Proceedings ArticleDOI

# A comparison between quantum and classical noise radar sources

AbstractWe compare the performance of a quantum radar based on two-mode squeezed states with a classical radar system based on correlated thermal noise. With a constraint of equal number of photons $N_S$ transmitted to probe the environment, we find that the quantum setup exhibits an advantage with respect to its classical counterpart of $\sqrt{2}$ in the cross-mode correlations. Amplification of the signal and the idler is considered at different stages of the protocol, showing that no quantum advantage is achievable when a large-enough gain is applied, even when quantum-limited amplifiers are available. We also characterize the minimal type-II error probability decay, given a constraint on the type-I error probability, and find that the optimal decay rate of the type-II error probability in the quantum setup is $\ln(1+1/N_S)$ larger than the optimal classical setup, in the $N_S\ll1$ regime. In addition, we consider the Receiver Operating Characteristic (ROC) curves for the scenario when the idler and the received signal are measured separately, showing that no quantum advantage is present in this case. Our work characterizes the trade-off between quantum correlations and noise in quantum radar systems.

Topics: Quantum radar (65%), Noise (electronics) (54%), Photon (54%), Quantum (52%)

### Introduction

• Amplification of the signal and the idler is considered at different stages of the protocol, showing that no quantum advantage is achievable when a large-enough gain is applied, even when quantum-limited amplifiers are available.
• The authors work characterizes the trade-off between quantum correlations and noise in quantum radar systems.
• A first proposal for implementing a microwave QI protocol was advanced in Ref. [11].
• Noise radar is an old concept that operates by probing the environment with a noisy signal and crosscorrelating the returns with a retained copy of the transmitted signal [15].
• The authors analysis shows that any quantum advantage is destroyed by the unavoidable noise added when amplifying either the signal or the idler.

### II. THEORY

• The authors introduce the models for the quantum and classical systems, within the quantum mechanical description.
• The authors emulate Refs. [14], [21], where the classical and quantum noise radar were first studied.

### A. Quantum preliminaries

• The commutation relation [q̂, p̂] = îI implies that the quadratures can not be measured simultaneously with arbitrary precision, due to the Heisenberg uncertainty relation.
• These mode designations are used interchangeably for both the quantum and classical system.
• The classicallycorrelated noise (CCN) system uses two sources of thermal noise, â0 and â1, at temperatures T0 and T1, respectively.
• In the following, the authors will refer as N0 (N1) the average number of photons for the mode â0 (â1).
• A TMSV state is closely related to a classicallycorrelated thermal noise, as also here both signal and idler photons are Bose-Einstein distributed.

### B. Covariance and correlation matrices

• As the states considered here are Gaussian, their statistics is entirely determined by the first and second order quadrature moments.
• (4) This is the case for both the classical and the entangled thermal noise states.
• (5) These coefficients, also referred to as Pearson’s correlation coefficients, characterize the linear dependence between the quadratures X̂i and X̂j .

### C. Quantum relative entropy

• The quantum relative entropy defines an information measure between two quantum states.
• This quantity is related to the performance in the asymmetric binary hypothesis testing via the quantum Stein’s lemma.
• The authors rely on quantum relative entropy computations and its variance in order to quantify the performance in the asymptotic setting, i.e., when M 1.
• This is in contrast to the original treatment based on the Chernoff bound [2], which provides an estimation of the average error probability when the prior probabilities of target absence or presence are equal.
• In a typical radar scenario, the prior probabilities are not the same, and may be even unknown.

### A. Probing the environment

• The signal mode is transmitted to probe the environment where an object may be present or absent.
• Here, η can be interpreted as the ratio between transmitted power and received power, including the effects of atmospheric attenuation, the antenna gain and the target radar cross section.
• The authors use a beamsplitter model to take into account the environmental losses.
• For the current calculation, the reflection coefficient is assumed to be non-fluctuating.
• The authors also assume 〈âI âB〉 = 0, i.e., the returned signal preserves some correlations with the idler mode only if the object is present.

