IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 86

Detecting a Target With Quantum Entanglement
As discussed in the section " Phase Space Distribution
and Gaussian States, " the entangled signal-idler pulses
needed for QI are produced in nonlinear devices through
the process of SPDC, which can happen only for a finite
range of frequencies known as the phase matching bandwidth
W. At optical frequencies (vs 100 THz), a typical
value of the phase matching bandwidth is W 1 THz,
while in the microwave regime (vs 10 GHz), one usually
has W 100 MHz. Accordingly, a pulse of duration
T will contain M ¼ WT independent signal-idler modes.
If we now consider that to have a quantum advantage
the number of photons NS (i.e., the energyhvNS) per
mode must be very low, in order to reach a significant
SNR ¼ MkNS=NB, the required time-bandwidth product
M ¼ TW must be very large. In particular, given the
bandwidths discussed earlier, to achieve a time-bandwidth
product M ¼ 106
one would require a pulse duration
T 1ms at optical wavelengths, and T 10 ms in the
microwave range. Such a long microwave pulse imply
that a target will move during the interrogation time
inducing a variable time-delay of the return signal. This is
an issue since to perform the joint quantum measurements
described earlier, the idler must be properly retarded,
rotated in polarization, and shifted in frequency in order to
match the range, polarization, and Doppler bin that one
needs to probe. Accordingly, a target moving over multiple
range bins during the interrogation time will cause a
temporal mismatch, which induces additional losses that
can be quantified by an overlap integral 0 < km 1.
These losses enter as a multiplicative factor in the SNR
for a quantum radar, but not for a classical one. In fact, for
a classical (coherent-state) radar a time-bandwidth limited
pulse (M ¼ WT91) with a high energy per mode
(NS 1) achieves the same performances of a pulse with
large time-bandwidth product (M ¼ WT 1) with a low
energy per mode (NS91). Therefore, contrary to its quantum
counterpart, CI is not forced to use large time-bandwidth
product.
Another consequence of the high time-bandwidth
product requirement ofQI is that the power of these pulses
will necessarily be extremely low. In fact, the pulse power
can be estimated as PhvsMNs=ThvsNsW, which
in the regime where the quantum advantage is observed
(NS ¼ 0:01) gives P 0:01 fW for microwaves. The
powers of classical radars range from the mWs, used in
extremely short-range applications, to the MWs, needed
to track planes. Therefore, the power range where QI provides
a quantum advantage is between 16 and 20 orders of
magnitude smaller than the typical values used for target
detection. Shorter pulses, and consequently higher powers,
could only be enabled by phase-matching bandwidths
W far beyond those currently available. However, considering
that the phase matching bandwidth W cannot be
larger than the signal frequency vs, the room for improvement
is very limited.
86
The phase-matching bandwidth problem is surely the
most severe limitation of QI. However, there is another
subtlety with the QI theory presented in the section
" Gaussian QI " that has a strong impact on the practical
implementation of the protocol. In particular, all receivers
for QI described in this tutorial require to store the idler
for the whole radar-to-target-to-radar propagation time. If
we consider nonideal idler storage, all error exponent will
be multiplied by a factor kI equal to the idler transmission
coefficient. On the other hand, in CI, there is nothing to be
stored, meaning, for example, that, in the Bayesian setting,
6 dB of idler storage losses will be enough to destroy the
full quantum advantage of QI. Storing the idler for time
long enough to preserve the quantum advantage poses
severe limitation on the range ofa quantum radar.
Moreover, in realistic scenarios (especially at optical
wavelengths), the amplitude and the phase of the returning
light are randomly modified. This effect, also known as
fading, nullifies the quantum advantage enabled by the
OPA receiver, and makes the quantum advantage enabled
by the FF-SFG receiver subexponential [42].
EXPERIMENTS ON QI
We conclude our discussion on quantum radars by presenting
what has been achieved so far experimentally at
optical and microwave frequencies. A summary of the
state ofthe art ofQI experiments in May 2020 is presented
in Table 1.
The first QI-like experiment was performed by
Lopaeva et al. [43]. In this experiment, the photon-counting
correlations induced by an SPDC source where
exploited to obtain an advantage over a correlated-thermal
state (correlated-noise radar). We used the adjective QIlike
to describe this experiment because it did not exploit
entanglement. As a consequence, the Lopaeva et al. setup
could only outperform a correlated-thermal state of the
same energy, and not a coherent state transmitter.
Zhang et al. [19] performed the first real QI experiment
using the protocol from Tan et al. [11] described in
the section " Gaussian QI " together with the OPA
receiver [12] discussed in the section " Practical Receivers
for QI. " Due to experimental imperfections, this experiment
could not achieve the full 3 dB quantum advantage,
but it demonstrated a 20% (0.8 dB) enhancement of the
error probability exponent. Up to today the work of Zhang
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