IEEE Aerospace and Electronic Systems Magazine - November 2020 - 47

Lanzagorta and Uhlmann
Table 1.

Impact of Clouds on the Attenuation Coefﬁcient x (1/
km) for Different Radar Wavelengths (Assuming an
Ambient Temperature in the Vicinity of 0  C)
Visibility
(m)

 ¼1.25 cm

 ¼3.2 cm  ¼10 cm

30

0.2878

0.0460

0.0046

90

0.0576

0.0092

0.0009

300

0.0104

0.0016

0.0002

brightness quantum sensing in stealth applications. More
specifically, when there is a need for low-brightness target
illumination, the value of entanglement-derived information increases.
~ % 9:2668 obtained earlier to maximize
Using log 10 M
the quantum advantage for the signal-to-noise ratio per
photon snr ¼ 10À9 , we obtain a signal-to-noise ratio of
SNR % 2:7 dB. Then, the respective upper bound detection error probabilities for the entanglement-based quantum sensor; coherent sensing with photon-by-photon
control; and a classical radar obeying the Albersheim
equation are respectively
Pq % 0:08
Pl % 0:32
Pc % 0:43:

(31)

Notice that in this specific case, entanglement information
gives the quantum sensor a 4Â improvement over lidar
and 5.38Â improvement over classical. However, as we
discussed before, in general entanglement-based quantum
radar offers a 6 dB advantage in the error-probability
exponent over the optimal coherent receiver that does not

Figure 8.
Detection error probability bounds Pq (solid line) and Pl (dashed
line) as a function of range r (in km) to the target.

use entanglement. Therefore, the maximum advantage
offered by a quantum radar over any other sensor that
does not use entanglement is approximately 6 dB. Calculating the maximum travel distance that satisfies our
required average photon number of Ns % 10À4 , we find
that attenuation from a 50 km round trip reduces the number of photons received at the detector to 1:85 Â 104 .
Figure 8 shows the detection error probability bounds
~ photons under low visibility conditions.
Pq and Pl with M
The quantum advantage is clearly significant in a window
around the 25-km range-to-target distance but decreases
rapidly beyond 60 km and is negligible beyond a distance
of 80 km, at which both sensors become essentially uninformative. For a target at 100 km, only 0.01% of the transmitted photons return to the detector, and at 132 km only
1 photon is expected to return.
The plots of Figure 8 suggest that the behaviors of Pq
and Pl are nearly identical up to a horizontal shift, i.e., up
to a difference in range-to-target distance. The reason can
be seen by considering
eÀMNs kð2rÞ=Nb
2

eÀMNs kð2rÞ=4Nb
2
(32)
where kð2rÞ represents the total photon attenuation as a
function of the entire traversal distance of returned photon
states (2r). Thus
~

Pq ðrÞ ¼

~

Pl ðrÞ ¼

Pq ðrÞ ¼ Pl ðr À zÞ

(33)

which is equivalent to
eÀMNs kð2rÞ=Nb ¼ eÀMNs kð2rÀ2zÞ=4Nb
~

~

(34)

and consequently
4kð2rÞ ¼ kð2r À 2zÞ ¼) ex2z ¼ 4

Figure 7.
 ¼ 3:2 cm radar attenuation as a function of range (in km)
through clouds with visibility 30 m (solid line), 90 m (dashed
line), and 300 m (dotted line).

NOVEMBER 2020

¼) z ¼

IEEE A&E SYSTEMS MAGAZINE

ln 4
2x

(35)

(36)

47
IEEE Aerospace and Electronic Systems Magazine - November 2020

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