# Bi-frequency illumination: a quantum-enhanced protocol

Mateo Casariego<EMAIL_ADDRESS>Physics of Information and
Quantum Technologies Group, Instituto de Telecomunicações, Portugal  Instituto
Superior Técnico, Universidade de Lisboa, Portugal  Yasser Omar
<EMAIL_ADDRESS>Physics of Information and Quantum Technologies Group,
Instituto de Telecomunicações, Portugal  Instituto Superior Técnico,
Universidade de Lisboa, Portugal  Mikel Sanz<EMAIL_ADDRESS>Department of
Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644,
E-48080 Bilbao, Spain  IQM, Nymphenburgerstr. 86, 80636 Munich, Germany
IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao,
Spain

###### Abstract

We propose a quantum-enhanced sensing protocol to measure the response of a
target object to the frequency of a probe in a noisy and lossy scenario. In
our protocol, a bi-frequency state illuminates a target embedded in a thermal
bath, whose reflectivity $\eta(\omega)$ is frequency-dependent. After a lossy
interaction with the object, we estimate the parameter
$\lambda=\eta(\omega_{2})-\eta(\omega_{1})$ in the reflected beam, which
captures information about the response of the object to different
electromagnetic frequencies. Computing the quantum Fisher information $H$
relative to the parameter $\lambda$ in an assumed neighborhood of $\lambda\sim
0$ for a two-mode squeezed state ($H_{Q}$), and a coherent state ($H_{C}$), we
show that a quantum enhancement in the estimation of $\lambda$ is obtained
when $H_{Q}/H_{C}>1$. This quantum advantage grows with the mean reflectivity
of the probed object, and is noise-resilient. We derive explicit formulas for
the optimal observables, and propose a general experimental scheme based on
elementary quantum optical transformations. Furthermore, our work opens the
way to applications in both radar and medical imaging, in particular in the
microwave domain.

Introduction. Quantum information technologies are opening very promising
prospects for faster computation, securer communications, and more precise
detection and measuring systems, surpassing the capabilities and limits of
classical information technologies (nielsen_chuang_2010, ; google-quantum-
supremacy, ; PhysRevLett.69.3598, ; Pirandola_2018, ; katori, ). Namely, in
the domain of quantum sensing and metrology (Giovannetti_2011, ), we are
currently witnessing a boost of applications to a wide spectrum of physical
problems: from gravimetry and geodesy (bongs, ; menoret, ; flury, ; grotti, ;
zeuthen, ), clock synchronisation (katori, ; okeke, ), thermometry
(thermometry, ) and bio-sensors (omar, ; plenio, ; biosensors, ; jensen, ), to
experimental proposals to seek quantum behavior in macroscopic gravity
(milburn, ), to name just a few. Most of these studies are focused on unlossy
(unitary) scenarios, while the more realistic, lossy case needs to be
investigated further lee ; changhun ; zhang ; zhang2 . It is thus important to
propose new sensing protocols that show a quantum enhancement even in the
presence of information losses. Quantum illumination Lloyd1463 ;
PhysRevLett.101.253601 ; guha2009 ; shapiroStory ; alsing2019 ;
pirandolaMicrowave ; shabir ; borre ; genovese ; gregory  is a particularly
interesting example of a lossy protocol where the use of entanglement proves
useful even in an entanglement-braking scenario. In a radar setting, the
detection of a low-reflectivity object in a noisy thermal environment with a
low-intensity signal is shown to be enhanced when the signal is entangled to
an idler that is kept for a future joint measurement with the reflected state.
The decision problem of whether there is an object or not can be rephrased as
a quantum estimation of the object’s reflectivity $\eta$, in order to
discriminate an absence ($\eta=0$) from a presence ($\eta\ll 1$) of a low-
reflectivity object Sanz_2017 .

The goal of quantum estimation (doi:10.1142/S0219749909004839, ) is to
construct an estimator $\tilde{\lambda}$ for certain parameter $\lambda$
characterizing the system. It is noteworthy that not every parameter in a
system corresponds to an observable, which may imply data post-processing.
However, the theory provides techniques to obtain an optimal observable –not
necessarily unique, i.e. whose mean square error is minimal. The estimator
$\tilde{\lambda}$ is nothing but a map from the results of measuring the
optimal observable to the set of possible values of the parameter $\lambda$.
One of the main results of this theory is the quantum Cramér-Rao (qCR) bound,
which sets the ultimate precision of any estimator. Whether this bound is
achievable or not depends on the data-analysis method used, and on the nature
of the experimental noise. In most practical situations, maximum likelihood
methods for unbiased estimators, together with Gaussian noise make the bound
achievable. In order to find the qCR –and the explicit form of the optimal
observable– one needs to compute the quantum Fisher information footnote1
(QFI), which roughly speaking quantifies how much information about $\lambda$
can be extracted from the system, provided that an optimal measurement is
performed. In general, computing the QFI involves diagonalisation of the
density matrix, which makes the obtention of analytical results challenging.
However, if one restricts to Gaussian states and Gaussian-preserving
operations, the so-called symplectic approach simplifies the task considerably
(Safranek_2018, ). As the QFI is by definition optimized over all POVMs, it
only depends on the initial state, often called probe. This means that a
second optimization of the QFI can be pursued, this time over all possible
probes. Moreover, this approach allows us to quantitatively compare different
protocols, e.g. with and without entanglement in the probe, since an increase
in the QFI when the same resources are used –which typically translates into
fixing the particle number or the energy– directly means an improvement in
precision.

Here, we propose a quantum-enhanced, lossy protocol to decide whether an
object embedded in a noisy environment has a frequency-dependent reflectivity
or not. In a radar-like setting, this can be seen as a second step after
quantum illumination: once the detection is completed, we may be interested in
extracting further information about the object, e.g. whether it is sensitive
to frequency changes in the light it is illuminated with. In the Supplemental
Material SuppMat  we test our methods to reproduce the results of quantum
illumination. The protocol comprises: a bi-frequency state sent to probe a
target, which is modeled as a beam splitter with a frequency-dependent
reflectivity $\eta(\omega)$. The object is embedded in a noisy environment,
which is assumed to be in a thermal state with equal expected photon number
for both frequencies $\omega_{1}$ and $\omega_{2}$, a good approximation
provided that we assume a short frequency distance. The goal is to obtain an
estimator for the parameter $\lambda=\eta(\omega_{2})-\eta(\omega_{1})$, that
captures information about the frequency dependence of the beam splitter. It
is assumed that the frequencies are sufficiently close so that we can work in
a neighborhood of $\lambda\sim 0$. This is not only a useful approach within
estimation theory, but also a requirement when we assumed that the two thermal
baths contain the same number of photons. For a more detailed and quantitative
discussion, see the Supplemental Material SuppMat . By imposing that the
expected photon number is the same in quantum and classical scenarios, we find
the QFI ratio between them, and analyze when it is greater than one. We find
that the maximum advantage is obtained for highly reflective targets, and
derive explicit limits in the highly noisy case. We also provide expressions
for the optimal observables, proposing a general experimental scheme described
in FIG. S2 of the Supplemental Material SuppMat , and motivating applications
in microwave technology (Sanz_2018, ).

Figure 1: An object reflects a bi-frequency beam (notice the similar colors of
the two beams coming out of the emitter), and mixes it with a thermal bath,
coming from the upper left. The transmitted signal and reflected thermal state
are lost, and a measurement $O_{\hat{\lambda}}$ is performed onto the
available part (lower right corner), whose expectation values converge, after
classical data processing, to an estimator of the parameter $\lambda$ encoded
in the object. In our case, $S_{\lambda}$ represents a multi-layered object
acting as a beam splitter, and $\lambda:=\eta_{2}-\eta_{1}$, where
$\eta_{i}=\eta(\omega_{i})$ are the reflectivities for the different
frequencies. The emitter can in principle entangle the two modes, which proves
advantageous in the parameter estimation.

Model, and fundamentals of quantum estimation theory. The model is synthesized
in FIG. 1: the target object, modeled as a beam splitter with a frequency-
dependent reflectivity is subject to an illumination with a bi-frequency
probe. For a single frequency, a beam splitter is characterised by a unitary
operator
${U}({\omega})\equiv\exp\left[\arcsin\left(\sqrt{\eta(\omega)}\right)(\hat{s}_{\omega}^{\dagger}\hat{b}_{\omega}e^{i\varphi}-\hat{s}_{\omega}\hat{b}_{\omega}^{\dagger}e^{-i\varphi})\right]$,
where $\eta(\omega)$ is a frequency-dependent reflectivity, related to
transmittivity $\tau$ via $\eta(\omega)+\tau(\omega)=1$. We assume for
simplicity that $\varphi=0$, i.e. there is no phase difference between
transmitted and reflected signals. This unitary maps states (density matrices)
that live in the density matrix space associated with Hilbert space
$\mathcal{H}$, $\mathcal{D}(\mathcal{H})$ to itself. Formulating the problem
from a density operator perspective, we have that the received state is
${\rho}_{\lambda}=\operatorname{Tr}_{S_{1}}\operatorname{Tr}_{S_{2}}\left[{U}_{\lambda}{\rho}{U}^{\dagger}_{\lambda}\right]$,
where ${\rho}\in\mathcal{H}_{S_{1},S_{2},B_{1},B_{2}}$ is a four-mode state
that includes the two signals (the two-mode state that we can control) and two
thermal environments of the form ${\rho}^{th}_{1}\otimes{\rho}^{th}_{2}$,
where the subscript indicates the frequency, i.e.
${\rho}^{th}_{a}=(1+N_{\text{th}})^{-1}\sum_{n=0}^{\infty}(N_{\text{th}}/(1+N_{\text{th}}))^{n}\ket{n}_{a}\bra{n}$
where
$N_{\text{th}}=\operatorname{Tr}({\rho}^{th}_{a}\hat{b}^{\dagger}_{\omega_{a}}\hat{b}_{\omega_{a}})$
is the average number of thermal photons, which we assume to be the same for
the two modes. This assumption is accurate for sufficiently small frequency
difference $\omega_{1}-\omega_{2}$ SuppMat . The interaction ${U}_{\lambda}$
comes from reparametrizing the four-mode unitary
${U}(\omega_{1})\otimes{U}(\omega_{2})$ using the difference of reflectivities
$\lambda\equiv\eta_{2}-\eta_{1}$.

