\sqrt{\cos\left(\phi-\phi_{c}\right)\cos\left(\phi+\epsilon-\phi_{c}\right)}\cos\left[\frac{\left(\omega_{0}+\Omega\right)d}{2c}\left\\{\sin\left(\phi+\epsilon\right)-\sin\phi\right\\}\right].$

Followed by the definition of QFI in Eq. (19), we derive the QFI for classical
and quantum radar as,

$\mathcal{F}^{\Omega}_{\text{C},\phi}\simeq\frac{1}{2}\mathcal{F}^{\Omega}_{\text{Q},\phi}\simeq\frac{S^{(n)}\left(\Omega\right)\kappa_{\phi,\phi_{c}}}{N_{B}}\left\\{\frac{d^{2}\left(\omega_{0}+\Omega\right)^{2}}{c^{2}}\cos^{2}\phi+\tan^{2}\left(\phi-\phi_{c}\right)\right\\}.$
(22)

Since QFI is additive across all the frequency modes $\Omega$, 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
$\mathcal{F}_{\text{C}(\text{Q}),\phi}=T_{d}\int_{-\infty}^{\infty}\mathcal{F}_{\text{C}(\text{Q}),\phi}^{\Omega}d\Omega$
to attain the gross QFIs of both scenarios $\mathcal{F}_{\text{C},\phi}\simeq
2\left(\text{SNR}\right)\Upsilon_{\phi}$ and
$\mathcal{F}_{\text{Q},\phi}\simeq 4\left(\text{SNR}\right)\Upsilon_{\phi}$,
where $\text{SNR}\equiv\kappa\mathcal{E}/N_{B}=\Delta\omega T_{d}\kappa
N_{S}/N_{B}$ and

$\Upsilon_{\phi}\equiv\frac{\cos\left(\phi-\phi_{c}\right)}{2}\left\\{\frac{d^{2}\left(\omega^{2}_{0}+\Delta\omega^{2}\right)}{c^{2}}\cos^{2}\phi+\tan^{2}\left(\phi-\phi_{c}\right)\right\\}.$
(23)

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 $\phi_{c}=\phi$ for evaluating
the CRB. Ultimately, we derive the quantum CRB (QCRB) and classical CRB (CCRB)
as $\delta\phi_{\text{CCRB}}^{2}=1/\mathcal{F}_{\text{C},\phi}$ and
$\delta\phi_{\text{QCRB}}^{2}=1/\mathcal{F}_{\text{Q},\phi}$, and plot them as
the red and blue dotted lines in Fig. 3.

####  2.3.2 Ziv-Zakai bound

Figure 3: The ZZBs and CRBs of quantum and classical cases normalized to
$\delta\phi^{2}_{\text{ref}}=\Delta\phi^{2}/12$. Blue solid and green dashed
curves denote the QZZBs derived from Eq. (26) and Eq. (27), the red solid
curve is derived from Eq. (27), red and blue dotted lines stands for the CCRB
and QCRB. $\omega_{0}/2\pi=100$ GHz, $\Delta\omega/2\pi=5$ GHz, $d=8$ m,
$\phi_{c}=\phi=0.1$ rad and $\Delta\phi=\pi/100$. The gray vertical dashed
line denotes the location of SNR threshold.

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 $P_{X}\left(\cdot\right)$, and then
the ZZB of the MSE can be evaluated as

$\displaystyle\delta\phi^{2}_{\text{ZZB}}$
$\displaystyle\equiv\int_{0}^{\infty}d\zeta\,\zeta
V\left\\{\int_{-\infty}^{\infty}dx\,\min\left\\{P_{X}\left(x\right),P_{X}\left(x+\zeta\right)\right\\}\text{Pr}\left(x;x+\zeta\right)\right\\},$
(24)

where $P_{X}\left(x\right)$ denotes the probability density of prior knowledge
at angle $x$, $V$ denotes the valley-filling operation, i.e.,
$Vf\left(\tau\right)=\max_{\eta\geq 0}f\left(\tau+\eta\right)$, and
Pr$\left(x;x+\zeta\right)$ denotes the minimum error probability to
distinguish two hypotheses,

$\mathcal{H}_{1}:X=x,\;\;\;\;\mathcal{H}_{2}:X=x+\zeta.$ (25)

Considering uniform prior-knowledge in the range
$[\phi-\Delta\phi/2,\phi+\Delta\phi/2]$, where $\Delta\phi$ denotes the
uncertainty range, we rewrite Eq. (24) as

