IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 69

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from a quantum interferometry perspective. However,
these works considered highly idealized scenarios, and
neglected the influence of thermal background. Since this
review is focused on the practicality ofquantum radars, we
will not further discuss this approach, and focus on the
other approach pioneered by Lloyd the same year [8], when
he studied how to use quantum light to detect a weakly
reflecting target embedded in thermal background [8].1 In
his work, Lloyd considered two protocols: the first interrogates
the target region with N independent single photons,
while the second protocol uses N photons are all entangled
with one another. Lloyd's results showed that in the entanglement-based
protocol the probability of making a wrong
decision about the presence of the target is dramatically
lower than in the single-photons' one. These results were
welcomed with excitation from the quantum optics community
because they seemed to suggest that entanglement
could revolutionize current radar technology.
However, the reader should be aware that both protocols
proposed by Lloyd are actually quantum, accordingly the
work in [8] does not prove that a quantum radar can outperform
any classical radar. On the contrary, both Lloyd's protocols
perform worse than a detection scheme using
coherent states [10] that as we will explain in Section " Phase
Space distribution and Gaussian States " can be considered
the quantum mechanical description of a traditional radar.
Fortunately, Tan et al. [11] had already proposed a more
sophisticated QI protocol that, in a very specific regime, can
provide a quantum advantage. This review aims at presenting
this protocol, its consequences and limitations.
In their works on QI, Lloyd [8] and Tan et al. [11]
considered the problem of discriminating between two
hypotheses: target present and target absent (see the section
" Discriminating Quantum States and Probability Distributions " ).
In this setting, by using tools from quantum
information theory (see the section " Bayesian Approach:
the Chernoff Bounds: Classical Chernoff Bounds " ), these
1A reader interested in the interferometric approach to quantum radar
is referred to the original papers, or to the review [9].
MAY 2022
authors determined the lowest probability of choosing the
wrong hypothesis. In particular, Tan et al. [11] proved
that the lowest error probability achievable with entangled
radiation is given by
PQI
e eMkNs=NB
1
2
(1)
where M is the number copies of the quantum state used
to probe the target, Ns is the average number of signal
photons in a single copy of the quantum state, and NB is
the average number of background photons present when
the return from a single-copy transmission is detected, k is
the roundtrip radar-to-target-to-radar transmissivity, such
that kNs
is the number of photons coming back to the
receiver when the target is present. As we will discuss in
this review, this expression requires the usual assumptions
of strong thermal background NB 1 and low roundtrip
transmissivity k 1, but also a low signal assumptions
NS 1. Accordingly, in order to have a satisfactory signal-to-noise
ratio (SNR) SNR ¼ MkNS=NB, M 1 is
required. In the best case of classical illumination (CI),
one has instead
PCS
e eMkNs=4NB
1
2
:
(2)
Therefore, following the protocol of Tan et al. [11], the
minimal error probability exponent achievable in the
quantum case is 6 dB smaller than the best obtainable in
the classical case. This quantum advantage stems from
entanglement, which provides stronger-than-classical correlations
between the signal transmitted to the target
region and an idler pulse retained in the radar detector.
However, these peculiar correlations cannot be detected
with standard receivers. Accordingly, at the time of the
work [11], no explicit detection scheme able to achieve
this lower bound was known.
The first receivers able to obtain a quantum advantage
in a QI scheme were proposed by Guha and Erkmen [12].
On the one hand, these two receivers have the advantage
of being implementable with current technology; on the
other hand, they do not achieve the ultimate limit allowed
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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