The quantum Fisher information $H(\lambda)$ quantifies the information about
the value of $\lambda$ that can be extracted from a state ${\rho}_{\lambda}$.
It assumes that an optimal measurement is performed. The usual basis-dependent
expression for the QFI is:

$H(\lambda)=2\sum_{m,n}\frac{\lvert\bra{\Phi_{m}}\partial_{\lambda}{\rho}_{\lambda}\ket{\Phi_{n}}\rvert^{2}}{\rho_{m}+\rho_{n}},$
(1)

where $\\{\ket{\Phi_{m}},\rho_{m}\\}_{m}$ are the solutions to the
eigenproblem of ${\rho}_{\lambda}$. Regardless of the basis one uses –the Fock
number basis or a coherent representation of states– the problem becomes hard
to solve analytically. However, in the framework of Gaussian states the
calculation becomes simpler. Gaussian states have a very convenient phase-
space representation: they can be fully characterised by their first and
second moments: the displacement vector $\vec{d}$ and the covariance matrix
$\Sigma$, respectively. The QFI of a $\lambda$-parametrised Gaussian state
$(\Sigma_{\lambda},\vec{d})$ can be computed as

$\displaystyle\begin{split}H(\lambda)&=\frac{1}{2(\det[A]-1)}\left[\det[A]\operatorname{Tr}\left[(A^{-1}\dot{A})^{2}\right]\right.\\\
&+\sqrt{\det[\text{1}_{2}+A^{2}]}\operatorname{Tr}\left[\left((\text{1}_{2}+A^{2})^{-1}\right)^{2}\right]\\\
&\left.-f(\nu_{+},\nu_{-})\right]+2\dot{\vec{d\,}}^{\dagger}\bm{\Sigma}^{-1}\dot{\vec{d}\,},\end{split}$
(2)

where
$f(\nu_{+},\nu_{-}):=4\left(\nu_{+}^{2}-\nu_{-}^{2}\right)\left(\frac{(\partial_{\lambda}\nu_{+})^{2}}{\nu_{+}^{4}-1}-\frac{(\partial_{\lambda}\nu_{-})^{2}}{\nu_{-}^{4}-1}\right)$,
the dot over $A$ and $\vec{d}$ denotes derivative with respect to $\lambda$,
and $\nu_{\pm}$ are the symplectic eigenvalues of ${\Sigma}_{\lambda}$,
defined as following Ref. (_afr_nek_2015, )

$2\nu_{\pm}^{2}:=\operatorname{Tr}[A^{2}]\pm\sqrt{\left(\operatorname{Tr}[A^{2}]\right)^{2}-16\det[A]},$
(3)

with the matrix $A$ given by
$A:=i{\Omega}{T}{\Sigma}_{\lambda}{T}^{\intercal}$,
$\Omega:=\text{antidiag}(\text{1}_{2},-\text{1}_{2})$, and
${T}_{ij}:=\delta_{j+4,2i}+\delta_{j,2i-1}$ is the matrix that changes the
basis to the quadrature basis
$(\hat{x}^{\text{th}}_{1},\hat{x}^{\text{S}}_{1},\hat{x}^{\text{th}}_{2},\hat{x}^{\text{S}}_{2},\hat{p}^{\text{th}}_{1},\hat{p}^{\text{S}}_{1},\hat{p}^{\text{th}}_{2},\hat{p}^{\text{S}}_{2})^{\intercal}$.
As mentioned above, this value of the QFI is obtained when an optimal
measurement is performed. In Ref. doi:10.1142/S0219749909004839  and
references therein it was proven that the qCR bound is saturated by

$\hat{O}_{\lambda}=\lambda\text{1}+\hat{L}_{\lambda}/H(\lambda),$ (4)

where 1 is the identity operator and $\hat{L}_{\lambda}$ is a symmetric
logarithmic derivative (SLD) fulfilling the equation
$\\{\hat{L}_{\lambda},{\rho}_{\lambda}\\}=2\partial_{\lambda}{\rho}_{\lambda}$,
where $\\{\cdot,\cdot\\}$ denotes the anticommutator. Assuming that the
estimator $\tilde{\lambda}$ is constructed using maximum likelihood methods so
that the qCR bound is asymptotically achieved, the optimal observable is the
one with the lowest possible variance, saturating the qCR inequality and
yielding an optimal estimator after classically processing the measurement
outcomes. For a Gaussian state
$\left({\Sigma}_{\lambda},\vec{d}_{\lambda}\right)$ written in the complex
basis, the symmetric logarithmic derivative given in Ref. (Safranek_2018, ) is

$\hat{L}_{\lambda}=\Delta\vec{\hat{A}}^{\dagger}{\mathcal{A}}_{\lambda}\Delta\vec{\hat{A}}-\operatorname{Tr}[{\Sigma_{\lambda}}{\mathcal{A}}_{\lambda}]/2+2\Delta\vec{\hat{A}}^{\dagger}{\Sigma}_{\lambda}^{-1}\partial_{\lambda}\vec{d}_{\lambda},$
(5)

where $\Delta\vec{\hat{A}}:=\vec{\hat{A}}-\vec{d}_{\lambda}$, $\vec{\hat{A}}$
the complex basis vector of bosonic operators,
${\mathcal{A}}_{\lambda}:={\mathcal{M}}^{-1}\partial_{\lambda}\vec{d}_{\lambda}$,
where
${\mathcal{M}}=\bar{{\Sigma}}_{\lambda}\otimes{\Sigma}_{\lambda}-{K}\otimes{K}$,
where the bar denotes complex conjugate, and
${K}:=\text{diag}\left(\text{1}_{2},-\text{1}_{2}\right)$. Note that when
$\lambda\rightarrow 0$ we have
$\hat{O}_{\lambda=0}\equiv\hat{O}=\hat{L}_{\lambda=0}/H(\lambda=0)$, since
both limits exist independently. This limit is of our interest because we work
in a neighbourhood of $\lambda\sim 0$, i.e. the measured value of the
parameter is expected to be small. In the next section, we apply the formulas
to the case of a two-mode squeezed vacuum state probe.

Two-mode squeezed vacuum state. The TMSV state is the continuous-variable
equivalent of the Bell state, being the Gaussian state that optimally
transforms classical resources (light, or photons) into quantum correlations.
The TMSV state is a cornerstone in experiments with quantum microwaves
(PhysRevLett.107.113601, ; PhysRevLett.109.183901, ; RevModPhys.77.513, ). In
our case, we are interested in states produced via nondegenerate parametric
amplification, in order to have two distinguishable frequencies. The state can
be formally written as: $\ket{\psi}_{\text{TMSV}}:=(\cosh
r)^{-1}\sum_{n=0}^{\infty}\left(-e^{i\phi}\tanh r\right)^{n}\ket{n,n}$, where
$r\in\mathbb{R}_{\geq 0}$ is the squeezing parameter . For simplicity we take
$\phi=0$. The total initial (real) covariance matrix –written in the real
basis
$(\hat{x}^{\text{th}}_{1},\hat{p}^{\text{th}}_{1},\hat{x}^{\text{S}}_{1},\hat{p}^{\text{S}}_{1},\hat{x}^{\text{th}}_{2},\hat{p}^{\text{th}}_{2},\hat{x}^{\text{S}}_{2},\hat{p}^{\text{S}}_{2})^{\intercal}$–
is given by

${\Sigma}=\left[\begin{array}[]{cccc}{\Sigma}_{\text{th}}&0&0&0\\\
0&{\Sigma}_{r}&0&{\varepsilon}_{r}\\\ 0&0&{\Sigma}_{\text{th}}&0\\\
0&{\varepsilon}^{\intercal}_{r}&0&{\Sigma}_{r}\end{array}\right],$ (6)

where ${\Sigma}_{\text{th}}=(1+2N_{\text{th}})\text{1}_{2}$ is the real
covariance matrix of a thermal state, ${\Sigma}_{r}=\cosh(2r)\text{1}_{2}$
corresponds to the diagonal part of one of the modes in a TMSV state, and
${\varepsilon}_{r}=\sinh(2r)\sigma_{Z}$ is the correlation between the two
modes, where $\sigma_{Z}$ is the Z Pauli matrix. We rewrite the covariance
matrix in terms of the mean photon number
$N_{\text{S}}\equiv\langle\hat{a}^{\dagger}_{S_{1}}\hat{a}_{S_{1}}\rangle=\langle\hat{a}^{\dagger}_{S_{2}}\hat{a}_{S_{2}}\rangle$,
using the relation $\sqrt{N_{\text{S}}}=\sinh r$ footnote2 . This gives
$\sinh(2r)=2\sqrt{2N_{\text{S}}(1+2N_{\text{S}})}$ and
$\cosh(2r)=1+4N_{\text{S}}$. The displacement vector of a TMSV state is
identically zero $\vec{d}_{\text{TMSV}}=\vec{0}$, so the last term of Eq. (2)
vanishes. Under the assumption that the object does not entangle the two
modes, we have that the symplectic transformation is
${S}(\eta_{1},\eta_{2})={S}_{\text{BS}}(\eta_{1})\oplus{S}_{\text{BS}}(\eta_{2})$
footnote3 , where

${S}_{\text{BS}}(x)=\left[\begin{array}[]{cc}\sqrt{x}\text{1}_{2}&\sqrt{1-x}\text{1}_{2}\\\
-\sqrt{1-x}\text{1}_{2}&\sqrt{x}\text{1}_{2}\end{array}\right]$ (7)

is the real symplectic transformation associated with a beam splitter of
reflectivity $x$. We define the parameter of interest as
$\lambda\equiv\eta_{2}-\eta_{1}$. With this, ${S}(\eta_{1},\eta_{2})$ becomes
a function of $\lambda$. For simplicity, we define
${S}_{\lambda}:={S}(\eta_{1},\eta_{1}+\lambda)$. The full state after the
signals get mixed with the thermal noise is given by
$\tilde{{\Sigma}}_{\lambda}\equiv{S}_{\lambda}{\Sigma}{S}_{\lambda}^{\intercal}$.
In covariance matrix formalism, partial traces are implemented by removing the
corresponding rows and columns (Adesso_2014, ); in our case the rows and
columns 1, 2, 5, and 6. The resulting received covariance matrix reads as
follows