### B. Cross-correlation coefficient

• The covariance matrix of the system composed of the received signal and the idler is easily computable using Eq. (4) and Eq. (8).
• For both the classical and the quantum noise radars considered here, applying Eq. (5) gives us the correlation matrix R = ( I κD(θ) κD(θ)T I ) , (9) where 0 ≤ κ ≤ 1 is the amplitude of the cross-correlation coefficient, and D is a matrix with determinant |D| = ±1.
• For a fair comparison between the quantum and classical systems, the authors introduce a constraint on the transmitted power of the signal modes, i.e., they set NS = ξN0 + (1− ξ)N1. (11) This constraint can be interpreted as giving both systems equal LPI properties.
• Eq. (12), for given noise transmitting power, defines the correlations of a class of classical noise radars, labeled by the beamsplitter parameter ξ.

• This setting as been used as benchmarking in the recent microwave quantum illumination experiments [12], [13].
• Show the functional behaviour of the correlation coefficients for the entangled and classically-correlated thermal noise source, depending on the transmitting power NS .
• The quantum advantage decays slowly with increasing NS , keeping an advantage also for finite NS , until virtually disappearing for NS ' 10.
• Here, the phase-shift of the signal mode due to the propagating path has been set to zero.
• This strategy shows the same asymptotic performance as in the case of known θ.

### D. The effect of amplification

• In the following, the authors consider three Gaussian amplifying schemes.
• 1) Amplification before transmitting the signal mode:.
• The classical and quantum covariance matrix transforms identically.
• In addition, in the weak signal regime where the quantum advantage is relevant, the effective SNR is low even before amplification.
• As can be intuitively understood, amplification at the receiver end can not increase correlations with the idler reference.

### E. Receiver Operating Characteristic performance

• The authors analyze the quantum and classical noise radar in the asymmetric setting, i.e., when the prior probabilities are not equal, under a different perspective.
• The authors derive the ROC curves in the case when the idler and the signal are separately measured with heterodyne detection.
• A substantial advantage can be also found for moderate values of NS (see Fig. 5).
• The variance of the quantum relative entropy provides the convergence rate of the error probability exponent to its asymptotic value, for the type-I error probability constrained to be lower than ε.
• The authors do not provide the explicit formula here.

### IV. CONCLUSION AND OUTLOOK

• The authors have compared the performance of a quantum noise radar based on two-mode squeezed states and a class of noise radars based on thermal states.
• The authors have found that, given a constraint on the transmitting power, a quantum advantage in the ROC curve and their asymptotic performances is possible.
• A quantum advantage appears when the idler and signal are allowed to be measured jointly.
• The authors results show that amplification is not a good strategy to overcome the technical difficulties that one must face in a practical quantum radar implementation.
• Interfacing a large bandwidth entangled microwave signal with the environment, and developing low-noise power detectors is crucial for performing a QI experiment with a quantum advantage.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
This material is protected by copyright and other intellectual property rights, and duplication or sale of all or
part of any of the repository collections is not permitted, except that material may be duplicated by you for
your research use or educational purposes in electronic or print form. You must obtain permission for any
other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not
an authorised user.
Jonsson, Robert; Di Candia, Roberto; Ankel, Martin; Strom, Anders; Johansson, Goran
A comparison between quantum and classical noise radar sources
Published in:
DOI:
Published: 21/09/2020
Document Version
Peer reviewed version
Jonsson, R., Di Candia, R., Ankel, M., Strom, A., & Johansson, G. (2020). A comparison between quantum and