$\delta\phi^{2}_{\text{ZZB}}=\int_{0}^{\Delta\phi}d\zeta\,\zeta\left\\{\frac{1}{\Delta\phi}\int_{\phi-\Delta\phi/2}^{\phi+\Delta\phi/2-\zeta}dx\,\text{Pr}\left(x;x+\zeta\right)\right\\}.$
(26)

Note that as $\text{Pr}\left(x;x+\zeta\right)$ decreases with $\zeta$
increasing, and the distribution is uniform, so the principle value is always
achieved at $\zeta$. Considering the limit of $\Delta\phi\ll 1$, we can
approximate the above results as

$\displaystyle\delta\phi^{2}_{\text{ZZB}}$
$\displaystyle\simeq\int_{0}^{\Delta\phi}d\zeta\,\zeta\left(1-\frac{\zeta}{\Delta\phi}\right)\text{Pr}\left(\phi;\phi+\zeta\right).$
(27)

We will justify this assumption later. The error probability
$\text{Pr}\left(\cdot\right)$ 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., $\text{Pr}\rightarrow
P^{(\text{QCB})}$). 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 A)

$\displaystyle P^{(\text{QCB})}_{\text{C}}\left(\phi;\phi+\zeta\right)$
$\displaystyle\leq\exp{\left[-\left(\text{SNR}\right)\bar{\Theta}_{\phi,\zeta}/2\right]}/2$
(28) $\displaystyle P^{(\text{QCB})}_{\text{Q}}\left(\phi;\phi+\zeta\right)$
$\displaystyle\leq\exp{\left[-2\left(\text{SNR}\right)\bar{\Theta}_{\phi,\zeta}\right]}/2,$

where

$\displaystyle\bar{\Theta}_{\phi,\zeta}\simeq$
$\displaystyle\cos\left(\phi-\phi_{c}\right)+\cos\left(\phi-\phi_{c}+\zeta\right)-2\sqrt{\cos\left(\phi-\phi_{c}\right)\cos\left(\phi-\phi_{c}+\zeta\right)}\;\times$
(29)
$\displaystyle\exp{\left[-\frac{d^{2}\Delta\omega^{2}}{8c^{2}}\left\\{\sin\left(\phi+\zeta\right)-\sin\phi\right\\}^{2}\right]}\cos\left[\frac{d\omega_{0}}{2c}\left\\{\sin\left(\phi+\zeta\right)-\sin\phi\right\\}\right],$

under the approximation of $N_{S},\kappa\ll 1$ and $N_{B}\gg 1$. 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
$\phi_{c}=\phi$ and plug the upper bounds of $P^{(\text{QCB})}_{\text{C}}$ and
$P^{(\text{QCB})}_{\text{Q}}$ from Eq. (28) into Eq. (27) to evaluate the ZZBs
normalized to $\delta\phi^{2}_{\text{ref}}=\Delta\phi^{2}/12$ (i.e.,
$\delta\phi^{2}_{\text{ref}}$ is the ZZB by assuming $\text{Pr}=1/2$) in Fig.
3.

Under appropriate parameter settings, Fig. 3 shows a huge quantum advantage
($\sim 30$ 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 $\Delta\phi\ll
1$ 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 $\Delta\phi$ (e.g.,
$\Delta\phi=\pi/100$).

###  2.4 Quantum advantage versus pulse duration and range

To understand the practical use scenario of quantum radar, it is necessary to
evaluate the trade-off between the quantum advantage with the pulse duration
at a given set of physical parameter (e.g., $\kappa$ 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${}_{\text{th}}$) of quantum
radar, manifested in Fig. 3. In the high SNR regime
($\text{SNR}\gg\text{SNR}_{\text{th}}$), the major contribution of the
integration over $\zeta$ in Eq. (27) comes from the values near the origin
and, thereafter, QZZB can be asymptotically derived as

$\delta\phi^{2}_{\text{QZZB}}\simeq
1/4\Upsilon_{\phi}\left(\text{SNR}\right),$ (30)

where $\Upsilon_{\phi}$ was defined in Eq. (23). On the other hand, the QZZB
in the low SNR regime ($\text{SNR}\ll\text{SNR}_{\text{th}}$) follows the
inequality,

$\delta\phi_{\text{ref}}^{2}\geq\delta\phi^{2}_{\text{QZZB}}\gtrsim\delta\phi_{\text{ref}}^{2}\exp{\left[-4\left(\text{SNR}\right)\right]},$
(31)

i.e., $0\leq\bar{\Theta}_{\phi,\zeta}\leq 2$. SNR${}_{\text{th}}$ 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.,
$\delta\phi_{\text{ref}}^{2}\exp{\left[-4\left(\text{SNR}\right)\right]}$),
and formulated as

$\text{SNR}_{\text{th}}\equiv
g\left[1/\Upsilon_{\phi}\delta\phi_{\text{ref}}^{2}\right]/4,$ (32)

where $g\left[y\right]$ is the inverse function of $ye^{-y}$, $\forall y>1$.