${\Sigma}_{\lambda}=\left[\begin{array}[]{cc}a\text{1}&b\sigma_{Z}\\\
b\sigma_{Z}&c_{\lambda}\text{1}\end{array}\right],$ (8)

with $a\equiv 1+2N_{\text{th}}+2\eta_{1}(2N_{\text{S}}-N_{\text{th}})$,
$b\equiv
2\sqrt{2}\sqrt{\eta_{1}}\sqrt{N_{\text{S}}(2N_{\text{S}}+1)}\sqrt{\eta_{1}+\lambda}$,
and $c_{\lambda}\equiv
1+2N_{\text{th}}+2(\eta_{1}+\lambda)(2N_{\text{S}}-N_{\text{th}})$. For this
state, the symplectic eigenvalues $\nu_{\pm}$ defined in Eq. (3) are strictly
larger than one for any value of the parameters $N_{\text{S}}$,
$N_{\text{th}}$ and $\eta_{1}$, so there is no need of any regularization
scheme (_afr_nek_2015, ). In what follows, we work in a neighborhood of
$\lambda\sim 0$ – which can be implemented by taking the limit
$\lambda\rightarrow 0$ in the derived expressions. This is a physical
assumption, since we are interested in probing regions of $\eta(\omega)$ that
do not change drastically, i.e. that are well approximated by a linear
function with either no slope or a small one. We obtain the function
$H_{Q}=H_{Q}(\eta_{1},N_{\text{S}},N_{\text{th}})$ from Eq. (2). An explicit
expression can be found in SuppMat . In the following section, we compute the
QFI of a classical probe.

Classical probe. Here we use a pair of coherent states as probe:
$\ket{\psi}=\ket{\alpha}\otimes\ket{\alpha}$. The total expected photon number
in this state is $2N_{\text{C}}:=2\left|\alpha^{2}\right|$. For simplicity we
take $\alpha\in\mathbb{R}$. The symplectic eigenvalues are always larger than
one in this case, so no regularization is required. The initial displacement
vector in the real basis is
$\vec{d}_{0}^{\intercal}=(0,0,\sqrt{2}\alpha,0,0,0,\sqrt{2}\alpha,0)$ which
leads –after the interaction and the trace of the losses– to
$\vec{d}^{\intercal}=\alpha(\sqrt{2\eta_{1}},0,\sqrt{2(\eta_{1}+\lambda)})$.
Inserting these in Eq. (2), and taking the limit $\lambda\rightarrow 0$, we
find that the QFI for the coherent state is

$H_{C}=\frac{4N_{\text{th}}^{2}\left((1+2N_{\text{th}}\tau_{1})^{2}+1\right)}{(1+2N_{\text{th}}\tau_{1})^{4}-1}+\frac{N_{\text{S}}}{\eta_{1}+2N_{\text{th}}\tau_{1}\eta_{1}},$
(9)

where $\tau_{1}=1-\eta_{1}$ is the transmittivity, and we have set the
resources $\left|\alpha^{2}\right|=N_{\text{S}}$ for a fair comparison with
the TMSV state. Having computed both the quantum and the classical QFIs, in
the next section we analyse their ratio $H_{Q}/H_{C}$, a quantifier for the
quantum enhancement.

Quantum advantage. We analyze the ratio between the TMSV state’s QFI ($H_{Q}$)
and the coherent pair’s QFI ($H_{C}$) for different situations. Finding values
of $(\eta_{1},N_{\text{th}},N_{\text{S}})$ such that the ratio $H_{Q}/H_{C}$
is larger than one means that one can extract more information about parameter
$\lambda$ using a TMSV than using a coherent pair, provided an optimal
measurement is performed in both cases. We plot the results for various values
of $\eta_{1}$ (see FIG. 2).

Figure 2: Values, represented by a non-linear color grading, of the quantum
enhancement given by the ratio $H_{Q}/H_{C}$ of the quantum Fisher information
of the two-mode squeezed vacuum state probe $H_{Q}$ by the quantum Fisher
information of the coherent states probe $H_{Q}$ as a function of the photon
numbers of the signal ($N_{\text{S}}$) and of the thermal bath
($N_{\text{th}}$), for a reflectivity $\eta_{1}$ of a) 0.75, b) 0.90, and c)
0.95. Equivalently, scales of squeezing, $r$, given by
$\sqrt{N_{\text{S}}}=\sinh r$, and temperature $T$ in Kelvin, are provided.
The relation between temperature and mean thermal photon number is obtained
via the usual Bose-Einstein distribution $N_{\text{th}}=1/(\exp(E/k_{B}T)-1)$
when the energy is set to $E=\hbar\omega=h\nu$, which requires a choice of the
frequency $\nu$. We have taken $\nu=5$ GHz, a typical frequency of microwaves.
White represents no quantum enhancement, i.e. $H_{Q}/H_{C}=1$. We clearly see
that as $\eta_{1}$ grows, the quantum advantage becomes not only more
significant, but also easier to achieve with less signal photons. Importantly,
as the reference reflectivity $\eta_{1}$ grows, the protocol becomes more
resilient to thermal noise. For the full range $\eta_{1}\in[0,1]$ see the
Supplemental Material SuppMat .

We can immediately see that the ratio gets larger for large values of
$\eta_{1}$, i.e. for highly reflective materials. In particular, we find the
high-reflectivity limit:

$\lim_{\eta_{1}\rightarrow
1}\frac{H_{\text{Q}}}{H_{\text{C}}}=\frac{N_{\text{S}}^{2}\left(8N_{\text{th}}(N_{\text{th}}+1)+4\right)+4N_{\text{S}}N_{\text{th}}^{2}+N_{\text{th}}^{2}}{N_{\text{th}}\left(N_{\text{S}}(4N_{\text{th}}+2)+N_{\text{th}}\right)}$
(10)

which converges to $1+8N_{\text{S}}^{2}/(4N_{\text{S}}+1)$ in the highly noisy
scenario $N_{\text{th}}\gg 1$. In the next sections we explicitly compute the
observables that lead to an optimal extraction of $\lambda$’s value for both
the classical and the quantum probes.

Optimal observable for the TMSV state probe. Computing the SLD in Eq. (2) and
inserting it in Eq. (4) we find
$\hat{O}_{Q}=L_{11}\hat{a}_{1}^{\dagger}\hat{a}_{1}+L_{22}\hat{a}_{2}^{\dagger}\hat{a}_{2}+L_{12}\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}+\hat{a}_{1}\hat{a}_{2}\right)+L_{0}\text{1}_{12}$,
where the general expressions for the coefficients can be found in SuppMat .
Here we focus on the high-reflectivity target, where the maximum quantum
enhancement is expected. Moreover, it is illustrative to study the noiseless
case, since this captures the essence of what is being measured:

$\displaystyle\begin{split}\lim_{\begin{subarray}{c}N_{\text{th}}\rightarrow
0\\\ \eta_{1}\rightarrow
1\end{subarray}}\hat{O}_{Q}&=-\mu^{2}\hat{a}_{1}^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}\\\
&+\mu\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}+\hat{a}_{1}\hat{a}_{2}\right)-\nu\text{1}_{12},\end{split}$
(11)

where $\mu^{2}\equiv\left(1+1/2N_{\text{S}}\right)$ and
$\nu\equiv\left(1+1/4N_{\text{S}}\right)$. We can rewrite this observable as
$\hat{b}^{\dagger}\hat{b}-1$, i.e. implementing photon-counting on the
operator $\hat{b}\equiv-i\left(\hat{a}_{2}^{\dagger}-\mu\hat{a}_{1}\right)$. A
detailed construction of these operators employing usual resources in
superconducting circuit technology, such as Josephson parametric amplifiers
finite-time ; cloaking  is provided in SuppMat .

Optimal observable for the coherent state probe. The optimal observable in
this case is given by
$\hat{O}_{C}=2A\text{1}_{(1)}\otimes\left[\left(\hat{a}_{2}^{\dagger}-\eta_{1}\sqrt{\alpha}\right)\Big{(}\hat{a}_{2}-\eta_{1}\sqrt{\alpha}\Big{)}+\frac{1}{2}\right]$,
where $A=1/2(\eta_{1}-1)(1-N_{\text{th}}(\eta_{1}-1))$ and $\text{1}_{(1)}$ is
the absence of active measurement of mode 1. The interpretation is simple:
because $\eta_{1}$ is known (it serves as a reference), there is nothing to be
gained by measuring the first mode in the absence of entanglement. Moreover,
the observable is separable, as one should expect, and the experimental
implementation is straightforward: photon-counting in the –locally displaced–
second mode. As discussed in the Supplemental Material SuppMat ,
implementations of these observables in microwave technology implies having
both efficient photon-counters dassonneville2020numberresolved  and good
digital filters PhysRevLett.117.020502 .

Conclusions. Quantum metrology in the presence of losses is a highly relevant
subject for realistic applications, with only a handful of known protocols
exhibiting quantum enhancement. Here, we have developed a method for achieving
a quantum enhancement in the decision problem of whether a target’s
reflectivity depends or not on the electromagnetic frequency, using only a bi-
frequency, entangled probe, and in the presence of both noise and losses. Such
measurement can be used to extract information about the electromagnetic
response of a reflective object to changes in frequency, and, consequently, it
can be applied to a wide spectrum of situations. It is important to stress
that although our results are general and platform-independent, the
atmospheric transparency window in the microwaves regime, together with the
naturally noisy character of open-air, makes applications in quantum
microwaves our first choice Sanz_2018 ; dassonneville2020numberresolved ;
GonzalezRaya2020 ; Zhang2020 ; munuera2020 ; dambach2107 . Namely, our results
could find applications in radar physics, where the protocol could be
understood as complementary to quantum illumination: quantum signals can
enhance the precision of measurement without increasing the intensity,
something very convenient when the emitter does not want to be detected.
Entanglement is thus seen as a resource in practical quantum metrology.
Moreover, accurate medical imaging with non-ionizing radiation is a permanent
goal in medicine. Microwaves, however, when applied in intense ways, can
damage the sample. Resorting to methods that increase the precision and/or
resolution while keeping a low intensity of radiation is thus crucial for non-
invasive imaging technologies. Our results suggest that this quantum advantage
will be more significant in the high reflectivity regime, which could prove
useful for contrast imaging of tissues with a low penetration depth. The
theoretical methods used are not unique to our protocol, and we hope our
results can boost efforts in the direction of lossy Gaussian quantum
estimation: these are powerful tools that allow us not only to find quantum-
enhanced protocols, but also to explicitly find the optimal observables,
making the transition to actual experimental proposals quite direct. Using
these techniques, we have proved that the scaling of the quantum Fisher
information (QFI) when using entanglement is faster than when only classical
(coherent) states are used. We have derived analytic expressions for the
optimal observables, which are the ones that allow the extraction of the
maximum available information of the parameter of interest, constructing an
optimal estimator after appropriate classical data processing. Our work paves
the way to extensions of the protocol to accommodate both presence of thermal
effects in the input modes, and continuous-variable frequency entanglement
PhysRevLett.84.5304 , where a more realistic model for a beam containing a
given distribution of frequencies could be used instead of sharp, ideal bi-
frequency states.

