# Entanglement-Based Quantum Information Technology

Zheshen Zhang  1* Chenglong You  2 Omar S. Magaña-Loaiza  2 Robert Fickler  3
Roberto de J. León-Montiel  4 Juan P. Torres  5,6 Travis Humble  7 Shuai Liu
1 Yi Xia  8 Quntao Zhuang9  1Department of Electrical Engineering and Computer
Science, University of Michigan, Ann Arbor, MI 48109, USA
2Quantum Photonics Laboratory, Department of Physics & Astronomy, Louisiana
State University, Baton Rouge, LA 70803, USA
3Tampere University, Photonics Laboratory, Physics Unit, Tampere, FI-33720,
Finland
4Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Apartado Postal 70-543, 04510 Cd. Mx., México
5ICFO - Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860
Castelldefels (Barcelona), Spain
6Department of Signal Theory and Communications, Campus Nord D3, Universitat
Politecnica de Catalunya, 08034 Barcelona, Spain
7Oak Ridge National Laboratory, Oak Ridge, Tennessee USA
8Institute of Physics, Swiss Federal Institute of Technology Lausanne (EPFL),
Lausanne, Switzerland
9Ming Hsieh Department of Electrical and Computer Engineering & Department of
Physics and Astronomy, University of Southern California, CA 90089, USA
<EMAIL_ADDRESS>
###### Abstract

Entanglement is a quintessential quantum mechanical phenomenon with no
classical equivalent. First discussed by Einstein, Podolsky, and Rosen and
formally introduced by Schrödinger in 1935, entanglement has grown from a
scientific debate to a radically new resource that sparks a technological
revolution. This review focuses on the fundamentals and recent advances in
entanglement-based quantum information technology (QIT), specifically in
photonic systems. Photons are unique quantum information carriers with several
advantages, such as their ability to operate at room temperature, their
compatibility with existing communication and sensing infrastructures, and the
availability of readily accessible optical components. Photons also interface
well with other solid-state quantum platforms. We will first provide an
overview on entanglement, starting with an introduction to its development
from a historical perspective followed by the theory for entanglement
generation and the associated representative experiments. We will then dive
into the applications of entanglement-based QIT for sensing, imaging,
spectroscopy, data processing, and communication. Before closing, we will
present an outlook for the architecture of the next-generation entanglement-
based QIT and its prospective applications.

††journal: osajournal††articletype: Review Article

###### Contents

  1. 1 Introduction
  2. 2 Foundation for Entanglement-Based Quantum Information Technology
    1. 2.1 Classical versus Entanglement-Based Protocols: an Architectural Perspective
    2. 2.2 Entanglement in a Nutshell: a Brief Chronicle, Basic Theory, and Experiments
      1. 2.2.1 Theory for Entanglement Generation via Spontaneous Parametric Down-Conversion
      2. 2.2.2 Continuous-Variable Entanglement Sources
      3. 2.2.3 Polarization Entangled-photon Sources
      4. 2.2.4 High-Dimensional Entangled Photons in Space and Time
      5. 2.2.5 Time-Energy Entanglement
      6. 2.2.6 Position-Momentum Entanglement
      7. 2.2.7 Entanglement in Multiple Degrees of Freedom
      8. 2.2.8 Integrated Entanglement Sources
  3. 3 Quantum Metrology Using Entangled States
    1. 3.1 Theoretical Description of Quantum Metrology
    2. 3.2 Quantum Metrology Using N00N States
    3. 3.3 Quantum Metrology Using Squeezed States
  4. 4 Quantum Illumination
    1. 4.1 Theory for Quantum Illumination
    2. 4.2 Quantum Illumination Experiments
    3. 4.3 Covert Sensing
  5. 5 Quantum Imaging with Entangled Photons
    1. 5.1 Entangled-Photon Sources for Quantum Imaging
    2. 5.2 Correlation-Based Quantum Imaging
    3. 5.3 Quantum Imaging Using Nonlinear Interference
    4. 5.4 Imaging with Entangled Photon-Number States
  6. 6 Entanglement-Enhanced Light-Matter Interactions and Spectroscopy
    1. 6.1 Theoretical Description of Entangled Two-Photon Absorption
    2. 6.2 Entangled-Photon Absorption Spectroscopy: First Approach
    3. 6.3 ETPA Spectroscopy with Varying Entanglement Time and Pump Wavelengths
    4. 6.4 ETPA Spectroscopy with Temperature-Controlled SPDC
  7. 7 Distributed Quantum Sensing
    1. 7.1 Distributed Quantum Sensing for Optical Phase Estimation
    2. 7.2 Distributed Quantum Sensing for Radio-Frequency Measurements
    3. 7.3 Distributed Quantum Sensing for Optomechanical Measurements
  8. 8 Quantum Machine Learning
    1. 8.1 Quantum Machine Learning for Classical Processing
    2. 8.2 Quantum Machine Learning for Quantum Processing
  9. 9 Entanglement-Assisted Communication
    1. 9.1 Superdense Coding with DV Entanglement
    2. 9.2 Communication Assisted by CV Entanglement
  10. 10 Outlook

##  1 Introduction

Quantum information technology (QIT) is revolutionizing the way information is
gathered, encoded, manipulated, transmitted, stored, and decoded. Among the
various genres of quantum-information carriers, photons feature robustness at
room temperature, compatibility with existing communication and sensing
infrastructure, and the availability of optical devices. Over the past three
decades, photonic QIT has led to a range of capabilities in communication,
sensing, and computing that surpass classical information technology.