A comparison between quantum and classical noise
Robert Jonsson
1
Department of Microtechnology
and Nanoscience
Chalmers University of Technology
2
G
¨
oteborg, Sweden
Email: robejons@chalmers.se
Anders Str
¨
om
G
¨
oteborg, Sweden
Roberto Di Candia
Department of Communications
and Networking
Aalto University
Helsinki, Finland
Email: roberto.dicandia@aalto.ﬁ
G
¨
oran Johansson
Department of Microtechnology
and Nanoscience
Chalmers University of Technology
G
¨
oteborg, Sweden
Martin Ankel
1
Department of Microtechnology
and Nanoscience
Chalmers University of Technology
2
G
¨
oteborg, Sweden
Abstract—We compare the performance of a quantum radar
based on two-mode squeezed states with a classical radar system
based on correlated thermal noise. With a constraint of equal
number of photons N
S
transmitted to probe the environment, we
ﬁnd that the quantum setup exhibits an advantage with respect
to its classical counterpart of
2 in the cross-mode correlations.
Ampliﬁcation of the signal and the idler is considered at different
stages of the protocol, showing that no quantum advantage
is achievable when a large-enough gain is applied, even when
quantum-limited ampliﬁers are available. We also characterize
the minimal type-II error probability decay, given a constraint
on the type-I error probability, and ﬁnd that the optimal decay
rate of the type-II error probability in the quantum setup
is ln(1 + 1/N
S
) larger than the optimal classical setup, in
the N
S
Operating Characteristic (ROC) curves for the scenario when the
idler and the received signal are measured separately, showing
that no quantum advantage is present in this case. Our work
characterizes the trade-off between quantum correlations and
I. INTRODUCTION
The quantum Illumination (QI) protocol [1]–[7] uses en-
tanglement as a resource to improve the detection of a low-
reﬂectivity object embedded in a bright environment. The
protocol was ﬁrst developed for a single photon source [1],
and it was then extended to general bosonic quantum states
and thermal bosonic channels [2]. Here, a
6
in the effective signal-to-noise ratio (SNR) is achievable
when using two-mode squeezed states instead of coherent
states. This gain has been recently shown to be optimal,
and reachable exclusively in the regime of low transmitting
power per mode [8], [9]. The QI protocol has possible
applications in the spectrum below the Terahertz band, as
here the environmental noise is naturally bright. In particular,
microwave quantum technology has been very well developed
in the last decades [10], paving the way of implementing these
ideas for building a ﬁrst prototype of quantum radar.
The possible beneﬁts of a quantum radar system are generally
understood to be situational. In an adversarial scenario, it
is beneﬁcial for a radar system to be able to operate while
minimizing the power output, in order to reduce the probability
for the transmitted signals to be detected. This property is
commonly referred to as Low Probability of Intercept (LPI),
and it is a common measure to limit the ability of the enemy
to localize and discover the radar. The low signal levels
required for QI are in principle excellent for acquiring good
LPI properties. However, there are several challenges to face
in order to achieve this goal. A ﬁrst proposal for implementing
a microwave QI protocol was advanced in Ref. [11]. The
protocol relies on an efﬁcient microwave-optical interface for
the idler storage and the measurement stage. This technology is
promising for this and other applications, however it is still in its
infancy. Furthermore, the signal generation requires cryogenic
technology, which must be interfaced with a room-temperature
environment. Recently, a number of QI-related experiments
have been carried out in the microwave regime [12], [13],
showing that some correlations of an entangled signal-idler
system are preserved after the signal is sent out of the dilution
refrigerator. While these results are a good benchmark for future
QI experiments, they strictly rely on the ampliﬁcation of the
signal and idler. This has been shown to rule out any quantum
advantage with respect to an optimal classical reference [14].
In this work, we discuss the role of quantum correlations
and ampliﬁcation in the QI protocol, providing a comparative
analysis of quantum and classical noise radars in different
scenarios. Noise radar is an old concept that operates by
probing the environment with a noisy signal and cross-
correlating the returns with a retained copy of the transmitted
signal [15]. A quantum noise radar operates similarly to its