The authors thank Vahid Salari and Gonçalo Frazão for useful discussions and
comments, and acknowledge the support from the EU H2020 Quantum Flagship
project QMiCS (820505). MC acknowledges support from the DP-PMI and FCT
(Portugal) through scholarship PD/BD/135186/2017. MC and YO thank the support
from Fundação para a Ciência e a Tecnologia (Portugal), namely through project
UIDB/50008/2020, as well as from project TheBlinQC supported by the EU H2020
QuantERA ERA-NET Cofund in Quantum Technologies and by FCT
(QuantERA/0001/2017). MS thanks support from the Spanish Government
PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE), and Basque Government IT986-16, the
project OpenSuperQ (820363) of the EU Flagship on Quantum Technologies, as
well as the EU FET Open project Quromorphic (828826). This work is supported
by the U.S. Department of Energy, Office of Science, Office of Advanced
Scientific Computing Research (ASCR) quantum algorithm teams program, under
field work proposal number ERKJ333.

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Supplemental Material for Bi-frequency illumination: a quantum-enhanced
protocol

##  I Quantum Fisher information: two-mode squeezed vacuum state

The real covariance matrix of the two-mode squeezed vacuum (TMSV) state,
written in the quadrature basis
$(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2})$ is

$\displaystyle\Sigma_{\text{TMSV}}=\left[\begin{array}[]{cccc}\cosh(2r)&0&\sinh(2r)&0\\\
0&\cosh(2r)&0&-\sinh(2r)\\\ \sinh(2r)&0&\cosh(2r)&0\\\
0&-\sinh(2r)&0&\cosh(2r)\\\ \end{array}\right],$ (S5)

where $r\in\mathbb{R}_{\geq 0}$ is the squeezing parameter. For convenience,
we rewrite this in terms of the expected photon number in either of the two
modes
$N_{\text{S}}=\langle\hat{a}_{1}^{\dagger}\hat{a}_{1}\rangle=\langle\hat{a}_{2}^{\dagger}\hat{a}_{2}\rangle$:

$\displaystyle\left[\begin{array}[]{cccc}4N_{\text{S}}+1&0&2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)}&0\\\
0&4N_{\text{S}}+1&0&-2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)}\\\
2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)}&0&4N_{\text{S}}+1&0\\\
0&-2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)}&0&4N_{\text{S}}+1\\\
\end{array}\right].$ (S10)

Assuming an equal expected thermal photon number for the two noisy
environments, the corresponding covariance matrix is
$\Sigma_{\text{th}}\oplus\Sigma_{\text{th}}$, where

$\displaystyle\Sigma_{\text{th}}=\left[\begin{array}[]{cc}2N_{\text{th}}+1&0\\\
0&2N_{\text{th}}+1\\\ \end{array}\right]$ (S13)

is the covariance matrix of one thermal mode. The assumption of equal thermal
photon number is accurate as long as the frequency difference
$\omega_{2}-\omega_{1}$ is sufficiently small. To make this statement more
quantitative, let us assume two different thermal photon densities, $N_{1}$
and $N_{2}$. The Bose-Einstein distribution for photons is $N_{i}\propto
1/e^{\beta\omega_{i}-1}$ where $\beta\equiv\hbar/k_{B}T$ is a function of the
temperature $T$. Then,

$\frac{N_{1}}{N_{2}}=\frac{e^{\beta\omega_{1}-1}}{e^{\beta\omega_{2}-1}}=\frac{1}{1+\frac{\beta\Delta\omega
e^{\beta\omega_{1}}}{e^{\beta\omega_{1}}-1}},$ (S14)

where $\Delta\omega\equiv\omega_{2}-\omega_{1}$. Up to first order in
$\beta\Delta\omega$, the last expression reduces to
$1-\Delta\omega/\omega_{1}$. This means that $N_{1}\approx N_{2}$ if
$\Delta\omega/\omega_{1}\ll 1$. In particular, for $T=300\text{ K}$ and
$\omega_{1}/2\pi=5\text{ GHz}$ the expected thermal photon number is roughly
1250. It is straightforward to check that for these frequencies and
temperatures, the above approximations are good (i.e. $\sim 4$% of relative
error) for frequency differences up to 20%.

To construct the whole initial (real) covariance matrix we will rearrange the
Hilbert subspaces by frequencies, in the following way: “bath + signal + bath
+ signal”. This is convenient because the interaction is by assumption
frequency non-mixing. The total covariance matrix of the input mode is then

$\displaystyle\left[\begin{array}[]{cccc}\Sigma_{\text{th}}&0&0&0\\\
0&(4N_{\text{S}}+1)\text{1}_{2}&0&2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)}\sigma_{Z}\\\
0&0&\Sigma_{\text{th}}&0\\\
0&2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)}\sigma_{Z}&0&(4N_{\text{S}}+1)\text{1}_{2}\\\
\end{array}\right],$ (S19)

where $\sigma_{Z}$ is the Pauli $Z$ matrix and $\text{1}_{n}$ is the $n\times
n$ identity matrix.

If the interaction with the object is frequency non-mixing, each mode in the
bi-frequency state sees a beam splitter with reflectivity $\eta(\omega_{i})$
with $i=1,2$ defining the two frequencies. In the symplectic formalism, the
total interaction is thus obtained by the direct sum of each beam-splitter
transformation: $S_{\text{T}}=S_{\text{BS}}(\eta_{1})\oplus
S_{\text{BS}}(\eta_{2})$, where

$\displaystyle
S_{\text{BS}}(\eta_{i}):=\left[\begin{array}[]{cc}\sqrt{\eta_{i}}\text{1}_{2}&\sqrt{1-\eta_{i}}\text{1}_{2}\\\
-\sqrt{1-\eta_{i}}\text{1}_{2}&\sqrt{\eta_{i}}\text{1}_{2}\\\
\end{array}\right].$ (S22)

The full, transformed state is finally characterised by the covariance matrix

$\displaystyle
S_{\text{T}}\Sigma_{\text{TMSV}}S_{\text{T}}^{\intercal}=\left[\begin{array}[]{cccc}A(\eta_{1})\text{1}_{2}&B(\eta_{1})\text{1}_{2}&f(1-\eta_{1},1-\eta_{2})\sigma_{Z}&f(1-\eta_{1},\eta_{2})\sigma_{Z}\\\
B(\eta_{1})\text{1}_{2}&C(\eta_{1})\text{1}_{2}&f(\eta_{1},1-\eta_{2})\sigma_{Z}&f(\eta_{1},\eta_{2})\sigma_{Z}\\\
f(1-\eta_{1},1-\eta_{2})\sigma_{Z}&f(\eta_{1},1-\eta_{2})\sigma_{Z}&A(\eta_{2})\text{1}_{2}&B(\eta_{2})\text{1}_{2}\\\
f(1-\eta_{1},\eta_{2})\sigma_{Z}&f(\eta_{1},\eta_{2})\sigma_{Z}&B(\eta_{2})\text{1}_{2}&C(\eta_{2})\text{1}_{2}\\\
\end{array}\right],$ (S27)

where

$\displaystyle f(x,y)$
$\displaystyle=2\sqrt{2N_{\text{S}}(2N_{\text{S}}+1)xy}$ $\displaystyle A(x)$
$\displaystyle=2x(N_{\text{th}}-2N_{\text{S}})+4N_{\text{S}}+1$ $\displaystyle
B(x)$ $\displaystyle=2\sqrt{\left(1-x\right)x}(2N_{\text{S}}-N_{\text{th}})$
$\displaystyle C(x)$
$\displaystyle=2x(2N_{\text{S}}-N_{\text{th}})+2N_{\text{th}}+1$

and $\sigma_{Z}$ is the fourth Pauli matrix. The received state is obtained by
partial tracing the losses, which in our construction correspond to the first
and third modes. In the symplectic formalism, partial traces are performed by
removing the corresponding rows and columns Adesso_2014 . We are thus left
with the following state:

$\displaystyle\left[\begin{array}[]{cc}C(\eta_{1})\text{1}_{2}&f(\eta_{1},\eta_{2})\sigma_{Z}\\\
f(\eta_{1},\eta_{2})\sigma_{Z}&C(\eta_{2})\text{1}_{2}\\\ \end{array}\right].$
(S30)

The reparametrization of the state in terms of $\lambda=\eta_{2}-\eta_{1}$ is
simply

$\displaystyle\Sigma_{\lambda}=\left[\begin{array}[]{cc}C(\eta_{1})\text{1}_{2}&f(\eta_{1},\eta_{1}+\lambda)\sigma_{Z}\\\
f(\eta_{1},\eta_{1}+\lambda)\sigma_{Z}&C(\eta_{1}+\lambda)\text{1}_{2}\\\
\end{array}\right].$ (S33)

Note that, in the absence of beam splitter, i.e., when $\eta_{1}=0=\lambda$ we
have that the received state is a pair of thermal modes, each with its
corresponding frequency: $C(0)=1+2N_{\text{th}}$ and $f(0,0)=0$. This is what
should be expected, since the received state is precisely the reflected part
of the signal. To compute the quantum Fisher information we follow
_afr_nek_2015 :

