IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 82

Detecting a Target With Quantum Entanglement
interest with a vacuum mode on a 50:50 beam splitter,
and then measuring via homodyne detections
the quadratures q and p of the two the output
modes.11 Accordingly, heterodyne allows us to
access both field quadratures at the same time. However,
we cannot escape the fact that ^p and ^q do not
commute, and therefore, by measuring both quadratures
at the same time, we incur in additional noise
as imposed by the uncertainty relation [23].
3) Photon counting consists in measuring the number
of photons in a certain mode, which after averaging
over several measurements, gives h^ni¼h^ay^ai.
For a coherent state transmitter, we can show that
homodyne detection allows us to saturate the Chernoff
bound (64) [12]. In fact, if we homodyne the return mode
in the CI protocol described in the section " Gaussian QI, "
the probability distributions pqk
ðQkjH0=1Þ of the measured
quadrature qk is a Gaussian with variance 2NB þ 1 and
meanqk ¼ 0 andqk ¼ 2
p
ffiffiffiffiffiffiffiffiffi
kNs
under hypotheses H0 and
H1, respectively. Here, k ¼ 1; .. . ;M labels the different
measurements on M copies of the return mode. These are
now classical probability distributions, and we can apply
classical decision theory do discriminate between hypotheses
H0 and H1. In particular, the Bayesian decision
rule (36) corresponds to evaluate the mean Qk of each
of the M homodyne measurements, to calculate q ¼
q1 þ þ qM, and decide in favor of H0 whenever Q<
ðM
p
ffiffiffiffiffiffiffiffiffi
kNs
Þ, and in favor of H1 otherwise [29]. Using the
abovementioned definitions for pqðQjH0=1Þ, and the fact
that Q is also a Gaussian random variable we can rewrite
the first line in (38) as a Gaussian integral and obtain [12]
PðhomÞ
e
¼ erfc
1
2
r
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kNsM
4NB þ 2
2
eMhom
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pMhom
(67)
where erfc is the complementary error function,hom ¼
kNs=ð4NB þ 2Þ, and the approximation holds for
kNsM 4NB þ 2. In particular, for NB 1, we have
hom kNs=4NB ¼C. Accordingly, homodyne detection
saturates the Chernoff bound for a coherent state
transmitter.
In QI, the information on the presence/absence of the
target is encoded in the phase-sensitive cross-correlations
[see the covariance matrices (63)] that depends on both
quadratures and, therefore, cannot be resolved by homodyne
or heterodyne detection without incurring in additional
noise. Photon-coincidence counting, which is
routinely used to detect the correlations between signal
and idler photons as generated in the SPDC process, is
also of no use in QI because of the strong thermal
11This procedure, known as dual homodyne is far more common at
optical frequencies and, because of the equivalence discussed earlier,
it is often referred to as heterodyne detection in the quantum
optics literature.
82
Figure 7.
Schematic representation of the OPA receiver. The mode returning
from the target region and the retained idler are fed to an
OPA, with a small gain G ¼ 1 þ2, and the number of photons
N in the amplified idler mode is counted. A decision in favor of
H0ð1Þ is made ifN is smaller (larger) than a threshold value Nth.
background. Finally, interferometrically combining the
return and the idler modes only allows us to access phaseinsensitive
correlations. As a consequence, it is not possible
to exploit the quantum advantage offered by QI by
using standard quantum optics measurements.
To overcome this problem, Guha and Erkmen proposed
a receiver that maps phase-sensitive cross-correlation
into photon counts [12].
This receiver uses an OPA12 (see Figure 7) that performs
the following transformation:
c^ðkÞ ¼
d^ðkÞ ¼
p
ffiffiffiffi
G
p
a^ðkÞ
ffiffiffiffi
G
I þ
a^ðkÞ
R þ
p
p
ffiffiffiffiffiffiffiffiffiffiffiffi
G 1
aR^yðkÞ
ffiffiffiffiffiffiffiffiffiffiffiffi
G 1
aI^yðkÞ
(68a)
(68b)
where G ¼ 1 þ2 is the gain of the OPA, and then counts
photons in the amplified idler mode Nk ¼h^cy
both hypotheses, the mode ^cðkÞ is in a thermal state rc
given by (30) with mean photon number [12]
N0 ¼ GNs þðG 1Þð1 þNBÞ
N1 ¼ GNs þðG 1Þð1 þNB þ kNsÞ
þ 2
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
GðG 1Þ
p
kNsðNs þ 1Þ
(69a)
(69b)
under hypotheses H0 and H1, respectively.
Given M copies of the amplified idler state, the optimal
strategy to distinguish between hypotheses H0 and
H1 consists in counting the total number of photons n and
comparing it with a threshold [12]. The probability distribution
of n under the two hypotheses can be calculated
from the quantum state rM
c
P0=1ðnÞ¼
, and yields
n þM 1
n
ðN0=1Þn
ð1 þN0=1ÞnþM :
(70)
In (70), the binomial coefficient takes into account the different
ways of distributing n photon counts within M
12We will talk here about OPAs, since this receiver is commonly
referred to as an OPA receiver. However, the mathematical
description presented here also apply to the microwave case where
JPAs are used instead. Note that these are the same kind of nonlinear
elements used to produced entangled photons.
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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