IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 84

Detecting a Target With Quantum Entanglement
threshold for the Neyman-Person test (53)
PðCIÞ
F
PðCIÞ
D
with d ¼ 2
p
¼ erfc p
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffi
MkNs
¼ erfc p
p
1
2
=
ffiffiffi
2
ffiffiffi
2
log
d
log
d
d
2
d
2
1
þ
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2NB þ 1
(75a)
(75b)
. From (75), we can plot
the ROC for the coherent state transmitter with homodyne
detection (blue curve in Figure 10).
Let us now move to QI starting from the OPA receiver
Figure 9.
Schematic representation of the FF-SFG receiver. The upper part
of the figure shows two successive cycles of the FF circuit. The
lower part shows the structure of the kth cycle where a part of
return mode is combined with the idler on the SFG device and
measurements on the sum-frequency mode and the transformed
signal mode are performed. The yellow blocks represent additional
two-mode nonlinear operations with tunable parameters,
which are conditioned on the results of measurements on the outputs
of the previous cycle. (Figure from [13].)
unrealistic) assumption that SFG has unit efficiency at the
single pair level, this receiver has the remarkable property
of being able to saturate the Chernoff bound for QI [see
(66)] and, therefore, to achieve the full 6 dB quantum
advantage enabled by quantum entanglement.
A detailed description of how this detector works is
quite involved and goes beyond the purpose ofthis review.
A qualitative description is presented in the caption of
Figure 9, and for further details, we refer the interested
readers to [13] and [39]. However, a rough approximation
of the performances of this device can be obtained in the
Ns 1 limit, where the output mode of the FF-SFG
receiver is found either in the vacuum j0i (H0 is true), or
in the coherent state j
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NskM=NB
PERFORMANCES: ROC FOR QI
Having introduced the OPA receiver, which is suboptimal,
but implementable with current technology, and the FF-
SFG receiver, which gives the ultimate performance for
QI, we are ready to compare the ROC for quantum and
CI [39].
Let us start by considering the coherent state transmitter
with homodyne detection. In the section " Practical
Receivers for QI, " we have seen that for NB 1 this is
the optimal CI setting, and that it maps the target detection
problem to the discriminations of two Gaussian functions
with identical variances, but different means. In this case,
we can derive the probability density for the likelihood
ratio L[29], and calculate the false alarm and detection
probabilities [see (54) and (55)] as functions of the
84
i (H1 is true) [39].
(see the section " Practical Receivers for QI " ). To calculate
the ROC for this receiver, we assume M 1, and we use
the central limit theorem in order to approximate the photon-counting
probability density (70) with a Gaussian
function [12]
PNjH0=1 ðnjH0=1Þ¼
e
ðnMN0=1Þ2=ð2Ms2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pMs2
0=1
with N0=1 defined in (69) and s2
0=1 ¼ N0=1ðN0=1 þ 1Þ as
discussed in the section " Practical Receivers for QI. " We,
therefore, have to calculate the false alarm and detection
probability for the discrimination between two Gaussian
functions. In this case, an analytical solution can be
obtained using the extended Van Trees receiving
0=1Þ
(76)
Figure 10.
ROC: the detection probability PD as a function of the false alarm
probability PF for different target detection schemes. In blue, the
optimal CI protocol: a coherent-state transmitter with homodyne
detection. In green, an entanglement-based scheme realizable
with current technology: QI with the OPA-receiver. In red, the
ultimate limit for QI, achieved with the FF-SFG receiver. The
number of signal photons Ns is chosen in the range where entanglement
provides a quantum advantage, while the value of NB
(number of photons in the thermal background) is typical for
microwave frequencies.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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