$\displaystyle\begin{split}H(\lambda)&=\frac{1}{2(\det[A]-1)}\left[\det[A]\operatorname{Tr}\left[(A^{-1}\partial_{\lambda}A)^{2}\right]+\sqrt{\det[\text{1}_{2}+A^{2}]}\operatorname{Tr}\left[\left((\text{1}_{2}+A^{2})^{-1}\right)^{2}\right]\right.\\\
&-\left.4\left(\nu_{+}^{2}-\nu_{-}^{2}\right)\left(\frac{(\partial_{\lambda}\nu_{+})^{2}}{\nu_{+}^{4}-1}-\frac{(\partial_{\lambda}\nu_{-})^{2}}{\nu_{-}^{4}-1}\right)+2\partial_{\lambda}\vec{d\,}^{\dagger}\Sigma_{\lambda}^{-1}\partial_{\lambda}\vec{d}\,\right],\end{split}$
(S34)

where the symplectic eigenvalues of $\Sigma_{\lambda}$ are defined by

$\nu_{\pm}:=\frac{1}{2}\sqrt{\operatorname{Tr}[A^{2}]\pm\sqrt{\left(\operatorname{Tr}[A^{2}]\right)^{2}-16\det[A]}},$
(S35)

where the matrix $A$ is given by $A:=i\Omega T\Sigma_{\lambda}T^{\intercal}$
and $T_{ij}:=\delta_{j+4,2i}+\delta_{j,2i-1}$ is the matrix that changes the
basis to the quadrature basis
$(\hat{x}^{\text{th}}_{1},\hat{x}^{\text{S}}_{1},\hat{x}^{\text{th}}_{2},\hat{x}^{\text{S}}_{2},\hat{p}^{\text{th}}_{1},\hat{p}^{\text{S}}_{1},\hat{p}^{\text{th}}_{2},\hat{p}^{\text{S}}_{2})^{\intercal}$,
and

$\bm{\Omega}:=\left[\begin{array}[]{cc}0&\text{1}_{2}\\\
-\text{1}_{2}&0\end{array}\right].$

We have that

$\displaystyle
A=i\left[\begin{array}[]{cccc}0&0&C(\eta_{1})&-f(\eta_{1},\eta_{1}+\lambda)\\\
0&0&-f(\eta_{1},\eta_{1}+\lambda)&C(\eta_{1}+\lambda)\\\
-C(\eta_{1})&-f(\eta_{1},\eta_{1}+\lambda)&0&0\\\
-f(\eta_{1},\eta_{1}+\lambda)&-C(\eta_{1}+\lambda)&0&0\\\ \end{array}\right],$
(S40)

and the symplectic eigenvalues

$\displaystyle\begin{split}\frac{\nu_{\pm}^{2}}{2\lambda(N_{\text{th}}-2N_{\text{S}})}=&\sqrt{4\lambda^{2}N_{\text{S}}^{2}+4N_{\text{S}}(\lambda-2\eta_{1}(\eta_{1}+\lambda-1))-2N_{\text{th}}(2\eta_{1}+\lambda-2)(2N_{\text{S}}(2\eta_{1}+\lambda)+1)+N_{\text{th}}^{2}(2\eta_{1}+\lambda-2)^{2}+1}\\\
&-8\eta_{1}^{2}N_{\text{S}}+8\eta_{1}N_{\text{S}}-2\lambda\left(2(2\eta_{1}-1)N_{\text{S}}(2N_{\text{th}}+1)-2(\eta_{1}-1)N_{\text{th}}^{2}+N_{\text{th}}\right)\\\
&-4(\eta_{1}-1)N_{\text{th}}(4\eta_{1}N_{\text{S}}+1)+2\lambda^{2}(N_{\text{th}}-2N_{\text{S}})^{2}+4(\eta_{1}-1)^{2}N_{\text{th}}^{2}+1.\end{split}$
(S41)

The symplectic eigenvalues are both larger than one for every $N_{\text{th}}$,
$N_{\text{S}}$ and $\eta_{1}$, so there is no need to use any regularization
procedure. Indeed, this is due to the mixedness of the received state:
regularization is only needed for pure states. The two-sided limit of the QFI
when the parameter $\lambda$ goes to zero is

$\begin{split}H_{Q}=k^{-1}&\left[8\tau_{1}\eta_{1}N_{\text{S}}^{3}(2N_{\text{th}}+1)+4N_{\text{S}}^{2}\left(-\eta_{1}+(\eta_{1}+3\eta_{1}N_{\text{th}})^{2}-\eta_{1}N_{\text{th}}(10N_{\text{th}}+7)+3N_{\text{th}}(N_{\text{th}}+1)+1\right)\right.\\\
&-2N_{\text{S}}N_{\text{th}}(-\eta_{1}+N_{\text{th}}(\eta_{1}(3\eta_{1}-8)+4\tau_{1}(\tau_{1}-\eta_{1})N_{\text{th}}+3)+1)\\\
&+\left.N_{\text{th}}^{2}(2\tau_{1}N_{\text{th}}(\tau_{1}N_{\text{th}}+1)+1)\right],\end{split}$
(S42)

where

$k=\tau_{1}(N_{\text{th}}(4N_{\text{S}}\eta_{1}+N_{\text{th}}\tau_{1}+1)+2N_{\text{S}}\eta_{1})\left(2N_{\text{th}}\tau_{1}(4N_{\text{S}}\eta_{1}+1)+4N_{\text{S}}\eta_{1}\tau_{1}+2N_{\text{th}}^{2}\tau_{1}^{2}+1\right),$
(S43)

and $\tau_{1}:=1-\eta_{1}$.

##  II Quantum Fisher information: coherent states

We reproduce the same calculations for the coherent pair now. Noting that
every coherent state, following our conventions, has a covariance matrix given
by the identity, the real covariance matrix in the quadrature basis is

$\displaystyle\left[\begin{array}[]{cccc}\Sigma_{\text{th}}&0&0&0\\\
0&\text{1}_{2}&0&0\\\ 0&0&\Sigma_{\text{th}}&0\\\ 0&0&0&\text{1}_{2}\\\
\end{array}\right],$ (S48)

while the displacement vector is
$\vec{d}^{\intercal}=\left[0,0,\sqrt{2}\alpha,0,0,0,\sqrt{2}\alpha,0\right]$,
where $\lvert\alpha\rvert^{2}=N_{\text{S}}$. The transformed displacement
vector is then

$S_{T}\vec{d}=\left[\alpha\sqrt{2\tau_{1}},0,\alpha\sqrt{2\eta_{1}},0,\alpha\sqrt{2}\sqrt{\tau_{1}-\lambda},0,\alpha\sqrt{2}\sqrt{\eta_{1}+\lambda},0\right]^{\intercal}$
(S49)

The transformed, and already reduced covariance matrix is (i.e. after partial
tracing the losses)

$\left[\begin{array}[]{cc}(1+2\tau_{1}N_{\text{th}})\text{1}_{2}&0\\\
0&1-2N_{\text{th}}(\lambda-\tau_{1})\text{1}_{2}\\\ \end{array}\right]$ (S50)

The matrix $A$ is

$A=i\left[\begin{array}[]{cccc}0&0&1+2N_{\text{th}}\tau_{1}&0\\\
0&0&0&1-2N_{\text{th}}(\lambda-\tau_{1})\\\ -1-2N_{\text{th}}\tau_{1}&0&0&0\\\
0&-1+2N_{\text{th}}(\lambda-\tau_{1})&0&0\\\ \end{array}\right]$ (S51)

The symplectic eigenvalues are

$\nu_{\pm}=\sqrt{\pm
2\sqrt{\lambda^{2}N_{\text{th}}^{2}(N_{\text{th}}(\lambda+\eta_{1}-\tau_{1})-1)^{2}}+2N_{\text{th}}\left(-\lambda+N_{\text{th}}\left(\lambda^{2}-2\lambda\tau_{1})+2\tau_{1}^{2}\right)-2\eta_{1}+2\right)+1}.$
(S52)

Again, the symplectic eigenvalues are always larger than one in the coherent
case, so no regularization of the QFI is needed. The QFI in the limit when
$\lambda\rightarrow 0$ is

$H_{C}=\frac{\alpha^{2}}{\eta_{1}(1+2N_{\text{th}}\tau_{1})}+\frac{N_{\text{th}}}{\tau_{1}(\tau_{1}N_{\text{th}}+1)}.$
(S53)

In the following we take $\alpha=\sqrt{N_{\text{S}}}$. The ratio of the two
QFIs converges to the following

$\lim_{\begin{subarray}{c}\lambda\rightarrow 0\\\ \eta_{1}\rightarrow
1\end{subarray}}H_{Q}/H_{C}=\frac{N_{\text{S}}^{2}(8N_{\text{th}}(N_{\text{th}}+1)+4)+4N_{\text{S}}N_{\text{th}}^{2}+N_{\text{th}}^{2}}{N_{\text{th}}(N_{\text{S}}(4N_{\text{th}}+2)+N_{\text{th}})}$
(S54)

In particular, in the noise-dominated regime $N_{\text{th}}\gg 1$ the above
expression converges to

$1+\frac{8N^{2}_{\text{S}}}{1+4N_{\text{S}}}.$ (S55)

When $N_{\text{S}}=0$ the ratio converges to one, meaning there is no
advantage in entanglement if there is nothing to entangle. The protocol is
then safe from any shadow-effect.

Figure S1: Quantum Fisher information ratio $H_{Q}/H_{C}$ as a function of the
noise factor $\beta:=N_{\text{S}}/N_{\text{th}}$ and the reflectivity
$\eta_{1}$ for an experimentally reasonable two-mode squeezing of
$N_{\text{S}}\sim 0.76$. As in previous diagrams, white represents no gain.
The protocol becomes more noise-resilient in the highly reflective region,
i.e. for smaller $\beta$.