Entanglement, a unique quantum mechanical phenomenon, has become a valuable
resource without any classical equivalent. The nonlocal and strong
correlations present in entangled objects are the backbone of various QIT
protocols. Photonic entanglement can now be routinely generated, processed,
and measured in quantum optics platforms in laboratory settings. With the
advent of integrated quantum photonics platforms compatible with the
established semiconductor processes, the path is open for photonic
entanglement to have a wide impact across various applications, including
sensing, imaging, spectroscopy, communication, networking, and computing.

This review article is dedicated to introducing the foundations of
entanglement-based QIT and reviewing its enabled applications. The article is
structured as follows. Sec. 2 will focus on the foundation for entanglement-
based QIT. Specifically, Sec. 2.1 will classify entanglement-based QIT
protocols into three major categories and compare them to their classical
counterparts from an architectural perspective. This classification sets the
stage for a discussion of QIT applications, as the sensing, imaging,
spectroscopy, data-processing, and communication protocols discussed in later
sections fall into these three classes. Next, Sec. 2.2 will delve into the
history, theoretical foundation, and experimental generation of entanglement.
The following seven sections will explore the applications of entanglement-
based QIT. Sec. 3 will discuss entanglement-based quantum metrology, first
introducing the Fisher information formalism (Sec. 3.1) followed by quantum-
metrology protocols based on N00N states (Sec. 3.2) and squeezed states (Sec.
3.3), which are closely linked to entanglement (see Sec. 2.2). The subsequent
sections will cover various entanglement-based QIT applications for sensing
and imaging, including quantum illumination (Sec. 4), quantum imaging (Sec.
5), light-matter interactions and spectroscopy (Sec. 6), and distributed
quantum sensing (Sec. 7). Sec. 8 will focus on emergent entanglement-based
quantum machine learning protocols and platforms, focusing on recent advances
in photonic quantum machine learning built on noisy intermediate-scale quantum
(NISQ) hardware platforms for classical (Sec. 8.1) and quantum processing
(Sec. 8.2). Sec. 9 will examine two types of entanglement-assisted
communication protocols based on discrete variables (Sec. 9.1) and continuous
variables (Sec. 9.2). Before closing, we will look ahead to the future
architecture of entanglement-based QIT, based on fast-advancing quantum
modules such as quantum memories, quantum transducers, and quantum gates, and
explore its potential capabilities and applications.

##  2 Foundation for Entanglement-Based Quantum Information Technology

This section will first elucidate the theoretical and experimental foundation
for entanglement-based QIT from an architectural perspective, describing three
classes of QIT protocols and comparing them with their classical counterparts.
Next, we will formally define entanglement, starting with a brief review on
the history of entanglement, from its discovery to the verification
experiments. We will then articulate the theory for the generation of two
important genres of entanglement widely employed in QIT, namely the
continuous-variable entanglement and the discrete-variable entanglement,
followed by an introduction to representative experiments for the generation
of the two genres of entangled states. Before closing this section, we will
introduce recent advances in entanglement generation in integrated platforms
as a route toward scalable entanglement-based QIT.

###  2.1 Classical versus Entanglement-Based Protocols: an Architectural
Perspective

Figure 1:  Three classes of entanglement-based QIT protocols and their
classical counterparts. (a) Class 1: signal and idler are entangled at the
transmitter. The signal undergoes a physical process while the idler is
retained in a quantum memory. A quantum receiver takes a joint measurement on
the signal and idler. For the classical counterpart in (d), the signal shares
classical correlations with a reference at the transmitter, and they are
measured by a classical receiver that conducts a coherent measurement. (b)
Class 2: multiple signals are entangled at the transmitter. The signals
undergo physical processes and are then measured by a quantum receiver that
performs separable measurements on the signals. For the classical counterpart
in (e), signals at the transmitter share classical correlations. (c) Class 3:
multiple signals are entangled at the transmitter. The signals undergo
physical processes and are then measured by a quantum receiver that performs a
joint measurement on the signals. For the classical counterpart in (f),
signals share classical correlations at the transmitter. In all figures, thick
solid green curve: entanglement; thin solid green curve: entanglement or
classical correlations; thin dashed green curve: classical correlations.