conventional counterpart, but differs in the use of a two-mode
entangled state as noise source [12], [13]. An advantage of
the quantum noise radar over the classical counterpart can be
declared if stronger correlations can be achieved, when both
systems illuminate the environment with equal power. In the
microwave regime, the two-mode squeezed state used for noise
correlations can be generated with superconducting circuits with
a Josephson Parametric Ampliﬁer (JPA) at
T ' 20
mK [16],
[17]. On the one hand, using quantum correlated signals
generated by a JPA enhances the signal-to-noise ratio in the
low transmitting power per mode regime. On the other hand,
Josephson parametric circuits are able to generate correlated
and entangled signals with large bandwidth [18]–[20]. This
allows, in principle, a system to operate in the low power-per-
Here, we analyze the performance of a JPA-based noise radar
in different scenarios which include different sources of noise.
Our analysis shows that any quantum advantage is destroyed
by the unavoidable noise added when amplifying either the
signal or the idler. We also show when the idler and signal are
measured separately, the entanglement initially present in the
signal-idler system is not properly exploited, and no quantum
advantage can be retained. The latter happens even without
amplifying the signal or the idler. Our work complements the
analysis done in Ref. [14] with the explicit calculations of the
cross-correlation coefﬁcients and the optimal asymptotic ROC
performance in the microwave regime.
II. THEORY
In this section, we introduce the models for the quantum and
classical systems, within the quantum mechanical description.
In this step, we emulate Refs. [14], [21], where the classical
and quantum noise radar were ﬁrst studied. In all expressions,
we assume the natural units (~ = 1, k
B
= 1).
A. Quantum preliminaries
A single, narrowband mode of the electric ﬁeld, at microwave
frequency
f
, is deﬁned with an operator (in suitable units)
as
ˆ
E = ˆq cos 2πft + ˆp sin 2πft,
where
ˆq
and
ˆp
are the in-
are related to the bosonic annihilation (
ˆa
) and creation (
ˆa
)
operators as
ˆq = a
a)/
2
and
ˆp = i(ˆa
ˆa)/
2
, where
a, ˆa
] =
ˆ
I
. The commutation relation
[ˆq, ˆp] = i
ˆ
I
implies
that the quadratures can not be measured simultaneously with
arbitrary precision, due to the Heisenberg uncertainty relation.
In the following, we represent the quadratures of the two-
modes of the electric ﬁeld by the vector
ˆ
X = (ˆq
S
, ˆp
S
, ˆq
I
, ˆp
I
)
T
,
where the indices S and I refer to the signal and idler mode,