##  III Optimal observable: two-mode squeezed vacuum state

In the main text we obtained an expression for the optimal quantum observable,
$\hat{O}_{Q}=L_{11}\hat{a}_{1}^{\dagger}\hat{a}_{1}+L_{22}\hat{a}_{2}^{\dagger}\hat{a}_{2}+L_{12}\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}+\hat{a}_{1}\hat{a}_{2}\right)+L_{0}\text{1}_{12}$,
and gave only the coefficient values in the high reflectivity and noiseless
case. Here we give the general expressions:

$\displaystyle\begin{split}L11&=-\frac{2\eta_{1}N_{\text{S}}\left(2N_{\text{S}}+1\right)\left(2N_{\text{th}}+1\right)}{-A+B-C+D}\\\
L22&=\frac{4\eta_{1}\left(2\eta_{1}-1\right)N_{\text{S}}^{2}\left(2N_{\text{th}}+1\right)+2N_{\text{S}}\left(\eta_{1}-2N_{\text{th}}\left(\left(\eta_{1}-3\right)\eta_{1}+\left(\eta_{1}-1\right)\left(3\eta_{1}-1\right)N_{\text{th}}+1\right)-1\right)+N_{\text{th}}\left(2\left(\eta_{1}-1\right)N_{\text{th}}\left(\left(\eta_{1}-1\right)N_{\text{th}}-1\right)+1\right)}{A-B+C-D}\\\
L12&=-\frac{\sqrt{2}\sqrt{N_{\text{S}}\left(2N_{\text{S}}+1\right)}\left(\eta_{1}^{2}\left(N_{\text{S}}\left(4N_{\text{th}}+2\right)-N_{\text{th}}^{2}\right)+N_{\text{th}}\left(N_{\text{th}}+1\right)\right)}{A-B+C-D}\\\
L_{0}&=-\frac{4\eta_{1}^{3}\left(N_{\text{th}}^{2}-2N_{\text{S}}\left(2N_{\text{th}}+1\right)\right){}^{2}-2\eta_{1}^{2}\left(6N_{\text{th}}+5\right)F\left(N_{\text{S}}\left(4N_{\text{th}}+2\right)-N_{\text{th}}^{2}\right)+4\eta_{1}\left(N_{\text{th}}+1\right)\left(N_{\text{S}}^{2}\left(8N_{\text{th}}+4\right)-2N_{\text{S}}\left(N_{\text{th}}\left(6N_{\text{th}}+5\right)+1\right)+N_{\text{th}}^{2}\left(3N_{\text{th}}+2\right)\right)+\left(2N_{\text{th}}+3\right)GF}{E-4\eta_{1}\left(2N_{\text{th}}+1\right)F\left(-4N_{\text{S}}N_{\text{th}}+N_{\text{S}}\left(2N_{\text{S}}-1\right)+N_{\text{th}}^{2}\right)-8N_{\text{S}}^{2}\left(3N_{\text{th}}\left(N_{\text{th}}+1\right)+1\right)+4N_{\text{S}}N_{\text{th}}\left(N_{\text{th}}\left(4N_{\text{th}}+3\right)+1\right)-2N_{\text{th}}^{2}G},\end{split}$

where

$\displaystyle\begin{split}A&\equiv
8\left(\eta_{1}-1\right)\eta_{1}N_{\text{S}}^{3}\left(2N_{\text{th}}+1\right)\\\
B&\equiv
4N_{\text{S}}^{2}\left(-\eta_{1}+\left(\eta_{1}+3\eta_{1}N_{\text{th}}\right){}^{2}-\eta_{1}N_{\text{th}}\left(10N_{\text{th}}+7\right)+3N_{\text{th}}\left(N_{\text{th}}+1\right)+1\right)\\\
C&\equiv
2N_{\text{S}}N_{\text{th}}\left(-\eta_{1}+N_{\text{th}}\left(\eta_{1}\left(3\eta_{1}-8\right)+4\left(\eta_{1}-1\right)\left(2\eta_{1}-1\right)N_{\text{th}}+3\right)+1\right)\\\
D&\equiv
N_{\text{th}}^{2}\left(2\left(\eta_{1}-1\right)N_{\text{th}}\left(\left(\eta_{1}-1\right)N_{\text{th}}-1\right)+1\right)\\\
E&\equiv
4\eta_{1}^{2}\left(-4N_{\text{S}}N_{\text{th}}+N_{\text{S}}\left(2N_{\text{S}}-1\right)+N_{\text{th}}^{2}\right)\left(N_{\text{S}}\left(4N_{\text{th}}+2\right)-N_{\text{th}}^{2}\right)\\\
F&\equiv 2N_{\text{S}}-N_{\text{th}}\\\ G&\equiv
2N_{\text{th}}\left(N_{\text{th}}+1\right)+1\end{split}$

In the high reflectivity case $\eta_{1}\rightarrow 1$ we find:

$\displaystyle\lim_{\eta_{1}\rightarrow 1}L_{11}$
$\displaystyle=-\frac{2N_{\text{S}}(2N_{\text{S}}+1)(2N_{\text{th}}+1)}{N_{\text{S}}^{2}(8N_{\text{th}}(N_{\text{th}}+1)+4)+4N_{\text{S}}N_{\text{th}}^{2}+N_{\text{th}}^{2}}$
$\displaystyle\lim_{\eta_{1}\rightarrow 1}L_{22}$
$\displaystyle=-\frac{4N_{\text{S}}(2N_{\text{S}}N_{\text{th}}+N_{\text{S}}+N_{\text{th}})+N_{\text{th}}}{N_{\text{S}}^{2}(8N_{\text{th}}(N_{\text{th}}+1)+4)+4N_{\text{S}}N_{\text{th}}^{2}+N_{\text{th}}^{2}}$
$\displaystyle\lim_{\eta_{1}\rightarrow 1}L_{12}$
$\displaystyle=\frac{2\sqrt{2}\sqrt{N_{\text{S}}(2N_{\text{S}}+1)}(N_{\text{S}}(4N_{\text{th}}+2)+N_{\text{th}})}{N_{\text{S}}^{2}(8N_{\text{th}}(N_{\text{th}}+1)+4)+4N_{\text{S}}N_{\text{th}}^{2}+N_{\text{th}}^{2}}$
$\displaystyle\lim_{\eta_{1}\rightarrow 1}L_{0}$
$\displaystyle=\frac{-2N_{\text{S}}(N_{\text{S}}(8N_{\text{th}}+4)+6N_{\text{th}}+1)-3N_{\text{th}}}{8N_{\text{S}}^{2}(2N_{\text{th}}(N_{\text{th}}+1)+1)+8N_{\text{S}}N_{\text{th}}^{2}+2N_{\text{th}}^{2}}$

Additionally, as shown in the main text, in the noiseless case we get

$\displaystyle\lim_{\begin{subarray}{c}N_{\text{th}}\rightarrow 0\\\
\eta_{1}\rightarrow 1\end{subarray}}\hat{O}_{Q}$
$\displaystyle=-\mu^{2}\hat{a}_{1}^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}+\mu\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}+\hat{a}_{1}\hat{a}_{2}\right)-\nu\text{1}_{12}$
$\displaystyle\equiv\hat{b}_{1}^{\dagger}\hat{b}_{1}-1,$

where $\mu^{2}\equiv\left(1+1/2N_{\text{S}}\right)$ and
$\nu\equiv\left(1+1/4N_{\text{S}}\right)$ and

$\hat{b}_{1}\equiv-i\left(\hat{a}_{2}^{\dagger}-\mu\hat{a}_{1}\right).$ (S56)

An scheme of the implementation of this observable can be seen in FIG. S2.
After the first beam splitter we have

$\displaystyle\begin{split}\hat{a}_{1}^{\prime}&=\cos\varphi\hat{a}_{1}+\sin\varphi\hat{a}_{2}\\\
\hat{a}_{2}^{\prime}&=-\sin\varphi\hat{a}_{1}+\cos\varphi\hat{a}_{2},\end{split}$
(S57)

then the Josephson parametric amplifiers (JPA) ideally are squeezing
operators, acting as

$\hat{a}_{i}^{\prime\prime}=S(r_{i},\theta_{i})^{\dagger}\hat{a}_{i}^{\prime}S(r_{i},\theta_{i})$
(S58)

where $S(r_{i},\theta_{i})$ is the squeezing operator, acting as

$\displaystyle\begin{split}S(r_{i},\theta_{i})^{\dagger}\hat{a}_{i}^{\prime}S(r_{i},\theta_{i})&=\cosh
r_{i}\hat{a}_{i}^{\prime}-e^{i\theta_{i}}\sinh r_{i}\hat{a}^{\dagger}_{i}\\\
S(r_{i},\theta_{i})^{\dagger}\hat{a}_{i}^{\prime\dagger}S(r_{i},\theta_{i})&=\cosh
r_{i}\hat{a}_{i}^{\prime\dagger}-e^{-i\theta_{i}}\sinh
r_{i}\hat{a}_{i}.\end{split}$

Assuming that the phase shifter $\phi$ acts as $\hat{c}\mapsto
e^{-i\phi}\hat{c}$ we find the following output modes

$\displaystyle\begin{split}e^{i\phi}\hat{b}_{1}&=\cos\theta\left(\cosh
r_{1}\hat{a}_{1}^{\prime}-e^{i\theta_{1}}\sinh
r_{1}\hat{a}_{1}^{\dagger}\right)+\sin\theta\left(\cosh
r_{2}\hat{a}_{2}^{\prime}-e^{i\theta_{2}}\sinh
r_{2}\hat{a}_{2}^{\dagger}\right)\\\ \hat{b}_{2}&=-\sin\theta\left(\cosh
r_{1}\hat{a}_{1}^{\prime}-e^{i\theta_{1}}\sinh
r_{1}\hat{a}_{1}^{\dagger}\right)+\cos\theta\left(\cosh
r_{2}\hat{a}_{2}^{\prime}-e^{i\theta_{2}}\sinh
r_{2}\hat{a}_{2}^{\dagger}\right).\end{split}$

We insert (S57) in the last expression and regroup, finding

$\displaystyle\begin{split}e^{i\phi}\hat{b}_{1}&=\cos\theta\left(\cosh
r_{1}(\cos\varphi\hat{a}_{1}+\sin\varphi\hat{a}_{2})-e^{i\theta_{1}}\sinh
r_{1}\hat{a}_{1}^{\dagger}\right)\\\ &+\sin\theta\left(\cosh
r_{2}(-\sin\varphi\hat{a}_{1}+\cos\varphi\hat{a}_{2})-e^{i\theta_{2}}\sinh
r_{2}\hat{a}_{2}^{\dagger}\right)\\\
&=\hat{a}_{1}\left(\cos\theta\cos\varphi\cosh r_{1}-\sin\theta\sin\varphi\cosh
r_{2}\right)\\\ &+\hat{a}_{2}\left(\cos\theta\sin\varphi\cosh
r_{1}+\sin\theta\cos\varphi\cosh r_{2}\right)\\\
&+\hat{a}_{1}^{\dagger}\left(-e^{i\theta_{1}}\cos\theta\cos\varphi\sinh
r_{1}+e^{i\theta_{2}}\sin\theta\sin\varphi\sinh r_{2}\right)\\\
&+\hat{a}_{2}^{\dagger}\left(-e^{i\theta_{1}}\cos\theta\sin\varphi\sinh
r_{1}-e^{i\theta_{2}}\sin\theta\cos\varphi\sinh r_{2}\right)\end{split}$

