IEEE Aerospace and Electronic Systems Magazine - November 2020 - 43

Lanzagorta and Uhlmann
Hypothesis H0 : there is no target within range of the
detector. In this case, the detector will only measure
noise photon states. In this case, the Wigner distribution covariance matrix is given by
0

ð0Þ

Gri

B
1B
0
B
¼ @
4 0
0

0
B
0
0

0
0
S
0

1
0
0C
C
0A
S

(11)

where
B ¼ 2Nb þ 1

(12)

with Nb representing the average number of noise
photons. From this, the entanglement criterion
function is
ð0Þ

fri ¼ 16Ns Nb ðNs þ 1ÞðNb þ 1Þ ! 0

(13)

which means that the state is not entangled.
Hypothesis H1 : the target is within range but only a
portion of the photons k return to the detector. In
this case, the Wigner distribution covariance matrix
is given by
0

ð1Þ

Gri

A
1B
0
B
¼ @
4 Cr
0

0
A
0
ÀCr

Cr
0
S
0

1
0
ÀCr C
C
0 A
S

(14)

where
A ¼ 2ðkNs þ Nb Þ þ 1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
¼ 2 kNs ðNs þ 1Þ

Cr
(15)

and the separability criterion function is given by
ð1Þ

fri ¼ À16Ns ðNs þ 1Þðk À Nb ÞðNb þ 1Þ

(16)

which means that the state is not entangled if
k Nb .
In the case of microwaves, the number of noise photons correspond to Nb % 104 . Indeed, solar radiation has a
strong contribution to noise in the microwave regime. In
ð1Þ
such a case, fri is always positive, which means that no
entanglement survives the noisy microwave environment
produced by solar radiation. Therefore, the states at the
detector may not be entangled under both hypotheses.
Indeed, the environment obliterates any degree of entanglement by the time the signal photon state returns to the
detector. Nonetheless, some of the initial quantum correlations expressed in the highly nonclassical state with
covariance matrix Gsi persist even after total annihilation
of the entanglement [34].
The next step is to discriminate between both hypotheses. In theory, this could be accomplished by measuring
^ given by
the operator A
NOVEMBER 2020

ð1Þ
ð0Þ
^¼r
^ri À r
^ri
A
ð1Þ

(17)
ð0Þ

^ri and r
^ri are the density matrices that correwhere r
spond to hypotheses 1 and 0, respectively [48]. These denð0Þ
sity matrices are related to the covariance matrices Gri
ð1Þ
and Gri . If the measurement yields a positive value then
the target is declared to be within range. On the other
hand, if the measurement yields a negative number, the
target is declared to be out of range. Needless to say, it is
^ As a consenot a simple task to find the eigenvalues of A.
quence, as will be shown in the following section, in the
most general case our theoretical analysis can only determine upper bounds on the detection error probability.

DETECTION ERROR PROBABILITY BOUNDS
The detection error probability represents the probability
that the sensor registers a false detection, i.e., registers a
detection when no target is present or fails to register a
detection when a target is present. In other words,
reflects the probability that the conclusion implied by a
particular sensor observation (detect or no-detect) is
wrong. Alternatively, ð1 À Þ represents the probability
that the conclusion implied by the observation is correct.
In the limit of a noninformative sensor that returns
essentially random results, the probability that the conclusion implied from a particular observation is incorrect is
equivalent to a coin toss, i.e., ¼ 0:5. This is a somewhat
counterintuitive measure in the sense that ¼ 0 and ¼ 1
represent limits that can only be achieved by a perfect sensor, albeit with the latter case representing a situation in
which the labels for detect and no-detect events have been
reversed. We choose this measure because it has mathematically convenient properties and is reasonable as long
as it is assumed that is less than or equal to 0.5, which
should be true for any realistic properly calibrated sensor.
As will be shown later, thermal background noise
tends to dominate over shot noise and dark counts in lowSNR scenarios for standoff detection, so to simplify our
analysis we will define a high-noise scenario as being in
the regime in which the joint impact on SNR of noise
sources other than background radiation is negligible.
In the case of Gaussian signal states described in the
previous section, the detection error probability can be
derived using conventional quantum optical models.
We will consider two cases, a " quantum " case in which
the signal and idler photons are entangled, and a
" coherent " case which does not use entangled photons. In
the second case, even though there is no entanglement, we
assume that we still have photon-by-photon control. As
such, this case can be understood as the best possible sensor that does not use entanglement. While such a coherent
sensor may be expensive and unnecessary in practice, it
offers a bound to better understand the performance of
other sensors that do not use entanglement.

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IEEE Aerospace and Electronic Systems Magazine - November 2020

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