respectively. These mode designations are used interchangeably
for both the quantum and classical system.
1) Classically-correlated thermal noise: The classically-
correlated noise (CCN) system uses two sources of thermal
noise,
ˆa
0
and
ˆa
1
, at temperatures
T
0
and
T
1
, respectively.
T
1
T
0
(ξ, ϕ)
ˆa
0
ˆa
1
ˆa
(C)
S
ˆa
(C)
I
Fig. 1.
Preparation of classically-correlated thermal noise.
A beamsplitter
with reﬂection coefﬁcient
ξ
and phase turning angle
ϕ
generates the signal
and idler modes from the modes
ˆa
0
and
ˆa
1
. These modes are in a thermal
state with
T
0
and
T
1
effective noise temperatures, respectively. The output
modes are correlated provided that T
0
6= T
1
.
In general, the quantum state of a thermal noise mode at
temperature T can be represented by the density operator
ρ
th
=
X
n=0
N
n
(N + 1)
n+1
|nihn|, (1)
where the average number
1
of photons is deﬁned by the thermal
equilibrium Bose-Einstein statistics at temperature
T
, i.e.,
N =
[exp (2πf/T ) 1]
1
. In the following, we will refer as
N
0
(
N
1
) the average number of photons for the mode
ˆa
0
(
ˆa
1
).
The thermal modes
ˆa
0
and
ˆa
1
pass through a beamsplitter, as
shown in Fig. 1. This generates a signal mode
ˆa
(C)
S
and an
idler mode ˆa
(C)
I
, related to the inputs as [22]
ˆa
(C)
S
ˆa
(C)
I
!
=
ξ
1 ξe
iϕ
1 ξe
iϕ
ξ
ˆa
0
ˆa
1
. (2)
Here,
ξ (0, 1)
is the reﬂection coefﬁcient and
ϕ
is the phase
turning angle of the beamsplitter, in the following set to be
zero. One can think of this process as a noise signal, generated
by a thermal source at temperature
T
0
, sent as input of a
power divider placed in an environment at temperature
T
1
.
The output modes
ˆa
(C)
S
and
ˆa
(C)
I
are in a thermal state with
ξN
0
+ (1 ξ)N
1
and
ξN
1
+ (1 ξ)N
0
average number of
photons, respectively. If
T
1
6= T
0
, or, equivalently,
N
1
6= N
0
,
then the outputs are classically-correlated, regardless of the
value of ξ.
2) Entangled thermal noise: A Two-Mode Squeezed Vac-
uum (TMSV) state |ψi
TMSV
is represented in the Fock basis
as
|ψi
TMSV
=
X
n=0
s
N
n
S
(N
S
+ 1)
n+1
|ni
S
|ni
I
, (3)
where
N
S
is the average number of photons in both the signal
and idler mode. A TMSV state is closely related to a classically-
correlated thermal noise, as also here both signal and idler
photons are Bose-Einstein distributed. However, as we will see,
the resulting correlations in the low signal-power regime are
stronger for the TSMV states.
1
All variables using the symbol
N
refer to mode quanta and should not be
confused with the noise ﬁgure of a microwave component, which often shares
the same symbol.