Entanglement is a quantum-mechanical property, with no classical description,
shared by two or multiple objects. Entanglement gives rise to a stronger-than-
classical nonlocal correlation, which can serve as a powerful resource for
achieving capabilities that surpass the limitations of classical physics. The
left panel of Fig. 1 illustrates the architectures for three classes of
entanglement-based QIT protocols with increasing technological complexity,
categorized by how entanglement is utilized and measurements are performed.
The classical counterparts for these QIT protocols are depicted in the right
panel as a comparison.

In Class 1 protocols, depicted in Fig. 1a, a quantum transmitter prepares
entangled signal and idler particles and sends the signal through a physical
process such as a communication channel or a sensor module. The idler,
sometimes referred to as the ancilla, is stored in a quantum memory without
undergoing any alteration. A quantum receiver combines the sigal and idler to
perform a joint measurement and capture the information carried on the signal.
QIT protocols falling into this class include quantum illumination [1, 2, 3,
4, 5, 6], entanglement-assisted communication [7, 8, 9, 10], and entanglement-
assisted absorption spectroscopy [11].

In Class 2 protocols, summarized in Fig. 1b, a quantum transmitter generates
multiple signals that are in an entangled state. Each signal undergoes a
distinct physical process and is individually measured at a quantum receiver.
This class of QIT protocols encompasses various applications, including
quantum metrology based on multipartite entanglement [12, 13], distributed
quantum sensing [14, 15, 16, 17, 18, 19], quantum machine learning for
distributed data processing [20], and quantum secret sharing [21, 22, 23, 24,
25].

Class 3 protocols, as illustrated in Fig. 1c, are similar to the second class,
with the distinction that a joint measurement is performed on the signals at
the quantum receiver. QIT protocols falling into this class include joint
encoding in quantum communication to achieve superadditivity [26, 27, 28, 29,
30], quantum machine learning with joint processing [31], entangled two-photon
absorption spectroscopy [32], and quantum error correction [33]. Table 1 lists
the QIT protocols belonging to each of the three classes.

Table 1: List of entanglement-based quantum information technology protocols
---
Class 1 |  • Quantum metrology using entangled states (Fig. 9, Sec. 3) • Quantum illumination and covert sensing (Fig. 12, Sec. 4) |  • Entanglement-assisted communication (Fig. 35, Sec. 9) • Quantum imaging (Fig. 18a and Fig. 19, Sec. 5)
Class 2 |  • Quantum key distribution [34] • Entanglement-enhanced light-matter interactions and spectroscopy (Fig. 21, Sec. 6) |  • Distributed quantum sensing (Fig. 25, Sec. 7) • Quantum machine learning with distributed processing (Fig. 30, Sec. 8)
Class 3 |  • Quantum machine learning with joint processing (Fig. 30, Sec. 8) • Hybrid quantum classical communication [35] |  • Quantum error correction [36, 37]

###  2.2 Entanglement in a Nutshell: a Brief Chronicle, Basic Theory, and
Experiments

The effect of entanglement was discussed in the seminal paper by Einstein,
Podolsky, and Rosen (EPR) in 1935 with the aim of showing that the quantum
mechanical description of correlated systems was seemingly incomplete [38],
pending the introduction of hidden-variable models. Later, Schrödinger
formally introduced the concept of entanglement in his response to EPR [39] to
describe systems whose correlation cannot be described by classical physics,
i.e., it is a nonclassical correlation. Even back then, Schrödinger described
entanglement as "not one but rather the characteristic trait of quantum
mechanics", emphasizing the important difference from classically known
correlations. Later, Bohm and Aharonov built upon the idea of the original EPR
argument and reformulated it using analog considerations for spin-1/2
particles, which are a prime example of two-level systems, i.e., qubits [40].
However, it was not until 1964 when Bell translated the mostly philosophical
discussion into a setting where experiments could show a clear difference
between classical correlations and their quantum counterparts [41], that the
testing of the counter-intuitive behavior of entangled systems incentivized
experimentalists.

The subsequent seminal work by Clauser, Horne, Shimony, and Holt (CHSH)
presented a bound, now called the CHSH inequality, for local hidden-variable
theories that could be tested using experimentally tangible polarization-
entangled photons. Pioneering experiments conducted between the 1970s and
1980s [42, 43, 44, 45] suggested that local hidden-variable theories were
incompatible with the observed correlations embedded in bipartite
entanglement, subject to several loopholes that needed to be addressed [46].

