IEEE Aerospace and Electronic Systems Magazine - November 2020 - 26

Noisy Receivers for Quantum Illumination
is equivalent to a heterodyne detection, the performance of
coherent state transmitters cannot be beaten. There is
indeed another argument to understand why this is the
case. Performing a Gaussian measurement on the idler
mode of a two-mode Gaussian state remotely prepares an
ensemble of Gaussian states on the signal mode [31]. In
particular, if the Gaussian state is a TMSV state and the
idler mode is heterodyned, then the signal mode is projected onto an ensemble of coherent states, whose average
state is thermal with mean number of photons equal to the
signal energy of the TMSV state.

CONCLUSION


V ¼ ÈN
k¼1


0 1
:
À1 0

(A1)

We can write the s-overlap as [27]
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
det Ps
Cs ¼ 2N
exp
det Ss

!
dT SÀ1
s d
À
2

(A2)

where d ¼ x0 À x1 . Here, Ps and Ss are defined as
È
Ps :¼ Gs ðVÈ
0 ÞG1Às ðV1 Þ

(A3)

Â À ÁÃ T
Â
À ÁÃ T
S0 þ S1 L1Às VÈ
S1
Ss :¼ S0 Ls VÈ
0
1

(A4)

introducing the two real functions

In this article, we have investigated the QI protocol for a
generically correlated Gaussian source, considering various receiver types. Keeping within the realms of Gaussian operations, we have paid attention to the PC receiver
and studied its performance in various cases of added
noise due to, for example, the action of a heterodyne
measurement on one or both of the modes. The potential
of performing a measurement on the idler, still retaining
a quantum advantage, would mitigate one of the major
problems associated with QI implementation: idler storage and later recombination with the returning signal.
This is of particular concern when the problem involves
target ranging, where alternative strategies should be
considered [32].
Under these considerations, we have modeled the PC
receiver for our generic source and have derived the
associated SNR in terms of the signal and idler energies,
and their cross correlations. Our SNR may be readily
adapted to include additional noise associated with
Gaussian measurements. Our results confirm that a heterodyne measurement performed on the idler mode leads
to a performance, which asymptotically approaches that
of a coherent state with homodyne detection, not surpassing it. Interestingly, if the signal mode is affected
by heterodyne but the idler mode is maintained clean,
the performance asymptotically approaches that of the
PC receiver without any added noise. Finally, let us
mention that it would be interesting to investigate these
aspects within the setting of unambiguous quantum
discrimination [33].

1
ðx þ 1=2Þs À ðx À 1=2Þs
ðx þ 1=2Þs þ ðx À 1=2Þs
Ls ðxÞ ¼
ðx þ 1=2Þs À ðx À 1=2Þs

Gs ðxÞ ¼

(A5)

N
k
calculated over the Williamson forms VÈ
i :¼ Èk¼1 ni 12 ,
È T
k
where VÈ
¼
S
V
S
for
symplectic
S
and
n
!
1=2
are
i i i
i
i
i
the symplectic spectra [28],[29].

APPENDIX B
SNR FOR THE PC RECEIVER
The PC receiver [8], see Figure 1 in the main text, phaseðkÞ
conjugates all M returning modes a^R;i , where 1 k M
and i ¼ 0; 1 (corresponding to the two hypotheses H0 and
H1 ), according to
a^PC;i ¼

pﬃﬃﬃ
2a^v þ a^{R;i

(B1)

where a^v is the vacuum operator. Since the creation and
annihilation operators are defined
pﬃﬃﬃ in terms of quadratures,
pﬃﬃﬃ
q^ and p^, via a^ ¼ ð^
q þ i^
pÞ= 2 and a^{ ¼ ð^
q À i^
pÞ= 2,
^ ¼ ð^
respectively, we may recast (B1) for X
q; p^ÞT . Thus,
the PC receiver transforms quadratures as
^ PC;i ¼
X

pﬃﬃﬃ
^ R;i
^ v þ ZX
2X

(B2)

and the corresponding conditional covariance matrices of
the return-idler states are given by


ðv þ 1Þ1 0
0
m1



V0PC;I ¼

1
2

QCB FOR MULTIMODE GAUSSIAN STATES

V1PC;I ¼

1
2

Consider two arbitrary N-mode Gaussian states,
^0 ðx0 ; V0 Þ and r
^1 ðx1 ; V1 Þ, with mean xi and CM Vi with
r
quadratures ^
x ¼ ðq^1 ; p^1 ; . . . ; q^N ; p^N ÞT and associated symplectic form

The individual phase-conjugated signal modes are then
mixed with the corresponding retained idler modes on a
50-50 beamsplitter whose outputs are given by

APPENDIX A

26

IEEE A&E SYSTEMS MAGAZINE



ðgpþ
ﬃﬃﬃ 1Þ1
kcZ

pﬃﬃﬃ 
kcZ
:
m1

(B3)

(B4)

NOVEMBER 2020
IEEE Aerospace and Electronic Systems Magazine - November 2020

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