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Jonsson, Robert; Di Candia, Roberto; Ankel, Martin; Strom, Anders; Johansson, Goran

A comparison between quantum and classical noise radar sources

Published in:

2020 IEEE Radar Conference, RadarConf 2020

DOI:

10.1109/RadarConf2043947.2020.9266597

Published: 21/09/2020

Document Version

Peer reviewed version

Please cite the original version:

Jonsson, R., Di Candia, R., Ankel, M., Strom, A., & Johansson, G. (2020). A comparison between quantum and

classical noise radar sources. In 2020 IEEE Radar Conference, RadarConf 2020 [9266597] (IEEE Radar

Conference; Vol. 2020-September). IEEE. https://doi.org/10.1109/RadarConf2043947.2020.9266597

A comparison between quantum and classical noise

radar sources

Robert Jonsson

1

Department of Microtechnology

and Nanoscience

Chalmers University of Technology

2

Radar Solutions, Saab AB

G

¨

oteborg, Sweden

Email: robejons@chalmers.se

Anders Str

¨

om

Radar Solutions, Saab AB

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

Radar Solutions, Saab AB

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

1 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.

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

dB advantage

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-

mode regime, where quantum radars show fully their advantage.

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-

phase and quadrature operators, respectively. The quadratures

are related to the bosonic annihilation (

ˆa

) and creation (

ˆa

†

)

operators as

ˆq = (ˆa

†

+ˆa)/

√

2

and

ˆp = i(ˆa

†

−ˆa)/

√

2

, where

[ˆa, 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

quadratures

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.

III. NOISE RADAR OPERATION

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

1−ξ

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.

C. Quantum advantage

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,

leading to an overestimation of the quantum radar advantage

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

. The quantum advan-

tage decays slowly with increasing

N

S

, keeping an advantage

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

an adaptive strategy where

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.