In the early 1990s, more intriguing properties of entanglement were unveiled
as studies incorporated multipartite systems. Notably, Mermin showed that
tripartite entanglement in the Greenberger-Horne-Zeilinger (GHZ) state
exhibited nonclassical correlations that could be verified without invoking
inequalities [47]. Then, in the late 1990s, the GHZ state was experimentally
prepared [48] and employed to demonstrate the unique quantum correlations
carried by the tripartite state [49]. Moving into the 21st century, with
pivotal technological advances in entangled-photon sources [50, 51, 52] and
single-photon detectors [53], tests of Bell’s inequalities culminated in a
series of unambiguous loophole-free experiments [54, 55, 56, 57].

The current focus of research on entanglement has shifted from its scientific
roots to the development of entanglement-based protocols, in tandem with the
experimental toolbox, to harness it as a resource for QIT. In recognition of
the remarkable trajectory of entanglement from fundamental studies to the
onset of a revolutionary technological era, the 2022 Nobel Prize in Physics
was awarded to John Clauser, Alain Aspect, and Anton Zeilinger for their
pioneering research on entangled quantum states. To gain a more in-depth
chronological review on the theoretical and experimental developments
pertaining to entanglement, readers may refer to a recent comprehensive review
article [58].

####  2.2.1 Theory for Entanglement Generation via Spontaneous Parametric
Down-Conversion

In general, entanglement is a type of correlation between two or more
subsystems of a composite quantum system. The $n$ subsystems are denoted by
indices $k=1,2,\cdots,n$ ($n\geq 2$). A state is considered entangled if it
cannot be written as a probabilistic mixture of separable states [59], i.e.,

$\displaystyle\rho\neq\sum_{i}p_{i}\rho_{1}^{i}\otimes\rho_{2}^{i}\otimes\cdots\otimes\rho_{n}^{i},$
(1)

where $\rho_{k}^{i}$ is the density matrix for subsystem $k$, and $p_{i}$ is
the probability for
$\rho_{1}^{i}\otimes\rho_{2}^{i}\otimes\cdots\otimes\rho_{n}^{i}$ to occur.
Here, a subsystem can refer to a photon in a system of multiple photons, an
optical mode in a multimode state, or sometimes different degrees of freedom
for the same photon. When all states are pure, the global state is considered
entangled if

$\displaystyle|\psi\rangle\neq|\psi\rangle_{1}\otimes|\psi\rangle_{2}\otimes\cdots\otimes|\psi\rangle_{n}.$
(2)

Quantifiers of entanglement can be derived from the framework of the quantum
resource theory [60]. For bipartite pure states $\psi_{12}$, the entanglement
entropy is given by

$\displaystyle E(\psi_{12})=S(\rho_{1})=S(\rho_{2}),$ (3)

where $\rho_{1}$ and $\rho_{2}$ are the reduced states of subsystems 1 and 2
for the joint pure state $\psi_{12}$. The quantification of entanglement in
mixed states or multipartite systems is more involved. Later in this section,
we will discuss the entanglement criteria for a specific class of continuous-
variable quantum states. Other entanglement quantifiers are beyond the scope
of this review. For further details on technical aspects of entanglement,
readers may refer to review articles in this area [59, 58].

We will now address the problem of entanglement generation. In the field of
QIT, two main classes of entanglement have been extensively exploited:
discrete-variable entanglement based on the properties of single photons, and
continuous-variable entanglement built on the field properties of optical
modes. We will first present a theoretical framework that unifies the
generation of these two classes of entangled states then discuss the processes
commonly used in QIT to produce bipartite entangled states, with the
understanding that significant progress has been made in generating both
continuous-variable [61, 62, 63] and discrete-variable [64, 65] multipartite
entangled states. These multipartite entangled states form the basis for one-
way optical quantum computing [66, 67, 68, 69, 70, 71], which is beyond the
scope of this review article.

One well-developed approach for generating optical entanglement is through the
process of spontaneous parametric down-conversion (SPDC). In SPDC, a small
portion of the higher-energy pump photons is converted into pairs of lower-
energy photons, known as the signal and idler photons, due to the nonlinearity
of an optical material such as lithium niobate (LiNbO3), potassium titanyl
phosphate (KTP), or beta barium borate (BBO), among others. We will first
consider non-degenerate SPDC, where the signal and idler photons are
distinguished by their physical properties such as their wavelengths,
polarization, or spatial modes. For the sake of simplicity, we will adopt a
single-mode interaction model that describes the signal and idler fields using
annihilation operators $\hat{a}_{S}$ and $\hat{a}_{I}$, respectively. A
complete theoretical treatment of multimode SPDC processes in continuous time-
frequency space can be found in Refs. [72, 73].

