IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 79

Sorelli et al.
Figure 5.
Pictorial representation ofa classical target detection scheme (on the left) and a quantum one based on entanglement (on the right).
certain conditions the output of the ultimate receiver for
QI can be approximated with a pure state, and therefore,
we will use (61) in the section " Performances:
ROC for Quantum Illumination " to roughly estimate
its ROC.
QI THEORY
This part represents the core of this review. Here, we
will use all the concepts introduced earlier to describe
in quite some details the theory of QI for target detection.
We will start by presenting the QI protocol based
on Gaussian states introduced by Tan et al. [11]. We
will then discuss different detection strategies that
allow us to exploit the quantum advantage enabled by
this protocol and we will quantify its performances.
Finally, we will comment on various critical points
and limitations of this protocol that severely limits its
practical usefulness.
GAUSSIAN QI
Let us now imagine that we transmit a signal to a
region that contains a strong thermal background and
may or may not contain a weakly reflecting target, and
by measuring the returned light we want to discriminate
between the hypothesis H0 (target absent) and the
hypothesis H1 (target present). When the hypothesis
H0 is true, the signal coming back from the target
region is defined by the annihilation operator ^aR ¼ ^aB,
where the mode
a^B is in a thermal state with average
photon number NB 1. We focus on the case of
strong thermal background (NB 1), since it is the
most relevant for microwave radar applications. Moreover,
it has been proved that quantum light gives no
benefit when the background is absent or weak [37].
When hypothesis H1 is true, the return-mode's annihilation
operator is given by
a^R ¼
p
MAY 2022
ffiffiffi
k
a^S þ
p
ffiffiffiffiffiffiffiffiffiffiffi
1 k
a^B
(62)
where k 1 is the radar-to-target round-trip transmissivity,
while the thermal state defined by ^aB now contains
NB=ð1 kÞ photons, such that the total number of
received background photons is the same under both
hypotheses.9 We will now determine how to discriminate
between these two hypotheses in the two scenarios illustrated
in Figure 5.
In the CI scenario, we send the coherent state j
p
ffiffiffiffiffiffi
Ns
i to
the target region and we measure the returned light, which
is either in a strong thermal state (H0 is true) or in a combination
ofa very weak coherent state and a much stronger
thermal state (H1 is true). Under both hypothesis, the
returned state is a Gaussian state of the form (23) with
mean vectors
x0 ¼ð0; 0Þ (under H0) and
ð2
p
ffiffiffiffiffiffiffiffiffi
kNs
x1 ¼
; 0Þ (under H1). The covariance matrix of the
return mode is the same under both hypothesis and is
given by V0 ¼ V1 ¼ B12, with B ¼ 2NB þ 1. In this scenario,
discriminating between the two hypotheses H0 and
H1 corresponds to distinguishing between a very broad
Gaussian distribution and an identical one which is
slightly shifted with respect to the first one.
In the QI scenario, we produce a two-mode squeezed
vacuum state, and we retain the idler at the sender, while
we transmit the signal to the target region. In order to
decide which one of the two hypotheses is true, we perform
this time a joint measurement on the return signal
and the retained idler. By joint measurement, we mean a
detection scheme that addresses the signal and the idler
together as a single quantum system and is, therefore, able
to witness their correlations. As we will discuss in the following,
measuring individually the signal and the
idler is not enough to reconstruct their correlations in
postprocessing.
Under both hypothesis, the quantum state of the return
and idler modes is Gaussian with mean vectorx0 ¼x1 ¼
ð0; 0; 0; 0Þ and covariance matrices
9It should be noted that for k 1 the difference between NB and
NB=ð1 kÞ is quite small. On the other hand, this assumption is
crucial to ensure that there is no passive signature of target presence
that could be sensed without transmission.
IEEE A&E SYSTEMS MAGAZINE
79

IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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