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

$\kappa=\frac{G_{T}}{4\pi L^{2}}\times\frac{\sigma A_{R}}{4\pi L^{2}}\ll 1,$
(33)

where $G_{T}=A_{R}/\left(2\pi c/\omega_{0}\right)^{2}$ is radar’s antenna
gain, $A_{R}$ is the antenna’s area, $\sigma$ is the target’s cross section
area. Plugging these parameters into our simulation model and fixing
$\text{SNR}=\text{SNR}_{\text{th}}$ by tuning $N_{S}$, we numerically
calculate the quantum advantage trade off in Fig. 4. Under this parameter
setting, the quantum advantage is appreciable ($\gtrsim 15$ dB) for distance
$L\sim 500$ m by setting a practical pulse duration of $T_{d}=0.1$ second.

Figure 4: The numerical calculation of quantum advantage at
SNR${}_{\text{th}}=4.88$ dB versus pulse duration and object’s distance.
$A_{R}=20$ m2, $\sigma=0.1$ m2, $N_{B}=32$ (i.e., $T_{B}=150$ K),
$\omega_{0}/2\pi=100$ GHz, $\Delta\omega/2\pi=5$ GHz, $d=10$ m, $\phi=0.1$ rad
and $\Delta\phi=\pi/100$.

##  3 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${}_{\text{th}}$) 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${}_{\text{th}}$ 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 $\phi$; however, we show that such a mode sorting approach only
brings marginal improvement even in the best-case scenario (see more details
in 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.

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.,
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$) is upper bounded by the QCB,

$P^{(\text{QCB})}\equiv\frac{1}{2}\underset{0\leq s\leq
1}{\text{inf}}\left\\{\text{Tr}\left[\hat{\rho}_{1}^{s}\hat{\rho}_{2}^{1-s}\right]\right\\},$
(34)

where $\hat{\rho}_{1}$ and $\hat{\rho}_{2}$ are the density operators
corresponding to $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$. In radar scenario,
the two hypotheses refer to parameters specified in Eq. (25), and the QCB
defined in Eq. (34) [32] under the approximation $\Delta\phi\ll 1$ can be
derived as

$P^{(\text{QCB})}\left(\phi;\phi+\zeta\right)=\prod_{\Omega}P^{\Omega}\left(\phi;\phi+\zeta\right),$
(35)

with each frequency contributing

$\displaystyle P^{\Omega}\left(\phi;\phi+\zeta\right)=\underset{0\leq s\leq
1}{\text{inf}}\left\\{P^{\Omega,s}\left(\phi;\phi+\zeta\right)\right\\},$ (36)

and

$P^{\Omega,s}\left(\phi;\phi+\zeta\right)=\frac{2^{N-1}\overset{N}{\underset{j=1}{\prod}}K^{\phi,\zeta}_{j,s}}{\sqrt{\det\Lambda_{\Omega,s}^{\phi,\zeta}}}\exp{\left[-\frac{1}{2}\delta\mathbb{v}_{\Omega}^{\phi,\zeta}\left(\Lambda_{\Omega,s}^{\phi,\zeta}\right)^{-1}\left(\delta\mathbb{v}_{\Omega}^{\phi,\zeta}\right)^{T}\right]}.$
(37)

Here $N$ denotes the number of involved mode (i.e., $N=2$ in classical and
$N=3$ in quantum),
$\delta\mathbb{v}_{\Omega}^{\phi,\zeta}=\langle\hat{\mathbb{q}}\rangle_{\Omega}^{\phi+\zeta}-\langle\hat{\mathbb{q}}\rangle_{\Omega}^{\phi}$
denotes the quadrature mean difference,
$G_{s}^{(\pm)}\left[y\right]=\sqrt{2}/\left[\left(y+1\right)^{s}\pm\left(y-1\right)^{s}\right]$,
i.e., $\forall y>1$,
$K^{\phi,\zeta}_{j,s}=G_{s}^{(-)}\left[\lambda^{\phi}_{j}\right]\times
G_{1-s}^{(-)}\left[\lambda^{\phi+\zeta}_{j}\right]$,
$\Lambda_{\Omega,s}^{\phi,\zeta}=\mathbb{C}^{\phi}_{\Omega,s}+\mathbb{C}_{\Omega,1-s}^{\phi+\zeta}$,