The Hamiltonian associated with the SPDC process is given by

$\displaystyle\hat{H}$
$\displaystyle=\hbar\omega_{S}\hat{a}^{\dagger}_{S}\hat{a}_{S}+\hbar\omega_{I}\hat{a}^{\dagger}_{I}\hat{a}_{I}+i\hbar\chi\left(\hat{a}_{S}^{\dagger}\hat{a}_{I}^{\dagger}e^{-i\omega_{P}t}-\hat{a}_{S}\hat{a}_{I}e^{i\omega_{P}t}\right)$
$\displaystyle=\hat{H}_{0}+\hat{H}_{I},$ (4)

where
$H_{0}=\hbar\omega_{S}\hat{a}^{\dagger}_{S}\hat{a}_{S}+\hbar\omega_{I}\hat{a}^{\dagger}_{I}\hat{a}_{I}$,
$\omega_{S}$ ($\omega_{I}$) is the angular frequency of the signal (idler)
photon, $\chi$ represents the strength of the nonlinear interaction, which is
determined by the nonlinearity of the material, the magnitude of the pump
field, and the power density of the fields. Here,
$\omega_{P}=\omega_{S}+\omega_{I}$ is the angular frequency of the pump
photon, dictated by energy conservation. In writing the Hamiltonian, we assume
that the pump is intense enough to be treated as a classical field. The term
$\hat{a}_{S}^{\dagger}\hat{a}_{I}^{\dagger}$ describes the process of a pair
of signal and idler photons being created by annihilating a pump photon, while
the term $\hat{a}_{S}\hat{a}_{I}$ corresponds to the inverse process.

The dynamics of the system can be derived in the interaction picture using the
following Heisenberg equations of motion [72]

$\displaystyle\frac{d\hat{a}_{S}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[\hat{H}_{I},\hat{a}_{S}\right],$
$\displaystyle\frac{d\hat{a}_{I}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[\hat{H}_{I},\hat{a}_{I}\right].$ (5)

Expanding the commutators above leads to the following two coupled operator
equations:

$\displaystyle\frac{d\hat{a}_{S}}{dt}$
$\displaystyle=\chi\hat{a}^{\dagger}_{I},$
$\displaystyle\frac{d\hat{a}_{I}}{dt}$
$\displaystyle=\chi\hat{a}^{\dagger}_{S}.$ (6)

Upon solving the equations, the dynamics of the annihilation operators are
given by

$\displaystyle\hat{a}_{S}$
$\displaystyle=\sqrt{g}\hat{a}_{S_{0}}+\sqrt{g-1}\hat{a}_{I_{0}}^{\dagger},$
$\displaystyle\hat{a}_{I}$
$\displaystyle=\sqrt{g}\hat{a}_{I_{0}}+\sqrt{g-1}\hat{a}_{S_{0}}^{\dagger},$
(7)

where $g\equiv\cosh^{2}\chi t$. Here, $\hat{a}_{S_{0}}$ and $\hat{a}_{I_{0}}$
represent the initial annihilation operators prior to the interaction. In the
context of SPDC, the initial modes are in the vacuum state. The parameter $g$
is commonly known as the gain of the parametric process and can be determined
experimentally.

To further characterize the process, we define the mean photon number as
$N_{S}\equiv\langle\hat{a}_{S}^{\dagger}\hat{a}_{S}\rangle=g-1$. This quantity
represents the average number of photons in the output mode $\hat{a}_{S}$ and
is indicative of the degree of amplification achieved.

We proceed by defining the quadrature operators as follows:

$\displaystyle\hat{q}$
$\displaystyle\equiv\frac{1}{2}\left(\hat{a}+\hat{a}^{\dagger}\right),$
$\displaystyle\hat{p}$
$\displaystyle\equiv\frac{1}{2i}\left(\hat{a}-\hat{a}^{\dagger}\right).$ (8)

Here, $\hat{q}$ ($\hat{p}$) represents the real (imaginary) part of the
annihilation operator and is often referred to as the amplitude (phase)
quadrature. These operators analogously correspond to the sine and cosine
components of a classical field relative to a phase reference. Experimentally,
these quadratures can be measured through homodyne detection, where the output
field is interfered with a strong reference field, commonly known as the local
oscillator, as depicted in Fig. 2a. For classical states, such as the initial
vacuum states of the signal and idler modes, the variance of the quadrature
measurements satisfies $\langle\Delta\hat{q}^{2}\rangle\geq 1/4$ and
$\langle\Delta\hat{p}^{2}\rangle\geq 1/4$. A state is considered nonclassical
if the variance of either of its quadratures falls below the classical limit
of $1/4$.

Now, let us consider two modes, $\hat{a}_{1}$ and $\hat{a}_{2}$, in a
classical state. It can be shown that
$\langle\Delta(\hat{q}_{1}-\hat{q}_{2})^{2}\rangle+\langle\Delta(\hat{p}_{1}+\hat{p}_{2})^{2}\rangle\geq
1$. The Duan-Giedke-Cirac-Zoller inseparability criterion, established in Ref.
[74], states that if
$\langle\Delta(\hat{q}_{1}-\hat{q}_{2})^{2}\rangle+\langle\Delta(\hat{p}_{1}+\hat{p}_{2})^{2}\rangle<1$,
the two modes are entangled. This criterion plays a crucial role in
entanglement detection.

