IEEE Aerospace and Electronic Systems Magazine - November 2020 - 49

Lanzagorta and Uhlmann

Figure 9.
Adversary sensor's interception error probability at a distance r
(in km) from a quantum radar (pq , solid line) and a lidar (pl ,
dashed line).

30 km of stealth sensing range, while at ranges less than
66 km both sensors satisfy the required level of stealth.
As should not be surprising from our previous examination of entangled-versus-non-entangled detection error
probabilities, the plots of pq and pl are the same up to a
simple shift
pq ðrÞ ¼ pl ðr þ zÞ

(39)

which is equivalent to
eÀMNs kðrÞ=4Nb ¼ eÀMNs kðrþzÞ=Nb
~

~

(40)

and thus
1
1
ln 4
kðrÞ ¼ kðr þ zÞ ¼) eÀxz ¼ ¼) z ¼
4
4
x

(41)

which in our case represents a shift of z % 30 km. The
conclusion to be drawn is that the analysis of an
adversary's interception detection probability exactly mirrors that of our previous analysis of sensor detection error
probability.

Jamming as an active detection countermeasure.

this article, we use this simple jamming model to compare
quantum and classical detection, but we acknowledge that
in reality the behavior of jammers is much more complex.
This equation relates a target that is characterized by its
assumed radar cross section s; electromagnetic jamming
waves with power Pj and gain Gj ; a transmitter that has
power Pt and gain Gt ; and a radar receiver with gain Gr
defined as a function of angular observation variables u
and f.
The signal power after being reflected by the target is
expressed by the following radar equation as
PrðtÞ ¼

Pt Gt Gr ðut ; ft Þ 2 s

(42)

ð4pÞ3 R4t

and a related equation can be derived that incorporates the
effect of a jamming signal of power PrðjÞ
PrðjÞ ¼

Pj Gj
4pR2j

!
Â

 2

 Gr ðuj ; fj Þ
:
4p

(43)

These two equations can then be used to define a signalto-jam ratio (SJR)

QUANTUM ROBUSTNESS TO JAMMING
Although slightly more complex to analyze, the quantum
advantage seen thus far in the cases of detection and interception error probabilities extends also to robustness
against active sensing countermeasures by an adversary,
i.e., active jamming of our sensor [32], [33]. This involves
adversarial transmission of noise photons toward our sensor in order to reduce the effective SNR available to our
system. Figure 10 shows a concrete example of an adversary aircraft transmitting jamming noise toward a sensor
to limit its ability to detect and/or track a target aircraft.
Jamming has been a major focus of classical radar
research for over 70 years and is commonly examined
using a simple mathematical expression referred to as the
jamming equation [51]. Following the general spirit of
NOVEMBER 2020

Figure 10.

SJR ¼

PrðtÞ

ðjÞ

Pr


Pt Gt
¼
Â
Pj Gj

R2j
R4t

!
Â

s
4p

Â

!
Gr ðut ; ft Þ
:
Gr ðuj ; fj Þ
(44)

The distance at which SJR% 1, i.e., when the jamming
system is effectively able to mask the target's return signal, is referred to as the burnthrough range [51]. In other
words, within the context of this simple model, jamming
by an adversary will be effective when SJR < 1 as the
radar will not be able to separate the return signal of the
target from the total noise entering the system. More

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