IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 83

Sorelli et al.
measurements, while the fraction gives the probability for
one of this way to occur. This probability distribution has
a mean value equal to MN0=1 and a variance Ms2
0=1 ¼ N0=1ðN0=1 þ 1Þ. Using the central limit theorem,
0=1, with
s2
for M 1, we can approximate (70) with a Gaussian
function, and then use the Bayesian decision rule (36) to
derive the threshold Nth ¼ Mðs1N0 þ s0N1Þ=ðs0 þ s1Þ,
such that we decide that H0 is true ifN< Nth and viceversa
otherwise. We will come back to this Gaussian
approximation later when we will calculate the ROC for
the OPA receiver.
We can now evaluate the mean error probability associated
with this strategy by calculating the classical Bhattacharyya
bound. Since n is a discrete random variable,
we do this by replacing the integral in (42) with a sum,
and we obtain [12]
Pe;OPA ðQBÞM
1
2
with
QB ¼ p
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 þN1Þð1 þN0Þ
2
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kNsðNs þ 1Þ
p
ffiffiffiffiffiffiffiffiffiffiffiffi
N0N1
(72)
For a low gain OPA (2 1), we have N1N0 ¼
1. We can, therefore, Taylor-expand
QB in powers of N1N0, taking into account that we
also have N1N0N0, and obtain
QB 1
ðN1N0Þ2
8N0ðN0 þ 1Þ
¼ 1B eB
Using the expressions for N0 and N1, we have
B ¼
2kNsðNs þ 1Þ
2NsðNs þ 1Þþ 22ð2Ns þ 1ÞðNB þNs þ 1Þ
(74)
which for Ns 1, k 1, and NB 1 is upper bounded
byB kNs=ð2NBÞ. This upper bound is reached when
2NBNs, e.g., when2 ¼ Ns=
p
ffiffiffiffiffiffiffi
NB
. The performances
ofQI with the OPA receiver are illustrated in Figure 8.
The OPA-receiver allows for an error exponentB,
which is twice the classical oneC, corresponding to 3 dB
quantum advantage, using only off-the-shelf components.
However, this is 3 dB less than the ultimate quantum
advantage attainable with QI. In the following section, we
will discuss how to fill this gap.
ULTIMATE RECEIVER: THE FEED-FORWARD SUMFREQUENCY-GENERATION
(FF-SFG) RECEIVER
The suboptimality of the OPA receiver discussed earlier
stems from the fact that it analyses the M return-idler
MAY 2022
:
(73)
mode pairs one by one. This strategy is known to be suboptimal
for the discrimination of two mixed quantum
states like those described by the covariance matrices
(63) [38]. In order to overcome this limit, Zhuang et al.
[13] proposed to use SPDC's inverse process: sum frequency
generation (SFG). SFG happens when a pair of
photons with frequencies vS and vI and wave vectors kS
and kI meet on a nonlinear device identical to the one
used for SPDC, and a photon in the pump mode, with frequency
vP ¼ vS þ vI and wave vector kP ¼ kS þ kI is
generated. Therefore, the receiver Zhuang et al. proposed
in [13] consists in going beyond the pair-by-pair analysis
of the return and idler modes by combining all M received
states that interrogated the target region via SFG such that
the information on the presence (absence) of the target
gets mapped in the presence (absence) of photons in the
pump mode.
Following this idea, Zhuang et al. proposed a configuration
that uses several cycles of SFG augmented by a
feed-forward (FF) circuit that implements additional nonlinear
operations conditioned on the results of measurements
on the output of the previous cycle (see Figure 9).
This combination of SFG with an FF circuit takes the
name of FF-SFG receiver. Under the (currently
IEEE A&E SYSTEMS MAGAZINE
83
Figure 8.
Upper bounds for the mean error probability Pe for (blue) CI
(Chernoff bound saturated by coherent state transmitter with
homodyne receiver), and a QI transmitter with (green) OPAreceiver
(classical Bhattacharyya bound) and (red) ideal receiver
(quantum Chernoff bound), as a function of the number of copies
M of the state used to interrogate the target region.
(71)
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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