$\mathbb{C}^{\phi^{\prime}}_{\Omega,s}\equiv\mathbb{S}_{\Omega}^{\phi^{\prime}}\left\\{\bigoplus_{j=1}^{N}\frac{G_{s}^{(+)}\left[\lambda^{\phi^{\prime}}_{j}\right]}{G_{s}^{(-)}\left[\lambda^{\phi^{\prime}}_{j}\right]}\otimes\mathbb{I}_{2}\right\\}\left(\mathbb{S}_{\Omega}^{\phi^{\prime}}\right)^{T},$
(38)

$\lambda^{\phi^{\prime}}_{j}$ is the $j$-th symplectic eigenvalue associated
with the symplectic matrix $\mathbb{S}^{x^{\prime}}_{\Omega}$ in parameter
$\phi^{\prime}=\left\\{\phi,\phi+\zeta\right\\}$.

Subsequently, we claim that the infimum in Eq. (36) occurs at $s=1/2$ and
support it with numerical justifications and perturbation theory. Fig. 5 shows
the plot of Eq. (37) in quantum case with $\Omega=0$ as a function of $s$
(i.e., $P^{0,s}_{\text{Q}}$). In Fig. 5(a), we fix $\kappa$ while changing
$N_{S}$; in Fig. 5(b), $N_{S}$ is fixed while $\kappa$ 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 $N_{S},\kappa\ll 1$ and $N_{B}\gg 1$, and easily
conclude that infinimum does occur at $s=1/2$ by the _Theorem 4_ of Ref. [34].

Therefore, we can calculate Eq. (36) in both cases as
$P^{\Omega}_{\text{C}}\left(\phi;\phi+\zeta\right)\leq\exp{\left[-\kappa
S^{(n)}\left(\Omega\right)\Theta^{\Omega}_{\phi,\zeta}/2N_{B}\right]}/2$ and
$P^{\Omega}_{\text{Q}}\left(\phi;\phi+\zeta\right)\leq\exp{\left[-2\kappa
S^{(n)}\left(\Omega\right)\Theta^{\Omega}_{\phi,\zeta}/N_{B}\right]}/2$ under
the approximation of $N_{S},\kappa\ll 1$ and $N_{B}\gg 1$, where
$\Theta^{\Omega}_{\phi,\zeta}$ was defined in Eq. (21), yielding the QCBs in
Eq. (28).

Figure 5: $P^{0,s}_{\text{Q}}$ versus $s$ with $\omega_{0}/2\pi=100$ GHz,
$\Delta\omega/2\pi=1$ MHz, $d=20$ m, $N_{B}=32$, $\phi=0$ rad and
$\zeta=\pi/3$. (a) Blue, red and black curves stand for different choices of
$\kappa$ and $T_{d}$ as $\left\\{\kappa,T_{d}\right\\}=\left\\{5\times
10^{-1},8\leavevmode\nobreak\ \text{s}\right\\}$, $\left\\{5\times
10^{-3},0.8\leavevmode\nobreak\ \text{ms}\right\\}$ and $\left\\{5\times
10^{-4},8\leavevmode\nobreak\ \text{ms}\right\\}$ with $N_{S}=0.1$. (b) Cyan,
magenta and gray curves stand for different choices of $N_{S}$ and $T_{d}$ as
$\left\\{N_{S},T_{d}\right\\}=\left\\{2,0.32\leavevmode\nobreak\ \text{\textmu
s}\right\\}$, $\left\\{0.63,10\leavevmode\nobreak\ \text{\textmu s}\right\\}$
and $\left\\{0.2,3.2\leavevmode\nobreak\ \text{\textmu s}\right\\}$ with
$\kappa=0.5$.

##  Appendix B Mode sorter

Figure 6: The scheme of single-receiver radar.

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. 6, 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 $\phi$, same as
in dual-receiver radar, but set the compensation angle to be zero $\phi_{c}=0$
(i.e., the face of radar is vertically directed, shown in Fig. 6) and consider
a soft-aperture located at the focal plane of the paraboloid antenna in Fig.
7(a). The received pulse is collected by the paraboloid antenna as an
elliptical Gaussian beam
$w\left(t;x,y\right)=s\left(t\right)\varepsilon\left(x,y\right)$, where

$\varepsilon\left(x,y\right)=\sqrt{\frac{\cos\phi}{2\pi
r^{2}}}\exp{\left[-\frac{x^{2}}{4r^{2}}-\frac{y^{2}}{4r^{2}/\cos^{2}\phi}\right]},$
(39)