Because we want to perform photon-counting over the operator in Eq. (S56), we
identify:

$\displaystyle i\mu$ $\displaystyle=\cos\theta\cos\varphi\cosh
r_{1}-\sin\theta\sin\varphi\cosh r_{2}$ (S59) $\displaystyle i$
$\displaystyle=e^{i\theta_{1}}\cos\theta\sin\varphi\sinh
r_{1}+e^{i\theta_{2}}\sin\theta\cos\varphi\sinh r_{2}.$ (S60)

Figure S2: Schematic circuit for the generation of mode $\hat{b}_{1}$, needed
to correctly implement the optimal observable in the two-mode squeezed vacuum
(TMSV) state. The original $\hat{a}_{i}$ modes mix at a $\varphi$ beam
splitter, and the outputs go through a single-mode squeezing operator
–Josephson parametric amplifier (JPA) in microwaves, with parameters $r_{i}$
and $\theta_{i}$ corresponding to squeezing and phase. Then they mix at a
second beam splitter $\theta$. A phase shift $\phi$ is applied at the end, to
cancel undesired terms.

##  IV Optimal observable: coherent states

The optimal observable in this case is given by

$\hat{O}_{C}=A\text{1}_{(1)}\otimes\left(2\hat{a}_{2}^{\dagger}\hat{a}_{2}-2\eta_{1}\sqrt{\alpha}(\hat{a}_{2}+\hat{a}_{2}^{\dagger})+1+2\eta_{1}^{2}\alpha\right),$
(S61)

where $A=1/2(\eta_{1}-1)(1-N_{\text{th}}(\eta_{1}-1))$ and $\text{1}_{(1)}$ is
the absence of active measurement of mode 1. This expression can be rearranged
as
$\hat{O}_{C}=2A\text{1}_{(1)}\otimes\left[\left(\hat{a}_{2}^{\dagger}-\eta_{1}\sqrt{\alpha}\right)\Big{(}\hat{a}_{2}-\eta_{1}\sqrt{\alpha}\Big{)}+\frac{1}{2}\right]$.
This operator can be experimentally performed with a displacement
$D(-\eta_{1}\sqrt{\alpha})$ and photon-counting in the resulting mode.

Test of the methods: Quantum Illumination As a test of the methods, we
reproduce known results from quantum illumination Sanz_2017 , providing some
new insight. In quantum illumination, an entangled bipartite state is
generated in the lab, and one of the parts (the signal) is sent towards a
noisy region where there may be a slightly reflective target. The other part
(the idler) is kept for future reference. The (possibly) reflected signal,
mixed with the environmental thermal noise is collected back, and a joint
measurement of this reflected signal together with the idler is performed. It
has been shown, both theoretically and experimentally, that this approach
achieves sensitivities which are impossible to get in the classical world,
i.e. when one does not use entanglement. One way to show this quantum
enhancement is by computing the quantum Fisher information for both cases, and
analyzing their ratio, provided that the same amount of energy is sent towards
the target. If, for example, one uses a two-mode squeezed vacuum state for
quantum illumination, the number of photons in the classical state should be
equal to half of the total photon number in the TMSV state. An important
remark is that one has to compare with the best classical protocol. It has
been shown that this corresponds to illumination with coherent states.

We shall proceed to prove these results with our approach. The initial (real)
covariance matrix is

$\Sigma_{\text{TMSV}}=\left[\begin{array}[]{cccccc}2N_{\text{th}}+1&0&0&0&0&0\\\
0&2N_{\text{th}}+1&0&0&0&0\\\
0&0&2N_{\text{S}}+1&0&2\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0\\\
0&0&0&2N_{\text{S}}+1&0&-2\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\\\
0&0&2\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0&2N_{\text{S}}+1&0\\\
0&0&0&-2\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0&2N_{\text{S}}+1\\\
\end{array}\right],$ (S62)

where we have ordered the modes as: thermal state, signal, and idler. Here
$N_{\text{S}}$ is the signal photon number:
$N_{\text{S}}:=\langle\hat{a}_{\text{S}}^{\dagger}\hat{a}_{\text{S}}\rangle$.

The symplectic transformation of quantum illumination is
$S=S_{\text{BS}}^{(\text{B, S})}(\eta)\oplus\text{1}^{(\text{I})}$, i.e., a
beam splitter of reflectivity $\eta$ mixing bath and signal, and no
interaction for the idler mode. In matrix form, this transformation reads

$S(\eta)=\left[\begin{array}[]{cccccc}\sqrt{\eta}&0&\sqrt{1-\eta}&0&0&0\\\
0&\sqrt{\eta}&0&\sqrt{1-\eta}&0&0\\\ -\sqrt{1-\eta}&0&\sqrt{\eta}&0&0&0\\\
0&-\sqrt{1-\eta}&0&\sqrt{\eta}&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
\end{array}\right],$ (S63)

meaning that the first two modes (thermal and signal) get mixed at a beam
splitter, and the idler remains intact. The total, transformed state is

$\left[\begin{array}[]{ccc}\left(-2(\eta-1)N_{\text{S}}+2\eta
N_{\text{th}}+1\right)\text{1}&2\sqrt{(1-\eta)\eta}(N_{\text{S}}-N_{\text{th}})\text{1}&2\sqrt{1-\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\sigma_{Z}\\\
2\sqrt{(1-\eta)\eta}(N_{\text{S}}-N_{\text{th}})\text{1}&\left(2\eta
N_{\text{S}}-2(\eta-1)N_{\text{th}}+1\right)\text{1}&2\sqrt{\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\sigma_{Z}\\\
2\sqrt{1-\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\sigma_{Z}&2\sqrt{\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\sigma_{Z}&\left(2N_{\text{S}}+1\right)\text{1}\\\
\end{array}\right]$ (S64)

After tracing out the losses, we receive the state:

$\Sigma_{\eta}=\left[\begin{array}[]{cccc}2\eta
N_{\text{S}}-2(\eta-1)N_{\text{th}}+1&0&2\sqrt{\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0\\\
0&2\eta
N_{\text{S}}-2(\eta-1)N_{\text{th}}+1&0&-2\sqrt{\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\\\
2\sqrt{\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0&2N_{\text{S}}+1&0\\\
0&-2\sqrt{\eta}\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0&2N_{\text{S}}+1\\\
\end{array}\right]$ (S65)

We can check this is correct since for $\eta=0$ we simply get two thermal
states with zero quantum correlations.

In order to agree with the original formulation of the problem, we will
substitute $\eta\rightarrow\eta^{2}$ in what follows. Therefore, the matrix
$A$ , defined as $A:=i{\Omega}{T}\Sigma_{\eta^{2}}{T}^{\intercal}$, where and
$\Omega:=\text{antidiag}(\text{1}_{2},-\text{1}_{2})$, and
${T}_{ij}:=\delta_{j+4,2i}+\delta_{j,2i-1}$ is the matrix that changes the
basis to the quadrature basis, results in

$A=i\left[\begin{array}[]{cccc}0&0&2\eta^{2}(N_{\text{S}}-N_{\text{th}})+2N_{\text{th}}+1&-2\eta\sqrt{N_{\text{S}}(N_{\text{S}}+1)}\\\
0&0&-2\eta\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&2N_{\text{S}}+1\\\
2\eta^{2}(N_{\text{th}}-N_{\text{S}})-2N_{\text{th}}-1&-2\eta\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&0&0\\\
-2\eta\sqrt{N_{\text{S}}(N_{\text{S}}+1)}&-2N_{\text{S}}-1&0&0\\\
\end{array}\right]$ (S66)

$\begin{split}\frac{\nu_{\pm}^{2}}{2\left(\eta^{2}-1\right)|N_{\text{S}}-N_{\text{th}}|}=&\sqrt{-2\eta^{2}\left(N_{\text{S}}^{2}+N_{\text{S}}+N_{\text{th}}^{2}+N_{\text{th}}\right)+\eta^{4}(N_{\text{S}}-N_{\text{th}})^{2}+(N_{\text{S}}+N_{\text{th}}+1)^{2}}\\\
&-2\eta^{2}\left(2N_{\text{S}}^{2}-2N_{\text{S}}N_{\text{th}}+N_{\text{S}}+2N_{\text{th}}^{2}+N_{\text{th}}\right)\\\
&+2\eta^{4}(N_{\text{S}}-N_{\text{th}})^{2}+2N_{\text{S}}(N_{\text{S}}+1)+2N_{\text{th}}(N_{\text{th}}+1)+1\end{split}$
(S67)

Figure S3: Two views of the same quantum Fisher information ratio
$H_{Q}/H_{C}$ for the problem of quantum illumination in the case of a low
reflective object. The red plane represents $H_{Q}=2H_{C}$, which is
asymptotically achieved in the highly noisy, and dim signal scenario, i.e.
when $N_{\text{th}}\rightarrow\infty$ and $N_{\text{S}}\rightarrow 0$.

Using this in equation (S34) and taking the limit $\eta\rightarrow 0$ we find

$\lim_{\eta\rightarrow 0}H_{Q}(\eta)\equiv
H_{Q}=\frac{4N_{\text{S}}(N_{\text{S}}+1)}{2N_{\text{S}}N_{\text{th}}+N_{\text{S}}+N_{\text{th}}+1}$
(S68)

confirming previously obtained analytical results. In the coherent probe case,
using equation (S34) we find

$H_{C}(\eta)=\frac{4N_{\text{S}}}{1-2N_{\text{th}}(\eta-1)}+\frac{4N_{\text{th}}\eta^{2}}{\left(\eta^{2}-1\right)\left(N_{\text{th}}\left(\eta^{2}-1\right)-1\right)},$
(S69)

taking the limit of low reflectivity $\eta\rightarrow 1$ we find
$H_{C}(\eta=0)\equiv H_{C}=4Ns/(1+2Nth)$. Computing the ratio of Eqs. (S68)
and (S69) we obtain:

$\frac{H_{Q}}{H_{C}}=\frac{(N_{\text{S}}+1)(2N_{\text{th}}+1)}{2N_{\text{S}}N_{\text{th}}+N_{\text{S}}+N_{\text{th}}+1}.$
(S70)

These results match the ones obtained in Ref. (Sanz_2017, ), which serves as a
confirmation that the methods are powerful (no diagonalisation was needed, for
instance).