B. Covariance and correlation matrices
As the states considered here are Gaussian, their statistics
is entirely determined by the ﬁrst and second order quadrature
moments. For zero-mean states, i.e., when
h
ˆ
X
i
i = 0
for all
i
,
the states are characterized entirely by the covariance matrix
Σ, with elements
Σ
i,j
=
1
2
h
ˆ
X
i
ˆ
X
j
+
ˆ
X
j
ˆ
X
i
i h
ˆ
X
i
ih
ˆ
X
j
i. (4)
This is the case for both the classical and the entangled
thermal noise states. Similarly, one can introduce the correlation
coefﬁcient matrix R, whose elements are
R
i,j
=
Σ
i,j
p
Σ
i,i
p
Σ
j,j
[1, 1]. (5)
These coefﬁcients, also referred to as Pearson’s correlation
coefﬁcients, characterize the linear dependence between the
ˆ
X
i
and
ˆ
X
j
.
C. Quantum relative entropy
The quantum relative entropy deﬁnes an information measure
between two quantum states. It is deﬁned as
D(ρ
1
||ρ
0
) = Tr ρ
1
(ln ρ
1
ln ρ
0
), (6)
for two density matrices
ρ
0
and
ρ
1
. This quantity is related to
the performance in the asymmetric binary hypothesis testing
via the quantum Stein’s lemma. The task is to discriminate
between
M
copies of
ρ
0
and
M
copies of
ρ
1
, given a bound
on the type-I error probability (probability of false alarm,
P
F a
)
of
ε (0, 1)
. In this discrimination, the maximum type-II error
probability (probability of miss, P
M
) exponent is
ln P
M
M
= D(ρ
1
||ρ
0
) +
r
V (ρ
1
||ρ
0
)
M
Φ
1
(ε) + O
ln M
M
, (7)
where
V (ρ
1
||ρ
0
) = Tr ρ
1
[ln ρ
1
ln ρ
0
D(ρ
1
||ρ
0
)]
2
is the
quantum relative entropy variance and
Φ
1
is the inverse
cumulative normal distribution [23]. In this work, we rely on
quantum relative entropy computations and its variance in order
to quantify the performance in the asymptotic setting, i.e., when
M 1
. This is in contrast to the original treatment based on
the Chernoff bound [2], which provides an estimation of the
average error probability when the prior probabilities of target
absence or presence are equal. In a typical radar scenario, the
prior probabilities are not the same, and may be even unknown.
In this section, we analyze the performance of the classical
and quantum noise correlated radars, based on the states deﬁned
in the previous section.
A. Probing the environment
The signal mode is transmitted to probe the environment
where an object (target) may be present or absent. This process
is modelled as a channel with reﬂection coefﬁcient
η
that is
non-zero and small when the target is present (
0 < η 1
)
and zero when the target is absent (
η = 0
), see Fig. 2. Here,
η
can be interpreted as the ratio between transmitted power and
Preparation
Detection
ˆa
I
ˆa
S
ˆa
R
ˆa
B
η
Fig. 2.
Scheme of the quantum and classical noise radar systems.
The
system probes a region of space with a signal
ˆa
S
to detect a possible object,
modelled as a channel with reﬂectivity parameter
η
. The returned signal
ˆa
R
is then used for detecting correlations with the idler mode
ˆa
I
, which has been
retained in the lab.
received power, including the effects of atmospheric attenuation,
the antenna gain and the target radar cross section. We use
a beamsplitter model to take into account the environmental
losses. In other words, the returned mode ˆa
R
is given by
ˆa
R
=
η ˆa
S
e
iθ
+
p
1 η ˆa
B
, (8)
where
ˆa
B
is a bright background noise mode with
hˆa
B
ˆa
B
i =
N
B
average power per mode, and where
θ
is a phase shift
relative to the idler. In the
1 10
GHz regime, where the
technology is advanced enough to apply these ideas in the
quantum regime, we have that
N
B
' 10
3
, which is assumed
for numerical computations. For the current calculation, the
reﬂection coefﬁcient is assumed to be non-ﬂuctuating. We also
assume
hˆa
I
ˆa
B
i = 0
, i.e., the returned signal preserves some
correlations with the idler mode only if the object is present.
This allows us to deﬁne a correlation detector able to detect
the presence or absence of the object.
B. Cross-correlation coefﬁcient
The covariance matrix of the system composed of the
received signal and the idler is easily computable using Eq.
(4)
and Eq.
(8)
. For both the classical and the quantum noise radars
considered here, applying Eq.
(5)
gives us the correlation matrix
R =
I κD(θ)
κD(θ)
T
I
, (9)
where
0 κ 1
is the amplitude of the cross-correlation
coefﬁcient, and
D
is a matrix with determinant
|D| = ±1
. The
cross-correlation coefﬁcient for the entangled TMSV source
can be derived directly from the deﬁnition,
κ
TMSV
=
p
ηN
S
(N
S
+ 1)
q
N
R
+
1
2
q
N
S
+
1
2
, (10)