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
$W\left(\Omega;x,y\right)=S\left(\Omega\right)\varepsilon\left(x,y\right)$,
where $s\left(t\right)$ and $S\left(\Omega\right)$ were denoted in Eq. (8) and
Eq. (9). The asymmetricity of the two axes comes from the angle deviation
$\phi$ between the incident plane and the imaging plane, yielding the y-axis,
shown in Fig. 7(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. 7(a). The
proposed HG mode has the eigenfunction

$\displaystyle\psi_{n,m}\left(x,y\right)=\frac{\beta H_{m}\left(\beta
x\right)H_{n}\left(\beta y\right)}{\sqrt{\pi
n!m!2^{n+m}}}\exp{\left[-\nu\beta^{2}\left(x^{2}+y^{2}\right)\right]},$ (40)

indexed by non-negative integers $n,m\in\mathbb{N}_{0}$, where
$\beta=\sqrt{2}\left(1+4D_{f}\right)^{1/4}/\delta$, $D_{f}=\left(2\pi
A_{\delta}cL/\omega_{0}\right)^{2}$ is the Fresnel number,
$A_{\delta}=\pi\delta^{2}/4$ is the aperture area (i.e., $\delta$ is its
diameter), $\nu=\left(1+i\omega_{0}/c\beta^{2}L\right)/2\simeq 1/2$, and
$H_{n}\left(\cdot\right)$ denotes the $n$-th order Hermite polynomial.

Figure 7: Scheme of single-receiver. (a) Trajectory of receiving signal. (b)
The schematics of $k$-series mode sorter.

The received signal can be regarded as a far field if $D_{f}\ll 1$ 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 $\phi$, as demonstrated in the inset of Fig. 7(a).

Figure 8: Occupation probability of $\psi_{0,0}$, $\psi_{0,1}$ and
$\psi_{1,0}$ in different combinations of $\delta/r$. $L=10$ km and
$\omega_{0}/2\pi=100$ GHz.

To begin our analyses, we evaluate the overlap of a Gaussian function with
each basis $\psi_{n,m}\left(x,y\right)$ at the image plane, leading to the
associate occupation probability

$\displaystyle p_{n,m}^{\phi}\equiv$
$\displaystyle\left\\{\frac{\left(2n-1\right)!!\left(2m-1\right)!!}{2^{n+m}n!m!}\right\\}\times\frac{4\chi\cos\phi}{\left(1+\chi\right)\left(\cos^{2}\phi+\chi\right)}\left(\frac{1-\chi}{1+\chi}\right)^{2n}\left(\frac{\cos^{2}\phi-\chi}{\cos^{2}\phi+\chi}\right)^{2m},$
(41)

where $\chi=2\beta^{2}r^{2}$ and this decomposition applies to our Gaussian
beam imaging system [36]. For smooth communication, we relabel the indices of
probability at each mode as
$\left\\{\mathcal{P}_{1}^{\phi},\mathcal{P}_{2}^{\phi},\mathcal{P}_{3}^{\phi},\mathcal{P}^{\phi}_{4},\mathcal{P}^{\phi}_{5},\mathcal{P}^{\phi}_{6},\cdots,\mathcal{P}^{\phi}_{k},1-\sum^{k}_{j=1}\mathcal{P}^{\phi}_{j}\right\\}=\left\\{p^{\phi}_{0,0},p^{\phi}_{1,0},p^{\phi}_{0,1},p^{\phi}_{2,0},p^{\phi}_{1,1},p^{\phi}_{0,2},\cdots\right\\}$
and plot the lowest three order modes in Fig. 8. This mathematical
decomposition process can be visualized by passing the incident Gaussian beam
through a $k$-series beamsplitters, shown in Fig. 7(b). The $j$-th mode (i.e.,
$1\leq j\leq k$) beamsplitter has the matrix form

$\mathbb{B}^{\phi}_{j}=\begin{pmatrix}\sqrt{\eta^{\phi}_{j}}&\sqrt{1-\eta^{\phi}_{j}}\\\
-\sqrt{1-\eta^{\phi}_{j}}&\sqrt{\eta^{\phi}_{j}}\end{pmatrix}\otimes\mathbb{I}_{2},$
(42)

where

$\eta^{\phi}_{j}=\begin{cases}\mathcal{P}^{\phi}_{j}\;\;,&\;\;\;\;\;\;j=1\\\
\mathcal{P}^{\phi}_{j}/\left(1-\sum_{l=1}^{j-1}\mathcal{P}^{\phi}_{l}\right),&2\leq
j\leq k\end{cases}.$ (43)

These $k$ beamsplitters entangle the $k$ vacuum modes $\hat{\mathcal{V}}$ 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 $\delta/r=2$ such that the
fundamental mode ($\psi_{0,0}$) dominates all other HG modes when the
concerning angle is small $\phi\ll 1$, shown in Fig. 8.

