IEEE - Aerospace and Electronic Systems - April 2020 - 11

Shapiro
where the ket vectors fjniKm : n ¼ 0; 1; 2; . . . ; g represent
states containing exactly n photons. Ideal photon counting
on the mth signal and idler modes measures the photonnumber operators N^Sm  a^ySm a^Sm and N^Im  a^yIm a^Im ,
respectively. It then follows that ideal photon counting on
the coherent state given in (B4), available online, yields a
Poisson-distributed output with mean NS , as expected
from the coherent state's photodetection statistics being
obtainable from semiclassical (shot-noise) theory [15].
The signal and idler's mth temporal modes have operator-valued quadrature components
Reð^
aKm Þ ¼

Figure 3.
Error-probability bounds for Tan et al.'s CS and QI radars [19]:
PrðeÞUB
CS
PrðeÞUB
QI
PrðeÞLB
CS

is the quantum Chernoff bound for the CS radar;
is the quantum Chernoff bound for the QI radar; and
is the Bhattacharyya lower bound for the CS radar.

lower bound on PrðeÞCS from [19]. It is important to
remember that quantum Chernoff bounds are known to be
exponentially tight upper bounds, while it is known that
the Bhattacharyya lower bound is always loose. Furthermore, Tan et al. showed that the CS radar affords the lowest error probability of any classical-state radar of the
same average transmitted energy. Thus, because Figure 3
LB
shows PrðeÞUB
QI becoming lower than PrðeÞCS for sufficiently high M values, it provides definitive proof that a
QI radar can, in principle, outperform all classical radars
of the same transmitted energy for the target-detection
scenario addressed by Tan et al. We will say more later
about why we refer to Figure 3's QI advantage over CS
operation as being "in principle." For now, it suffices to
point out that when [19] appeared there was no known
receiver that provided any QI performance advantage over
CS operation. Why that was so and how it was overcome
require some understanding of quantum photodetection
theory and the quantized electromagnetic field's Gaussian
states. So those topics are next on our agenda.
The mth temporal modes of Tan et al.'s QI signal and
idler from (B1) and (B2), available online, have associated
photon annihilation operators a^Sm and a^Im whose adjoints
are the photon creation operators a^ySm and a^yIm . These
names originate from the operators' actions on their mode's number states, which, for K ¼ S; I, obey
a^Km jniKm ¼

 pﬃﬃﬃ
n jn À 1iKm ;
0;

for n ¼ 1; 2; . . .
for n ¼ 0

(8)

and
a^yKm jniKm

APRIL 2020

a^Km þ a^yKm
2

and

Imð^
aKm Þ ¼

a^Km À a^yKm
2j
(10)

for K ¼ S; I, and ideal (quantum-limited) optical homodyne detection with the appropriate local oscillator fields
measures these operators [15]. The positive-operatorvalued measurements (POVMs) associated with a^Sm and
a^Im [21] can be realized by ideal (quantum-limited) optical heterodyne detection [15]. Although heterodyne detection provides information about both quadratures, the
Heisenberg uncertainty principle forces this measurement
to incur extra noise on each quadrature that is not present
in homodyne measurement of a single quadrature [15].
Gaussian states of the mth signal and idler modes a^Sm
and a^Im are the quantum analogs of classical, complex-valued Gaussian random variables aSm and aIm . Thus, Gaussian states of these two modes are completely characterized
by knowledge of their first and second moments [21] h^
aKm i,
hD^
ayKm D^
aJm i, and hD^
aKm D^
aJm i, where K ¼ S; I, J ¼ S;
I, hÁi denotes ensemble average, and D^
aKm  a^Km À
h^
aKm i. The coherent state from (B4), available online, is a
pﬃﬃﬃﬃﬃﬃﬃ
Gaussian state with h^
aSm i ¼ NS , and hD^
a Sm i ¼
aySm D^
2
hD^
aSm i ¼ 0. The entangled signal-idler state from (B3),
available online, is also Gaussian. It is a two-mode
squeezed vacuum (TMSV) state [21], whose measurement
statistics are completely characterized by h^
aSm i ¼ h^
aIm i ¼
h^
aySm a^Sm i ¼ h^
ayIm a^Im i ¼ NS , h^
a2Sm i ¼ h^
a2Im i ¼ 0,
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
y
h^
aSm a^Im i ¼ 0, and h^
aSm a^Im i ¼ NS ðNS þ 1Þ.
The preceding brief introduction to Gaussian states
provides enough information to understand the origin of
Tan et al.'s QI advantage and why conventional optical
receivers-direct detection, homodyne detection, and heterodyne detection-do not realize any of that advantage.
The cross correlations of all zero-mean classical signalidler states must obey [22]

0,

jh^
aySm a^Im ij

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
h^
aySm a^Sm ih^
ayIm a^Im i

(11)

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
h^
aySm a^Sm ih^
ayIm a^Im i:

(12)

and
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
¼ n þ 1 jn þ 1iKm ; for n ¼ 0; 1; 2; . . .

(9)

jh^
aSm a^Im ij

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