Applying this criterion to the two output modes, $\hat{a}_{S}$ and
$\hat{a}_{I}$, from the SPDC process, it can be derived that
$\langle\Delta(\hat{q}_{S}-\hat{q}_{I})^{2}\rangle+\langle\Delta(\hat{p}_{S}+\hat{p}_{I})^{2}\rangle=(\sqrt{g}-\sqrt{g-1})^{2}<1$
for any gain $g\neq 1$. Hence, it confirms that the signal and idler modes
obtained from SPDC are entangled. This entangled state is known as the two-
mode squeezed vacuum (TMSV) state or the EPR state [38]. It serves as a
fundamental resource for numerous continuous-variable QIT protocols.

We next consider degenerate SPDC in which the signal and idler modes are
indistinguishable in all physical degrees of freedom. Mathematically, we let
$\hat{a}_{S}=\hat{a}_{I}$ and $\hat{a}_{S_{0}}=\hat{a}_{I_{0}}$ and transform
Eq. (2.2.1) into

$\hat{a}_{S}=\sqrt{g}\hat{a}_{S_{0}}+\sqrt{g-1}\hat{a}_{S_{0}}^{\dagger}.$ (9)

The quantum state associated with $\hat{a}_{S}$ is known as the single-mode
squeezed vacuum state, with
$\langle\Delta\hat{p}_{S}^{2}\rangle=(\sqrt{g}-\sqrt{g-1})^{2}/4$ being the
variance of the squeezed quadrature below the classical limit, and
$\langle\Delta\hat{q}_{S}^{2}\rangle=(\sqrt{g}+\sqrt{g-1})^{2}/4$ being the
variance of the anti-squeezed quadrature. We next suppose that a second
single-mode state associated with the annihilation operator
$\hat{a}_{S}^{\prime}$ is squeezed in its $\hat{p}_{S}^{\prime}$ quadrature.
As Fig. 2b illustrates, mixing $\hat{a}_{S}$ and $\hat{a}_{S}^{\prime}$ on a
symmetric beam splitter [75] characterized by the transfer matrix

$\hat{U}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&i\\\ i&1\end{bmatrix},$ (10)

followed by applying a $\pi/2$ phase shift on one of the output arms yields
the two output modes

$\displaystyle\hat{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-\hat{a}_{S}^{\prime}+i\hat{a}_{S}\right),$
$\displaystyle\hat{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{a}_{S}^{\prime}+i\hat{a}_{S}\right).$
(11)

One then obtains

$\displaystyle\hat{p}_{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-\hat{p}_{S}^{\prime}-\hat{q}_{S}\right)$
$\displaystyle\hat{q}_{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(-\hat{q}_{S}^{\prime}+\hat{p}_{S}\right)$
$\displaystyle\hat{p}_{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{p}_{S}^{\prime}-\hat{q}_{S}\right)$
$\displaystyle\hat{q}_{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{q}_{S}^{\prime}+\hat{p}_{S}\right).$
(12)

It would not be difficult to verify that
$\langle\Delta(\hat{p}_{a}-\hat{p}_{b})^{2}\rangle+\langle\Delta(\hat{q}_{a}+\hat{q}_{b})^{2}\rangle<1$,
therefore proving $\hat{a}$ and $\hat{b}$ are entangled. The correlations
embedded in the quadratures of $\hat{a}$ and $\hat{b}$ are very similar to
those in $\hat{a}_{S}$ and $\hat{a}_{I}$, suggesting that $\hat{a}$ and
$\hat{b}$ are also in a TMSV state.

Figure 2: Four types of sources for continuous-variable entanglement. The
marginal states at the two arms are depicted in the phase space in purple. (a)
Entanglement produced from a non-degenerate process. (b) Entanglement produced
by combining two amplitude-squeezed states on a symmetric beam splitter with a
transfer matrix given by Eq. (10). (c) Entanglement produced by combining an
amplitude and a phase-squeezed state on a beam splitter with a transfer matrix
given by Eq. (13). (d) Entanglement produced by splitting a single-mode
squeezed state.