###  B.1 Classical radar

In classical single-receiver scheme, the global state, at the angle $\phi$,
has the CM
$\mathbb{V}_{\text{C}}=\left(2N_{B}+1\right)\bigoplus_{\Omega}\mathbb{I}_{4}$
and quadrature mean
$\langle\hat{\mathbb{q}}\rangle_{\text{C}}^{\phi}=\bigoplus_{\Omega}\mathbb{B}^{\phi}_{1}\langle\hat{\mathbb{q}}\rangle_{\text{C}}^{\phi,\Omega}$,
where
$\langle\hat{\mathbb{q}}\rangle^{\phi,\Omega}_{\text{C}}=\sqrt{2S^{(n)}\left(\Omega\right)\kappa_{\phi,0}}\left(\cos\Xi,\sin\Xi,0,0\right)^{T}$
is the quadrature at frequency $\omega_{0}+\Omega$ in the basis of
$\left(\hat{q}_{S},\hat{p}_{S},\hat{q}^{\bot}_{S},\hat{p}^{\bot}_{S}\right)$,
where $\Xi\in(0,2\pi]$ is the overall phase of the return signal,
$\left\\{\hat{q}_{S},\hat{p}_{S}\right\\}$ and
$\left\\{\hat{q}^{\bot}_{S},\hat{p}^{\bot}_{S}\right\\}$ are the quadrature
pairs of signal, projected on $\psi_{0,0}$ (i.e., occupation probability
$\mathcal{P}^{\phi}_{1}$), and the residual HG modes $\psi^{\bot}$ (i.e.,
occupation probability $1-\mathcal{P}^{\phi}_{1}$). Akin to calculating CRB
and ZZB of dual-receiver radar in the main text, under the approximation
$N_{S},\kappa\ll 1,N_{B}\gg 1$, we asymptotically and analytically derive the
CCRB (red dotted line in Fig. 9),

$\delta\phi^{2}_{\text{CCRB}}\simeq\left\\{\frac{\text{SNR}}{2\cos\phi}\left\\{\sin^{2}\phi+\frac{\cos^{2}\phi}{\mathcal{P}_{1}^{\phi}\left(1-\mathcal{P}_{1}^{\phi}\right)}\left(\frac{d\mathcal{P}_{1}^{\phi}}{d\phi}\right)^{2}\right\\}\right\\}^{-1}.$
(44)

Similarly, the QCB for distinguishing the hypotheses in Eq. (25) can be
obtained as
$P^{(\text{QCB})}_{\text{C}}\left(x;x+\zeta\right)\leq\exp{\left[-\left(\text{SNR}\right)\Gamma_{x,\zeta}/4\right]}/2$,
where

$\Gamma_{x,\zeta}\simeq\cos
x+\cos\left(x+\zeta\right)-2\sqrt{\cos\left(x\right)\cos\left(x+\zeta\right)}\left(\sqrt{\mathcal{P}^{x}_{1}\mathcal{P}^{x+\zeta}_{1}}+\sqrt{\left(1-\mathcal{P}^{x}_{1}\right)\left(1-\mathcal{P}^{x+\zeta}_{1}\right)}\right).$
(45)

rior knowledge in $\left[\phi-\Delta\phi/2,\phi+\Delta\phi/2\right]$, CZZB is
numerically derived by plugging the upper bound of
$P^{(\text{QCB})}_{\text{C}}$ into Eq. (27) with the approximation
$\Delta\phi\ll 1$ and is plotted as the red solid curve in Fig. 9.

In Eq. (44), note that the size of the receiver antenna comes in via the
transmissivity $\kappa$ in the SNR (see Eq. (33)). The area of aperture
$A_{\delta}$ 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.

Figure 9: The ZZBs and CRBs of quantum and classical cases in single-receiver
scenario. $\Delta\phi=\pi/100$ and $\phi=\pi/2-\Delta\phi/2$.

###  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 $x^{\prime}=\left\\{x,x+\zeta\right\\}$ is

$\displaystyle\mathbb{V}_{\text{Q}}=\left(\mathbb{I}_{2}\oplus\mathbb{B}^{x^{\prime}}_{1}\right)\begin{pmatrix}A_{\Omega}\mathbb{I}_{2}&\mathcal{C}^{\Omega}_{x^{\prime}}\mathbb{R}_{x^{\prime}}&\mathbb{0}_{2}\\\
\mathcal{C}^{\Omega}_{x^{\prime}}\mathbb{R}_{x^{\prime}}&\mathcal{B}^{\Omega}_{x^{\prime}}\mathbb{I}_{2}&\mathbb{0}_{2}\\\
\mathbb{0}_{2}&\mathbb{0}_{2}&\mathcal{B}^{\Omega}_{x^{\prime}}\mathbb{I}_{2}\end{pmatrix}\left(\mathbb{I}_{2}\oplus\mathbb{B}^{x^{\prime}}_{1}\right)^{T},$
(46)