with
N
R
= ηN
S
+ (1 η)N
B
. For a fair comparison between
the quantum and classical systems, we introduce a constraint
on the transmitted power of the signal modes, i.e., we set
N
S
= ξN
0
+ (1 ξ)N
1
. (11)
This constraint can be interpreted as giving both systems equal
LPI properties. Eq.
(11)
yields an expression for the classical
cross-correlation amplitude as
κ
CCN
=
η(N
S
N
1
)
q
N
R
+
1
2
q
N
S
N
1
+
ξ
1ξ
N
1
+
1
2
, (12)
where
N
S
> N
1
follows directly from Eq.
(11)
and the
assumption
N
0
> N
1
. This quantity is maximal in the
N
1
1
regime, where Eq.
(11)
reduces to
N
S
= ξN
0
. We assume
N
1
1
, which corresponds to classically-correlated thermal
noise generated in a noise-free environment. At microwave
frequencies this is achievable at mK temperatures. Eq.
(12)
,
for given noise transmitting power, deﬁnes the correlations of
a class of classical noise radars, labeled by the beamsplitter
parameter ξ.
It is easy to see that the quantity
κ
2
is linearly proportional
to the effective SNR in the likelihood-ratio tests. A larger value
of
κ
2
means a stronger discrimination power. Therefore, we
deﬁne a ﬁgure of merit
Q
A
, quantifying the advantage of the
quantum over the classical noise radar, as
Q
A
κ
2
TMSV
κ
2
CCN
=
N
S
+ 1
N
S
+
1
2
1 +
ξ
2N
S
(1 ξ)
, (13)
which can be evaluated for different values of the free parameter
ξ
. Restricting the constraint to equal power in both the signal
and idler mode is equivalent to applying a 50-50 beamsplitter
to the thermal noise source in Eq.
(2)
, or, in other words, it
corresponds to setting ξ = 1/2. This gives
Q
A
(ξ = 1/2) = 1 +
1
N
S
, (14)
which is unbounded for
N
S
0
. This setting as been used as
benchmarking in the recent microwave quantum illumination
experiments [12], [13]. However, this choice of
ξ
is not optimal,
2
.
A strongly asymmetric beamsplitter must be applied to a
very bright noise source (
N
0
1
) in order to maximize
the correlations in the classical case, while maintaining the
constraint on transmitted power
N
S
= ξN
0
. It can be seen in
Eq.
(13)
, that
Q
A
is maximized in the
ξ/(1 ξ) N
S
limit.
In this setting, the CCN idler has a much better SNR than in
the symmetric conﬁguration, and we get
Q
A
(ξ/(1 ξ) N
S
) 2
1
N
S
2N
S
+ 1
. (15)
In the low transmitting power limit we have that
lim
N
S
0
Q
A
= 2
,
i.e., a
2
advantage in the correlation coefﬁcient. In Fig. 3 we
2
This criticism was already raised by J. H. Shapiro in Ref. [14].
10
-2
10
-1
10
0
10
1
0
0.5
1
10
-2
10
-1
10
0
10
1
0
1
2
3
4
5
Fig. 3.
Performance of the quantum and classical noise radars in terms
of the cross-correlation coefﬁcient.
(Upper) The cross-correlation coefﬁcients
for the quantum and classical noise radars, as functions of the transmitted
quanta, rescaled such that
η = 1
. (Lower) The quantum advantage for the
cases considered above. In both plots, we have considered the
ξ = 10
3
and
ξ = 0.5
settings. The ﬁrst setting achieves a close to optimal cross-correlation
coefﬁcient. The second setting is suboptimal, and it has been used as classical
reference in recent microwave illumination experiments [12], [13].
show the functional behaviour of the correlation coefﬁcients for
the entangled and classically-correlated thermal noise source,
depending on the transmitting power
N
S
tage decays slowly with increasing
N
S
also for ﬁnite
N
S
, until virtually disappearing for
N
S
' 10
.
Note that in the limit where the advantage is maximized, the
modes also become uncorrelated (
κ 0
) in both systems. A
signal-idler system with a large operating bandwidth is needed
to compensate the low amount of correlations per mode in the
N
S
1 limit.
Here, the phase-shift of the signal mode due to the prop-
agating path has been set to zero. In other words, we work
in a rotated frame, where the inter-mode phase angle
θ
is
applied only to the idler mode, i.e.,
ˆa
I
7→ ˆa
I
e
iθ
. Note that,
in general, the phase
θ
is unknown. The original QI protocol
assumes the knowledge of the inter-mode phase angle
θ
. In
this case, the complex conjugate receiver deﬁned in Ref. [24]
saturates the quantum advantage given in Eq.
(13)
with a
likelihood-ratio test [25]. If
θ
is unknown, then one can deﬁne
O(
M)
copies are used to get
a rough estimate of
θ
, and then
M O(
M)
are used to
perform the discrimination protocol in the optimal reference
frame maximizing the Fisher information [7]. This strategy
shows the same asymptotic performance as in the case of
known θ.
D. The effect of ampliﬁcation
In the following, we consider three Gaussian amplifying
schemes. We show how the quantum advantage deﬁned in
terms the cross-correlation coefﬁcients rapidly disappears when
an ampliﬁcation is involved at any stage of the protocol.