The authors thank the support from the EU H2020 Quantum Flagship project QMiCS
(820505). MC acknowledges support from the DP-PMI and FCT (Portugal) through
scholarship PD/BD/135186/2017. MC and YO thank the support from Fundação para
a Ciência e a Tecnologia (Portugal), namely through project UIDB/50008/2020,
as well as from project TheBlinQC supported by the EU H2020 QuantERA ERA-NET
Cofund in Quantum Technologies and by FCT (QuantERA/0001/2017). MS thanks
support from the Spanish Government PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE),
and Basque Government IT986-16, the project OpenSuperQ (820363) of the EU
Flagship on Quantum Technologies, as well as the EU FET Open project
Quromorphic (828826). This work is supported by the U.S. Department of Energy,
Office of Science, Office of Advanced Scientific Computing Research (ASCR)
quantum algorithm teams program, under field work proposal number ERKJ333.

## References

  * (1) G. Adesso, S. Ragy, and A. R. Lee. Continuous variable quantum information: Gaussian states and beyond. Open Systems & Information Dynamics, 21(01n02):1440001 (2014).
  * (2) D. Šafránek, A. R. Lee, and I. Fuentes. Quantum parameter estimation using multi-mode Gaussian states. New Journal of Physics, 17(7):073016 (2015).
  * (3) M. Sanz, U. Las Heras, J. J. García-Ripoll, E. Solano, and R. Di Candia. Quantum estimation methods for quantum illumination. Phys. Rev. Lett., 118, 7 (2017).


# Racial/ethnic and socioeconomic disparities of Covid-19 attacks rates in
Suffolk County communities.

Daniel Dobin  Candlewood Middle School, New York, USA  Alexander Dobin
<EMAIL_ADDRESS>Cold Spring Harbor Laboratory, New York, USA

###### Abstract

We investigated the dependence of Covid-19 attack rates on demographic and
socioeconomic factors for the communities in Suffolk County (Long Island, New
York State), presently the 5th most-affected county in the United States.
Confirming the previous observations that minorities are disproportionately
impacted by the Covid-19 disease, we found that the attack rate is strongly
correlated with the minority population proportion, with an alarmingly high
$\sim 4$-fold attack rate increase for Black and Hispanic populations.

##  1 Introduction.

The Covid-19 disease first emerged in late 2019 and has since risen to a
global health crisis and was declared a pandemic on March 11th, 2020. Covid-19
is caused by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2)
which is transmitted from person to person in close contact by small
respiratory droplets. As of , this highly contagious disease has infected more
than 2 million people and has caused over 150 thousand deaths in more than 200
countries.

Socioeconomic and racial/ethnic inequities frequently affect health and
healthcare access, resulting in a higher burden of disease and mortality in
vulnerable social groups [1, 2, 3, 4]. Previous studies showed that multiple
diseases, both non-infectious (e.g. cardiovascular diseases, cancer, obesity,
diabetes) and infectious (influenza, tuberculosis, pneumonia), have a higher
incidence rates in the minority populations [5, 6, 7, 8, 9, 10].

There have been multiple reports of significant impacts of racial/ethnic
disparities on the severity of Covid-19. For instance, the CDC COVID-NET
survey of 580 hospitalized Covid-19 patients from 14 US states [11] found that
$33\%$ of the patients were African Americans, which is significantly higher
than the percentage of African Americans ($18\%$) in the COVID-NET catchment
population. In New York City, African Americans account for $28\%$ of all
fatalities while accounting for 22% of the total population, and Hispanic
Americans account for $34\%$ of the total cases while accounting for 29% of
the population [12] .

Measuring the effects of socioeconomic and racial disparities can help inform
the public and policymakers about at-risk populations, resulting in a more
efficient allocation of resources and personnel to effectively mitigate the
spread of Covid-19. It could also benefit the most affected communities by
increasing their awareness of the Covid-19 dangers and preventative measures.

In this work, we investigated the dependence of Covid-19 attack rate on
racial/ethnic composition and several socioeconomic metrics for the
communities in Suffolk County (Long Island, New York state), presently the 5th
most affected county in the United States. We found that the attack rate is
strongly correlated with the minority population percentage, confirming the
previous observation that minorities are much stronger affected by the
Covid-19 disease.

##  2 Methods.

We have collected the Covid-19 attack rate data on the Suffolk county
Department of Health Services interactive web-map on April 10-11 [13]. The
attack rate is defined as the total number of positive cases in a community
(at the collection time) divided by the total population of the community.
Next, we collected demographic and economic data for each community from the
censusreporter.org website [14]. This information was sourced from the
2014-2018 American Community Survey [15]. The following metrics were
considered: proportions of White, Black, Hispanic and Asian residents, median
age, per capita income, median household income, proportion of persons below
poverty level, mean travel time to work. We restricted our analyses to the 128
communities with populations greater than 1,000 persons. We utilized standard
statistical and plotting functions in Matlab; confidence intervals for all
estimates were calculated by bootstrapping with 10,000 samples. The data table
and code are available on GitHub:
https://github.com/alexdobin/Covid19_disparities.

##  3 Results.

In Figure 1 we plotted the Covid-19 attack rate vs. the minority population
percentage in each community. Attack rate correlates strongly with the
minority population proportion: Pearson correlation coefficient $R=0.61$,
$95\%$ confidence interval $[0.41,0.75]$. Performing the linear least-squares
fit to the data, we estimate that the attack rate for the minorities is 4.1
times higher than that for the whites ($95\%$ confidence interval
$[2.6,6.1]$). We note that this ratio is higher than ratios that can be
derived from the previous reports of rascal/ethnic composition of fatalities
in New York City $(28\%+34\%)/(22\%+29\%)\approx 1.2$ [12] and hospitalized
patients across the US $33\%/18\%\approx 1.8$ [11].

In Figure 2A we plotted Covid-19 attack rate vs. per capita income. We observe
that low-income communities have higher attack rates than high-income
communities. However, as shown in figure 2B, the per capita income is itself
strongly anti-correlated with the minority population proportion. For
communities with minority population $<20\%$, we found no significant
correlation between attack rate and per capita income: $R=0.14$, $95\%$
confidence interval $[-0.06,0.45]$.

The multicollinearity is also present for other demographic and economic
predictor variables in our study. Figure 3A shows pairwise correlation
coefficients between the predictor variables, as well as attack rate and
predictor variables. We see a strong negative correlation of minority
population percentage with per capita income, mean household income, and age,
and a positive correlation with proportion of persons below the poverty line,
and the same trends are observed in the dependence of attack rate on these
predictors. Figure 3B shows that the minority population proportion has the
highest absolute correlation with the attack rate. The attack rates are
correlated with Black and Hispanic population proportions equally strongly.
The Asian population proportion and travel time to work do not correlate
significantly with the attack rate, though this may be owing to the narrow
range of these variables in the communities under study. We also investigated
multilinear regression of the attack rates with all pairs of predictor
variables, and found that none of them increased the correlation statistically
significantly, which indicates that our study is not powered to discern
separate contributions of these demographic and socioeconomic variables.

##  4 Discussion.

In this study, we investigated the dependence of the Covid-19 attack rate on
the several demographic and economic factors in the Suffolk County
communities. Among all the variables, the percentage of the minority
population in a community was the best predictor of Covid-19 attack rate.
Unlike other approaches, which utilize the demographics of the infected
patients, our study estimates the attack rate dependence on race/ethnicity by
correlating it with the racial/ethnic composition of communities. While the
former approaches are more direct, our approach is more informative for
assessing hardships and disparities affecting the minority communities.

Our work does not utilize any genetic information and hence does not point to
any genetic basis in the racial/ethnic disparities of the Covid-19 infection.
The racial/ethnic and socioeconomic inequities have been widely documented for
other diseases and are attributed to a multitude of health and environmental
causes, such as limited access to healthcare, lack of health insurance, pre-
existing conditions, substandard housing, and hazardous work. In the case of
Covid-19 pandemic, these hardships are further exacerbated by the higher
economic stress for minority communities resulting in diminished capacity for
social distancing which is imperative to suppress the spread of the novel
coronavirus. In addition, minorities constitute a large proportion of
essential workers, further increasing their exposure to Covid-19 infection.

Our main finding is that the Covid-19 attack rate for Black and Hispanic
populations is alarmingly (4.1-fold) higher than that for the White
population. In line with previous observations, this indicates that the
minorities are affected the greatest by Covid-19, and thus minority
communities require the most assistance in order to lessen the distress caused
by the Covid-19 pandemic. We notice that the New York State and Suffolk County
governments have begun taking first steps to address this crisis. The new
Covid-19 testing sites have been opened on April 8-16 in minority communities
that are Covid-19 hot spots: Brentwood (3-fold higher attack rate than Suffolk
County average), Huntington Station (2.4-fold), Wyandanch (2.8-fold). We hope
that our study will increase public and government awareness of the severity
of the Covid-19 impact on minorities, and result in more decisive actions
taken to protect the minority communities in the United States and around the
world.

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## Figures

Figure 1: Covid-19 attack rate vs. minority population percentage. The circles
represent communities, the solid red line is a linear least squares fit to the
data, the dashed magenta lines represent the 95% confidence bands of the fit
obtained by bootstrapping (10,000 samples).

Figure 1

Figure 2: A: attack rate vs. per capita income. Circles represent communities,
their color scale corresponds to the percentage of the minority population in
each community. B: Per capita income vs. minority population percentage. The
curved red line represents the quadratic least squares fit.

Figure 2A

Figure 2B

Figure 3: A: Matrix of pairwise Pearson correlation coefficients between the
predictor variables and attack rate. B: Pearson correlation of attack rate
with different predictors. Error bars represent the 95% confidence intervals
calculated using bootstrapping (10,000 samples).

Figure 3A

Figure 3B