in quadrature basis
$\left(\hat{q}_{I},\hat{p}_{I},\hat{q}_{S},\hat{p}_{S},\hat{q}^{\bot}_{S},\hat{p}^{\bot}_{S}\right)$,
where $\left\\{\hat{q}_{I},\hat{p}_{I}\right\\}$ denotes the quadrature pair
of idler mode,
$\mathcal{B}^{\Omega}_{x^{\prime}}=2N_{B}+2\kappa_{x^{\prime},0}S^{(n)}\left(\Omega\right)+1$,
$\mathcal{C}^{\Omega}_{x^{\prime}}=2\sqrt{\kappa_{x^{\prime},0}}S^{(p)}\left(\Omega\right)$.
The QCRB (blue dotted line in Fig. 9) is asymptotically and analytically
derived as,

$\delta\phi^{2}_{\text{QCRB}}\simeq\left\\{\frac{\text{SNR}}{\cos\phi}\left\\{\sin^{2}\phi+\frac{\cos^{2}\phi}{\mathcal{P}_{1}^{\phi}\left(1-\mathcal{P}_{1}^{\phi}\right)}\left(\frac{d\mathcal{P}_{1}^{\phi}}{d\phi}\right)^{2}\right\\}\right\\}^{-1}$
(47)

and QCB, for distinguishing $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ in Eq.
(25),
$P^{(\text{QCB})}_{\text{Q}}\left(x;x+\zeta\right)\leq\exp{\left[-\left(\text{SNR}\right)\Gamma_{x,\zeta}\right]}/2$
under the approximation $N_{S},\kappa\ll 1$ and $N_{B}\gg 1$. With uniform
prior knowledge in $\left[\phi-\Delta\phi/2,\phi+\Delta\phi/2\right]$, QZZBs
can be numerically calculated with or without the approximation $\Delta\phi\ll
1$ by Eq. (27), which are plotted as the blue solid and green dashed curves in
Fig. 9. Same conclusion as dual-receiver, the concurrence of two curves
justifies the formula of Eq. (27) and they both coincide with the QCRB in high
SNR regime. In Fig. 9, we plot ZZBs in both classical and quantum scenarios by
fixing the uncertainty tolerance $\Delta\phi=\pi/100$ and setting
$\phi=\pi/2-\Delta\phi/2$. The choice of $\phi=\pi/2-\Delta\phi/2$ has the
maximal distinguishability between $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$,
because the occupation probability in HG fundamental mode varies dramatically
as target angle approaches $\pi/2$. However, in Fig. 9, despite the optimal
choice of $\phi$, the noticeable reduction of
$\delta\phi^{2}/\delta\phi^{2}_{\text{ref}}$ 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., $\sim
1$ dB).

