IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 85

Sorelli et al.
characteristic approximation [35]. The results of this
approximation give the green ROC curve in Figure 10.
Finally, we consider QI with its ultimate receiver: the
FF-SFG receiver that we discussed in the section
" Ultimate Receiver: The Feed-Forward Sum-Frequency
Generation (FF-SFG) receiver. " An approximation of the
performance of this receiver is given by the ROC for discriminating
the coherent state j
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NskM=NB
i from the
vacuum. Given that both a coherent state and the vacuum
are pure states, the latter is given by (61) with h ¼
1 expðNskM=NBÞ (red curve in Figure 10). This
approximation allows us to obtain simple analytical results
that describe the performance of what is in fact a complicated,
structured receiver. The price that we have to pay
for this simplification is in the accuracy ofsuch an approximation.
In particular, as the false alarm probability PF !
0, the QI detection probability PD should also vanish,
while the coherent state approximation gives PD ! h,
which can be significantly different from zero for modest
SNRs SNR ¼ NskM=NB. A comparison between the
approximated ROC for discriminating a coherent state
from the vacuum and the one of a numerically simulated
FF-SFG is presented in [39].
Figure 10 shows the performances of QI, as quantified
by the ROC, in the Ns 1 regime, where an entanglement-based
transmitter provides a significant advantage
over a coherent state transmitter. However, in this regime,
since each state contains a very low average photon number
per pulse, in order to obtain a detection probability
PD, which is close to one, we need millions of copies of
the entangled states we use to interrogate the target region
(see M in Figure 10).
To better draw our conclusions, let us now consider a
different way to visualize the performance of a targetdetection
protocol, namely, we plot the probability of
detection PD as a function of the SNR for different fixed
values of the probability of false alarm PF.13 In the present
context, we can define the SNR as the number of
received photons divided by the number of thermal photons:
SNR ¼ MkNs=NB. It is interesting to notice that
the optimal CI performances, as described by (75), depend
on the system parameters only through d, which for NB
1 can be approximated as d
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SNR=2
. Analogously, as
discussed in the case of the ROC, the performances of the
FF-SFG receiver are well approximated by considering
the problem of discriminating the coherent state j
p
ffiffiffiffiffiffiffiffiffiffi
SNR
i
and the vacuum. The performances of these optimal classical
and QI schemes, according to the new metric introduced
earlier, are plotted in Figure 11.
Figure 11 shows an interesting feature of QI in the
Neyman-Pearson scenario, namely the fact that the quantum
advantage in this case is not limited to 6 dB. We can
13We thank Roch Settinieri for suggesting us this way to visualize QI
performances.
MAY 2022
Figure 11.
Detection probability PD as a function of the SNR: SNR ¼
MkNs=NB, for different values of the false alarm probability
PF ¼ 102 (solid line), 104 (dashed line) and 106 (dotted line).
Blue lines represent the performances of the optimal CI protocols:
coherent state transmitter and homodyne detection. Red lines correspond
to the QI with the optimal FF-SFG receiver.
see this by noting that the red curves (for QI) get further
apart from the blue ones (for CI) while decreasing the
false alarm probability PF. Quantitatively, the SNR necessary
to achieve an 80% detection probability with false
alarm probabilities PF ¼ 102, 104 and 106 with QI
(red curves) is smaller than the one in the optimal classical
case (blue curves) by, respectively, 6, 8.2, and 8.9 dB14.
In fact, by using an entropic approach [40],[41], Wilde
et al. [36] proved that in the Neyman-Pearson setting the
ratio between the classical and the quantum detection
probabilities, i.e., the quantum advantage, can be made
arbitrary large by making Ns arbitrary small. However,
for Ns ! 0, both the abovementioned probabilities go to
zero and one needs to consider infinitely many copies of
the transmitted state (M!1) in order to obtain a finite
detection probability. As we will see in the following, this
represents an important drawback of QI.
CRITICALITIES AND LIMITATIONS
In this section, we report the main criticalities and limitations
of the QI protocol as they are known in the quantum
optics literature [4], [5]. Additionally, we quantitatively
characterise the quantum advantage regime in terms of
pulse power and duration, in order to give an answer to
the question: Is QIpractically usefulfor target detection?
14For some applications an 80% detection probability may not be
enough. If we require a 99% detection probability, we obtain the
more modest quantum advantages of 3.23, 4.42, and 4.59 dB, at
PF ¼ 102; 104,and 106, respectively
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