IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 70

Detecting a Target With Quantum Entanglement
by quantum mechanics as discussed by Tan et al., but they
only provide a 3 dB advantage. A receiver able to achieve
this ultimate bound has been found in 2017 only by
Zhuang et al. [13]. Such a receiver is extremely complicated
and far beyond the capability of state of the art
experiments. However, knowing its blueprint has great
theoretical relevance since it allows us to determine the
receiving operating characteristic (ROC), namely the
probability of target detection as a function of the
false alarm probability (see the section " Neyman-Pearson
Approach: The Receiving Operating Characteristic: Classical
Neyman-Pearson Approach " ).
The first theoretical proposal for performing QI with
microwaves caught the attention of the radar community
in 2015 in [14]. This work suggested to use an electrooptomechanical
device to produce a microwave signal
beam entangled with an optical idler beam that is stored
for subsequent joint measurement with the radiation
returned from the region under interrogation. This first
idea had the merit to open QI research to microwaves,
which are more suitable for target detection, but it has
never been used in experimental implementations. However,
the same work also suggested to use more practical
superconducting devices, such as the Josephson parametric
amplifiers (JPA). Indeed, such devices have been used
in several experiments thereafter [15]-[18].
Regarding the experimental implementations, it is
worth mentioning that the only work demonstrating the
quantum advantage of the Tan et al. protocol [11] with the
Guha-Erkmen receiver was performed at optical frequencies
in [19]. Several experiments in the microwave regime
have been reported in the last few years [15]-[18].
However, in the attempt to work around some technical
difficulties, all these experiments modified the QI protocol
and ended up in regimes where it can be proved that
quantum entanglement does not provide any quantum
advantage [5].
PRELIMINARIES
In this section, we provide the background knowledge
needed to understand QI. More specifically, we deal
with two important subjects: the quantum mechanical
description of the electromagnetic field and the problem
of discriminating between two different quantum
states.
QUANTUM OPTICS CRASH COURSE
This section is particularly intended for radar engineers with
familiarity with classical electromagnetic theory, but no
background in quantum optics. We will start by considering
the standard procedure to quantize the electromagnetic
field [20], [21]. We will then introduce those quantum states
70
that are used to describe classical radiation and thermal noise
in quantum optics. Finally, the concept ofentanglement, and
the particular entangled state used in QI will be presented.
Readers familiar with quantum optics can safely skip to the
section " Discriminating Quantum States and Probability
Distributions. "
QUANTIZATIONOFTHEELECTROMAGNETICFIELD
Let us start by defining a set of modes ofthe electromagneticfield
[22] fiðr;tÞfgas solutions of the wave equation
r2
1 @2
c2 @t2
fiðr;tÞ¼ 0
satisfying the transversality and orthonormality conditions
r fiðr;tÞ¼ 0
1
V
Z
f
i ðr;tÞfjðr;tÞd3r ¼ di;j
(3)
(4a)
(4b)
where denotes complex conjugation, di;j is the Kronecker
delta function, and V is an arbitrarily large volume
containing the full physical system under consideration.
Given an arbitrary orthonormal mode basis fiðr;tÞfg, any
solution of Maxwell's equations, e.g., the electric field
Eðr;tÞ, can be expressed (in SI unit) as
Eðr;tÞ¼
XN
i¼1
1=2
20 V
hvi
aifiðr;tÞþ a
i f
i ðr;tÞ
(5)
with vi the central frequency associated with the quasimonochromatic
modes fi and ai ¼ðqi þ ipiÞ=2 are complex
numbers, where qi and pi, are known as fields
quadratures.2
We can now quantize the field Eðr;tÞ by replacing the
coefficients in (5) with operators according to
ai ! ^ai
a
i ! ^iay
(6a)
(6b)
wherey denotes Hermitian conjugation. The operators ^iay
and ^ai are known as creation and annihilation operators
and take their names from their action on photon-number
states. In fact, if we denote with jniii, the quantum state
of the electromagnetic field containing ni photons whose
spatial and temporal profile is defined by the mode
fiðr;tÞ, the action of the annihilation (creation) operator
on such a state is to remove (add) a photon
a^ijniii ¼
p
ffiffiffi
n
jni 1ii
(7a)
2In this review, we will use the typical notation ofthe quantum optics
literature ai ¼ðqi þ ipiÞ=2. However, in the radar literature, it is
more common to use the notation ai ¼ðIi þ iQiÞ=2. We invite
those reader more familiar with the second notation to be aware of
the change ofmeaning of the letter " q. "
IEEE A&E SYSTEMS MAGAZINE
MAY 2022
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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