## References

## References

  * [1] H. L. Van Trees, _Detection, Estimation, and Modulation Theory, Part I: Detection, Estimation, and Linear Modulation Theory_ (Wiley, New York, 2001).
  * [2] H. L. Van Trees, _Detection, Estimation, and Modulation Theory, Part III: Radar-Sonar Signal Processing and Gaussian Signals in Noise_ (Wiley, New York, 2001).
  * [3] A. Mallinckrodt and T. Sollenberger, _Optimum pulse-time determination_ , IRE Trans. Inf. Theory 3, 151 (1954).
  * [4] M. I. Skolnik, _Theoretical accuracy of radar measurements_ , IRE transactions on aeronautical and navigational electronics ANE-7, 123 (1960).
  * [5] M. I. Skolnik, _Introduction to Radar Systems, Third Edition_ (McGraw-Hill, New York, 2002).
  * [6] _Radar Handbook_ , edited by M. I. Skolnik (McGraw Hill, New York, 2008).
  * [7] J. Marcum, _A statistical theory of target detection by pulsed radar_ , IRE Trans. Inf. Theory 6, 59 (1960).
  * [8] V. Giovannetti, S. Lloyd, and L. Maccone, _Quantum enhanced positioning and clock synchronization_ , Nature (London) 412, 417 (2001).
  * [9] V. Giovannetti, S. Lloyd, and L. Maccone, _Quantum enhanced measurements: Beating the standard quantum limit_ , Science 306, 1330 (2004).
  * [10] J. H. Shapiro, _Quantum pulse compression laser radar_ , Proc. SPIE Int. Soc. Opt. Eng. 6603, 660306 (2007).
  * [11] L. Maccone and C. Ren, _Quantum Radar_ , Phys. Rev. Lett. 124, 200503 (2020).
  * [12] M. Lanzagorta, _Quantum Radar Synthesis Lectures on Quantum Computing_ (Morgan, Claypool, San Rafael, 2012).
  * [13] R. G. Torromé, N. B. Bekhti-Winkel, and P. Knott, _Introduction to quantum radar_ , arXiv:2006.14238 (2020).
  * [14] J. H. Shapiro, _The quantum illumination story_ , IEEE Trans. Aerosp. Electron. Syst. 35, 8 (2020).
  * [15] S.-H. Tan _et al_ , _Quantum Illumination with Gaussian States_ , Phys. Rev. Lett. 101, 253601 (2008).
  * [16] Q. Zhuang, Z. Zhang, and J. H. Shapiro, _Optimum Mixed State Discrimination for Noisy Entanglement-Enhanced Sensing_ , Phys. Rev. Lett. 118, 040801 (2017).
  * [17] S. Lloyd, _Enhanced sensitivity of photodetection via quantum illumination_ , Science 321, 1463 (2008).
  * [18] Q. Zhuang and J. H. Shapiro, _Ultimate Accuracy Limit of Quantum Pulse-Compression Ranging_ , Phys. Rev. Lett. 128, 010501 (2022).
  * [19] Q. Zhuang, _Quantum Ranging with Gaussian Entanglement_ , Phys. Rev. Lett. 126, 240501 (2021).
  * [20] M. Tsang, _Ziv-Zakai Error Bounds for Quantum Parameter Estimation_ , Phys. Rev. Lett. 108, 230401 (2012).
  * [21] M. S. Ebrahimi, S. Zippilli, D. Vitali, _Feedback-enabled Microwave Quantum Illumination_ , Quantum Sci. Technol. 7 035003 (2022)
  * [22] S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro, and S. Pirandola, _Microwave Quantum Illumination_ , Phys. Rev. Lett. 114, 080503 (2015).
  * [23] J. N. Sanders-Reed and S. J. Fenley, _Visibility in degraded visual environments (dve)_ , Proc. SPIE Int. Soc. Opt. Eng. 10642, 106420S (2018).
  * [24] M. Caris, W. Johannes, S. Sieger, V. Port, and S. Stanko, _Detection of small UAS with W-band radar_ , in Proceedings of the 18th International Radar Symposium , pp. 1–6 (IEEE, New York, 2017).
  * [25] H. L. Van Trees and K. L. Bell, _Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking_ , Wiley-IEEE, Piscataway, New York, (2007).
  * [26] J. Ziv and M. Zakai, _Some lower bounds on signal parameter estimation_ , IEEE Trans. Inform. Theor. 15, 386 (1969).
  * [27] L. P. Seidman, _Performance limitations and error calculations for parameter estimation_ , Proc. IEEE 58, 644 (1970).
  * [28] D. Chazan, M. Zakai, and J. Ziv, _Improved Lower Bounds on Signal Parameter Estimation_ , IEEE Trans. Inform. Theor. 21, 90 (1975).
  * [29] S. Bellini and G. Tartara, _Bounds on Error in Signal Parameter Estimation_ , IEEE Trans. Commun. 22, 340 (1974).
  * [30] E. Weinstein, _Relations between Belini-Tartara, Chazan-Zakai-Ziv, and Wax-Ziv lower bounds_ , IEEE Trans. Inform. Theor. 34, 342 (1988).
  * [31] L. Banchi, S. L. Braunstein, and S. Pirandola, _Quantum Fidelity for Arbitrary Gaussian States_ , Phys. Rev. Lett. 115, 260501 (2015).
  * [32] S. Pirandola and S. Lloyd, _Computable bounds for the discrimination of Gaussian states_ , Phys. Rev. A 78, 012331 (2008).
  * [33] J. L. Pereira, L. Banchi, S. Pirandola, _Symplectic decomposition from submatrix determinants_ , Proc. R. Soc. A. 477, 20210513 (2021).
  * [34] M. R. Grace and Saikat Guha, _Perturbation theory for quantum information_ , arXiv:2106.05533 (2021).
  * [35] M. Tsang, R. Nair, and X.-M. Lu, _Quantum Theory of Superresolution for Two Incoherent Optical Point Sources_ , Phys. Rev. Applied 6, 031033 (2016).
  * [36] J. Shapiro, S. Guha, and B. Erkmen, _Ultimate channel capacity of free-space optical communications_ , J. Opt. Netw. 4, 501 (2005).