IEEE Aerospace and Electronic Systems Magazine - November 2020 - 28

Noisy Receivers for Quantum Illumination
where we have used the following identity for higher order
Gaussian moments

REFERENCES
[1] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook,

^n i ¼ s n ðn À 1Þ!!
hO

(B20)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
^2 i is the standard deviation and !! denotes
where s ¼ hO
the double factorial.
Rewriting the final term ðÃÞ in terms of quadrature
components, (B16), we find that

and S. Lloyd, " Advances in photonic quantum sensing, "
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1 2 2
2 2
2
2 2
ðÃÞ ¼ ðh^
q^À i þ h^
p^À i
q i À h^
qþ
qþ
ih^
p2À i À h^
qþ
q ih^
2 þ À
2
2 2
þ h^
p2þ ih^
p^þ iÞ
p2À i À h^
p2þ p^2À i þ h^
qÀ
ih^
p2þ i À h^
qÀ

[4] Z. Zhang, S. Mouradian, F. N. C. Wong, and J. H. Shapiro,

¼ Àðh^
qþ q^À i2 þ h^
qþ p^À i2 þ h^
pþ p^À i2 þ h^
qþ p^þ i2 Þ

[5] E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni,
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(B21)

" Entanglement-enhanced sensing in a lossy and noisy environment, " Phys. Rev. Lett., vol. 114, 2015, Art. no. 110506.

realization of quantum illumination, " Phys. Rev. Lett.,

where we have used the following identity for multivariate
higher order Gaussian moments:
^2 O
^2 ^2
^ ^ 2
^2
hO
i j i ¼ hOii ihOjj i þ 2hOi Oj i

vol. 110, 2013, Art. no. 153603.
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(B22)

2013, Art. no. 010501.

^i O
^j i denotes the covariance of Gaussian variawhere hO
^j .
^
bles Oi and O
Computing the required variances and covariances, we
find

[7] S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali, J. H. Shapiro,
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2
2
h^
qþ;0
i2 ¼ h^
qÀ;0
i2 ¼ h^
p2þ;0 i2 ¼ h^
p2À;0 i2 ¼ a2þ

[9] J. Calsamiglia, J. I. de Vicente, R. Mu~noz-Tapia, and

h^
qþ;0 q^À;0 i2 ¼ h^
pþ;0 p^À;0 i2 ¼ a2À
2
h^
qþ;1
i2
2
h^
qÀ;1
i2

¼ h^
p2þ;1 i2 ¼ b2þ
¼ h^
p2À;1 i2 ¼ b2À
h^
qþ;1 q^À;1 i2 ¼ h^
pþ;1 p^À;1 i2

E. Bagan, " Local discrimination of mixed states, " Phys.

(B23)

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¼ g Ã2 :

[11] Q. Zhuang, Z. Zhang, and J. H. Shapiro, " Optimum
mixed-state

Inserting these into (B18), (B19), and (B21), we obtain the
photon number variances

for

noisy

entanglement-

Art. no. 040801.
[12] Q.

1
hDN^02 i ¼ ðmð1 þ vÞÞ
2

discrimination

enhanced sensing, " Phys. Rev. Lett., vol. 118, 2017,
Zhuang,

Z.

Zheshen,

and

J.

H.

Shapiro,

" Entanglement-enhanced Neyman-Pearson target detection using quantum illumination, " J. Opt. Soc. Amer. B,

(B24)

vol. 34, pp. 1567-1572, 2017.

hDN^12 i ¼

Á
1À 2
kc þ mð1 þ gÞ :
2

[13] Q. Zhuang, Z. Zheshen, and J. H. Shapiro, " Quantum illutargets, " Phys. Rev. A, vol. 96, 2017, Art. no. 020302.

Finally, we find that the single-mode SNR, (B15) of the
PC receiver is given by
kc2
SNRPC ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2
kc2 þ mð1 þ gÞ þ mð1 þ vÞ

28

[14] S. Barzanjeh, S. Pirandola, D. Vitali, and J. M. Fink,
" Microwave

quantum

illumination

using

a

digital

receiver, " Sci. Adv., vol. 6, 2020, Art. no. eabb0451.
[15] D. Luong, C. W. Chang, A. M. Vadiraj, A. Damini,

(B26)

C. M. Wilson, and B. Balaji, " Receiver operating characteristics for a prototype quantum two-mode squeezing

which directly relates to its error probability after M uses,
for equally likely hypotheses, satisfying[8]
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
ðMÞ
PPC ¼ erfc
MSNRPC :
2

mination for enhanced detection of Rayleigh-fading

(B25)

radar, " IEEE Trans. Aerosp. Electron. Syst., vol. 56, no. 3,
pp. 2041-2060, Jun. 2020.
[16] C. Eichler, D. Bozyigit, and A. Wallraff, " Characterizing
quantum microwave radiation and its entanglement with

(B27)

IEEE A&E SYSTEMS MAGAZINE

superconducting qubits using linear detectors, " Phys. Rev.
A,, vol. 86, 2012, Art. no. 032106.

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IEEE Aerospace and Electronic Systems Magazine - November 2020

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