The TMSV state can also be generated by interfering two modes squeezed,
respectively, in the amplitude and phase quadratures as sketched in Fig. 2c.
To show this, let us consider $\hat{a}_{S}$ squeezed in the amplitude
quadrature with $\langle\Delta\hat{q}_{S}^{2}\rangle<1/4$ and
$\hat{a}_{S}^{\prime}$ squeezed in the phase quadrature. Mixing them on a beam
splitter with the transfer matrix of

$\hat{U}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix},$ (13)

yields the following output modes:

$\displaystyle\hat{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{a}_{S}+\hat{a}_{S}^{\prime}\right),$
$\displaystyle\hat{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{a}_{S}-\hat{a}_{S}^{\prime}\right).$
(14)

A quick derivation gives

$\displaystyle\hat{p}_{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{p}_{S}+\hat{p}_{S}^{\prime}\right),$
$\displaystyle\hat{q}_{a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{q}_{S}+\hat{p}_{S}^{\prime}\right),$
$\displaystyle\hat{p}_{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{p}_{S}-\hat{p}_{S}^{\prime}\right),$
$\displaystyle\hat{q}_{b}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{q}_{S}-\hat{p}_{S}^{\prime}\right).$
(15)

Hence,
$\langle\Delta(\hat{p}_{a}-\hat{p}_{b})^{2}\rangle+\langle\Delta(\hat{q}_{a}+\hat{q}_{b})^{2}\rangle<1$,
proving the state is entangled.

A different type of continuous-variable entangled state, often referred to as
the multimode squeezed state or the v-class entanglement, can be generated
using a series of beam splitters to divert a single-mode squeezed vacuum state
into multiple modes $\hat{a}_{S_{1}},\hat{a}_{S_{2}},...,\hat{a}_{S_{N}}$, as
sketched in Fig. 2d. One can readily show that this state is entangled by
using the Fisher information approach [76].

The dynamics of non-degenerate SPDC can also be solved in the Schrödinger
picture. The initial quantum states are vacuum for both the signal and idler,
denoted as $|0\rangle_{S}|0\rangle_{I}$. The evolution of the quantum state is
given by [72]

$|\psi(t)\rangle=\exp\left[\chi
t\left(\hat{a}_{S}^{\dagger}\hat{a}_{I}^{\dagger}-\hat{a}_{S}\hat{a}_{I}\right)\right]|0\rangle_{S}|0\rangle_{I}.$
(16)

The output state of SPDC, i.e., the TMSV state, expressed in the number basis
reads

$|\psi\rangle_{SI}=\sum_{n=0}^{\infty}\sqrt{\frac{(g-1)^{n}}{g^{n+1}}}|n\rangle_{S}|n\rangle_{I}.$
(17)

One can verify that the TMSV state described by Eq. (17) is inseparable, i.e.,
cannot be written in the form of Eq. (1). In the limit of $g-1\ll 1$, i.e.,
with a weak pump, the only dominant non-vacuum contribution is from the
single-pair state
$|1\rangle_{S}|1\rangle_{I}=\hat{a}_{S}^{\dagger}\hat{a}_{I}^{\dagger}|0\rangle_{S}|0\rangle_{I}$
while all the multi-pair contributions with $n\geq 2$ can be neglected. This
is the regime for the generation of photon pairs. In tandem with auxiliary
components, one can produce discrete-variable entanglement in the
polarization, time-energy, time bin, or path degree of freedom, as we will
articulate in the next sections.

####  2.2.2 Continuous-Variable Entanglement Sources

As discussed in Section 2.2.1, TMSV states can be generated through a non-
degenerate SPDC process or by interfering two single-mode squeezed states.
Both approaches have been pursued in experimental studies. The generation of
continuous-variable EPR entanglement traces back to a pioneering experiment
conducted with a type-II KTP crystal in an optical cavity [77], which
demonstrated the EPR paradox [38]. However, the quality of the produced
entangled state was affected by dispersion among the three interacting modes
inside the optical cavity.

Figure 3:  Experiments to generate continuous-variable entanglement. (a) A
source based on non-degenerate SPDC. The signal and idler modes are
represented by two polarization modes at the output of the optical cavity,
where a type-II PPKTP crystal resides. The signal and idler modes are
separated by a PBS and then directed to two balanced homodyne detectors (BHDs)
for characterization. MC: mode cleaner; PZT: piezo-electric transducer; EOM:
electro-optic modulator; ISO: isolator; HWP: half-wave plate; BHD: balanced
homodyne detectors; NOPA: non-degenerate optical parametric oscillator; M:
mirror; PBS: polarizing beam splitter; D: detector; SA: spectrum analyzer. (b)
Characterization data on the amplitude and phase quadratures of the entangled
state from (a), showing that the variance of the sum of the amplitude
quadrature $\langle\Delta(\hat{q}_{1}+\hat{q}_{2})^{2}\rangle$ and the
variance of the difference of the phase quadrature
$\langle\Delta(\hat{p}_{1}-\hat{p}_{2})^{2}\rangle$ both fall below the shot-
noise level, thereby fulfilling the Duan-Giedke-Cirac-Zoller inseparability
criterion [74]. (c) A source based on interfering single-mode squeezed states
produced from two independent degenerate SPDC processes. Phase-locked loops
are implemented to stabilize the phase between the reference and the entangled
modes. PD: photo diode; DBS: dichroic beam splitter; AOM: acousto-optic
modulator; FI: Faraday Isolator; PS: phase shifter. (d) Characterization data
showing that the variances for the sum of the amplitude quadrature
$\hat{X}_{A}^{\rm vac}+\hat{X}_{B}^{\rm vac}$ and the difference of the phase
quadrature $\hat{P}_{A}^{\rm vac}-\hat{P}_{B}^{\rm vac}$ fall below the
corresponding vacuum noise levels by more than 10 dB. (a, b) reproduced from
Ref. [78]. (c, d) reprinted from Ref. [79].

