IEEE Aerospace and Electronic Systems Magazine - November 2020 - 65

Bourassa and Wilson
states with an average photon number NS ¼ ðcoshð2rÞ À
1Þ=2. However, their covariance CQ ¼ sinhð2rÞ=2 can be
greater than what is possible classically.
The quantum effect of the enhanced covariance is to
^
reduce the
pﬃﬃﬃ noise in the combined-quadraturepﬃﬃIﬃÀ ¼
^ þ ¼ ðQ
^a þ Q
^b Þ= 2). If
ðI^a À I^b Þ= 2 (and similarly in Q
the measurement noise is subtracted, we find the variance
of the pure quantum state is
Var½I^À ¼

eÀ2r
2

(23)

that is, a factor eÀ2r below vacuum. As can be seen in
Figure 4(c), the JPA squeezes the noise along the
combined-quadrature I^À , with the adverse effect of
^À to
amplifying it in the conjugate quadrature Q
2r
^
^
Var½QÀ ¼ e =2 (as well as Iþ ) in order to maintain the
Heisenberg inequality (11). Experimentally, the squeezing of the signal-idler correlated noise power of the output of the JPA [13], has been demonstrated to be as high
as À12 dB below vacuum noise [12], which is one of the
largest amounts of squeezing ever observed across the
whole electromagnetic spectrum.

QUANTUM ADVANTAGE OF QENR
As we have mentioned earlier, improving the signal-idler
correlations leads to better target detection probabilities in
noise radar systems. As such, quantum sources of TMSS
could provide a quantum advantage in target detection
efficiency as they can provide larger correlations than
classically allowed. Indeed, looking back at (22) and using
a bit of hyperbolic trigonometry, the quantum covariance
can be written as
CQ ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
NS ðNS þ 1Þ:

(24)

Importantly, classical and quantum signals are affected by
noise and loss exactly the same way. As such, the target
reflectivity and atmospheric noise and loss will affect the
TMSS covariance matrix exactly as it did in the classical
case. In the absence of a target, the covariance matrix
Rabsent
remains the same as before in (19). If the target
Q
is present, the only difference is that the
measured signal-
pﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
idler covariance now reads CQ ¼ h NS ðNS þ 1Þ.
The quantum enhancement QE is obtained by comparing the quantum covariance CQ relative to the classical
value CC in (18) for the same output power NS . For ideal
classical and quantum sources of correlated noise signals
the quantum enhancement is found to be
rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1
QE ¼
¼ 1þ
NS
CC
CQ

(25)

and was verified experimentally in superconducting circuits in [8]. Most notably, it is independent of the signal
NOVEMBER 2020

Figure 6.
Quantum enhancement QE and detection time gain of QENR over
classical noise radar as a function of the source brightness NS .
The quantum advantage grows as the source brightness is reduced
due to the benefits of the extra correlations introduced by quantum
entanglement. In turn, the extra quantum correlations are always
beneficial for all source brightness, leading to large gains in detection time.

losses and depends only on the strength of the correlations
at the source.
As we can see in Figure 6, in the ideal case the
enhancement is always present as QE ! 1 for all values of source brightness. Meaning that the ideal quantum source will always provide more correlations
than the ideal classical source under the same noise
and loss conditions. The quantum enhancement
diverges as the source brightness diminishes NS ( 1,
highlighting the unique contributions of the quantum
correlations when the source power lies below the vacuum noise level.
As was stated earlier in the section, " Target Detection
With Classical Noise Radar, " increasing the correlations
at the source reduces the number of required measurements for a successful detection. Using a quantum source
of entangled microwaves providing an enhancement of
the correlations by a factor QE would then directly translates into a reduction in the number of measurements by a
factor 1=Q2E . In Figure 6, we have also plotted the gain in
measurement time ðQ2E À 1Þ Â 100% with respect to a
comparable classical source and it clearly shows that very
large gains can be obtained in the very low signal regime.
This behavior of the quantum enhancement is exactly why
QI protocols performs best at very low transmitting
powers, making it an attractive technology for low-signal,
high-noise situations or when the observer does not want
to be seen.
In contrast to joint-measurement protocols, quantumenhanced noise radar has the advantage of being more
practical as heterodyne measurements do not require lossless and adjustable delay lines for detection and ranging.
Postprocessing of the measurement records allows for
automated adjustments of the relative signal-idler phase,
whereas the target range is recovered from time-binning
of the covariance signal. Given the measurement back-

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IEEE Aerospace and Electronic Systems Magazine - November 2020

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