##### Citations
More filters

Journal ArticleDOI
Abstract: (b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

1,753 citations

Proceedings ArticleDOI
14 May 2017
Abstract: We propose a structured receiver for optimum mixed-state discrimination in quantum illumination target detection, paving the way for entanglement-enhanced minimum-error-probability sensing in an entanglement-breaking environment.

35 citations

##### References
More filters

Book
01 Jan 1973
Abstract: Preface 1. Planck's radiation law and the Einstein coefficients 2. Quantum mechanics of the atom-radiation interaction 3. Classical theory of optical fluctuations and coherence 4. Quantization of the radiation field 5. Single-mode quantum optics 6. Multimode and continuous-mode quantum optics 7. Optical generation, attenuation and amplification 8. Resonance fluorescence and light scattering 9. Nonlinear quantum optics Index

3,032 citations

Journal ArticleDOI
, Tim Duty2
17 Nov 2011-Nature
TL;DR: The dynamical Casimir effect is observed in a superconducting circuit consisting of a coplanar transmission line with a tunable electrical length and two-mode squeezing in the emitted radiation is detected, which is a signature of the quantum character of the generation process.
Abstract: One of the most surprising predictions of modern quantum theory is that the vacuum of space is not empty. In fact, quantum theory predicts that it teems with virtual particles flitting in and out of existence. Although initially a curiosity, it was quickly realized that these vacuum fluctuations had measurable consequences-for instance, producing the Lamb shift of atomic spectra and modifying the magnetic moment of the electron. This type of renormalization due to vacuum fluctuations is now central to our understanding of nature. However, these effects provide indirect evidence for the existence of vacuum fluctuations. From early on, it was discussed whether it might be possible to more directly observe the virtual particles that compose the quantum vacuum. Forty years ago, it was suggested that a mirror undergoing relativistic motion could convert virtual photons into directly observable real photons. The phenomenon, later termed the dynamical Casimir effect, has not been demonstrated previously. Here we observe the dynamical Casimir effect in a superconducting circuit consisting of a coplanar transmission line with a tunable electrical length. The rate of change of the electrical length can be made very fast (a substantial fraction of the speed of light) by modulating the inductance of a superconducting quantum interference device at high frequencies (>10 gigahertz). In addition to observing the creation of real photons, we detect two-mode squeezing in the emitted radiation, which is a signature of the quantum character of the generation process.

755 citations

Journal ArticleDOI

Abstract: In the past 20 years, impressive progress has been made both experimentally and theoretically in superconducting quantum circuits, which provide a platform for manipulating microwave photons. This emerging field of superconducting quantum microwave circuits has been driven by many new interesting phenomena in microwave photonics and quantum information processing. For instance, the interaction between superconducting quantum circuits and single microwave photons can reach the regimes of strong, ultra-strong, and even deep-strong coupling. Many higher-order effects, unusual and less familiar in traditional cavity quantum electrodynamics with natural atoms, have been experimentally observed, e.g., giant Kerr effects, multi-photon processes, and single-atom induced bistability of microwave photons. These developments may lead to improved understanding of the counterintuitive properties of quantum mechanics, and speed up applications ranging from microwave photonics to superconducting quantum information processing. In this article, we review experimental and theoretical progress in microwave photonics with superconducting quantum circuits. We hope that this global review can provide a useful roadmap for this rapidly developing field.

688 citations

Journal ArticleDOI
12 Sep 2008-Science
TL;DR: It is shown that for photodetection, quantum illumination with m bits of entanglement can in principle increase the effective signal-to-noise ratio by a factor of 2m, an exponential improvement over unentangled illumination.
Abstract: The use of quantum-mechanically entangled light to illuminate objects can provide substantial enhancements over unentangled light for detecting and imaging those objects in the presence of high levels of noise and loss. Each signal sent out is entangled with an ancilla, which is retained. Detection takes place via an entangling measurement on the returning signal together with the ancilla. This paper shows that for photodetection, quantum illumination with m bits of entanglement can in principle increase the effective signal-to-noise ratio by a factor of 2 m , an exponential improvement over unentangled illumination. The enhancement persists even when noise and loss are so great that no entanglement survives at the detector.

568 citations

Journal ArticleDOI

TL;DR: By making the optimum joint measurement on the light received from the target region together with the retained spontaneous parametric down-conversion idler beam, the quantum-illumination system realizes a 6 dB advantage in the error-probability exponent over the optimum reception coherent-state system.
Abstract: An optical transmitter irradiates a target region containing a bright thermal-noise bath in which a low-reflectivity object might be embedded. The light received from this region is used to decide whether the object is present or absent. The performance achieved using a coherent-state transmitter is compared with that of a quantum-illumination transmitter, i.e., one that employs the signal beam obtained from spontaneous parametric down-conversion. By making the optimum joint measurement on the light received from the target region together with the retained spontaneous parametric down-conversion idler beam, the quantum-illumination system realizes a 6 dB advantage in the error-probability exponent over the optimum reception coherent-state system. This advantage accrues despite there being no entanglement between the light collected from the target region and the retained idler beam.

453 citations