To overcome this challenge, more recent experiments have utilized wedged type-
II periodically poled KTP (PPKTP) crystals to realize EPR sources, as shown in
Fig. 3a [78, 80]. These experiments engineered the poling periods of the KTP
crystals to tailor the wavelengths of the entangled beams at 1550 nm [78] and
1080 nm [80]. By adjusting the position of the PPKTP crystal inside the
optical cavity, tri-resonance of the pump, signal, and idler modes was
achieved, significantly enhancing the parametric interaction. As a result,
correlations exceeding 8 dB below the classical limit were achieved for both
the phase and amplitude quadratures in these experiments, as shown in Fig. 3b.

The aforementioned experiments produced entangled states that exhibited
quantum correlations in both the amplitude and phase quadratures. Another
study by Ref. [81] explored the generation of entangled states with
correlations in general quadratures using non-degenerate SPDC. The experiment
introduced tunable coupling between the signal and idler modes in the optical
cavity, enabling the generation of a more diverse set of continuous-variable
entanglement.

Continuous-variable entanglement can also be generated by interfering two
single-mode squeezed states from degenerate SPDC processes, as schematically
sketched in Fig. 2b and 2c. Such a configuration has been exploited to produce
continuous-variable entanglement in both free-space [82, 79, 83] and optical
fiber [84] platforms. In the experiment sketched in Fig. 3c, the outputs from
two independent single-mode squeezed-light sources interfered on a 50:50 beam
splitter to produce a two-mode entangled state. The two entangled modes were
characterized by two balanced homodyne detectors. A key ingredient in the
experiment was the implementation of phase-locked loops, which involves
creating and detecting the locking signals at a series of different RF
frequencies. The measurement data presented in Fig. 3d show that the variances
of both the sum of the amplitude quadratures ($\hat{X}_{A}+\hat{X}_{B}$) and
the difference of the phase quadratures ($\hat{P}_{A}-\hat{P}_{B}$) situate
more than 10 dB below the vacuum noise levels ($\hat{X}_{A}^{\rm
vac}+\hat{X}_{B}^{\rm vac}$ and $\hat{P}_{A}^{\rm vac}-\hat{P}_{B}^{\rm
vac}$). This allowed for the validation of the Duan-Giedke-Cirac-Zoller
inseparability criterion [74] and the Reid EPR paradox criterion [85].

In optical fibers, the Kerr nonlinearity is leveraged, in lieu of SPDC, to
generate continuous-variable entangled states. In an experiment reported in
Ref. [84], a pump was launched into a Sagnac loop in a polarization-
maintaining fiber at approximately $45^{\circ}$ relative to the fiber axes,
generating amplitude-squeezed states in the two orthogonal $p$ and $s$
polarizations. A 50:50 beam splitter with the transformation matrix presented
in Eq. (10) then interfered the two squeezed states to generate a two-mode
entangled state.

Figure 4:  Multimode squeezed-state source. (a) Experimental setup for the
generation of a multimode squeezed state by splitting a single-mode squeezed
state into two modes. PD: photodiode. (b) The conditional variances for the
amplitude quadratures ${\rm Var_{A|B}}(X_{A},X_{B})$ and phase quadratures
${\rm Var_{A|B}}(P_{A},P_{B})$ are shown in blue. The red curves represent the
Reid criterion for the EPR paradox, surpassing the classical limit of 1.
Dashed curves: idealized theoretical model. Solid curves: theoretical model
with additional excess noise factored in. Dots: experimental measurements.
Figures reprinted from Ref. [86].

Finally, as depicted in Fig. 2d, multimode squeezed state or v-class
entanglement can be generated by splitting a single-mode squeezed state using
a network of beam splitters. This type of entangled state has been
experimentally verified [86] to satisfy the Reid criterion for the EPR paradox
[85]. In the experimental setup shown in Fig. 4a, a single-mode squeezed state
produced from a PPKTP crystal within an optical cavity is evenly split into
two arms by a 50:50 beam splitter, resulting in the generation of two
entangled modes. Balanced homodyne detectors, supplied with local oscillators
as phase references, are used to measure the quadratures of the entangled
modes. The measured conditional variances of the amplitude quadratures ${\rm
Var_{A|B}}(X_{A},X_{B})$ and the phase quadratures ${\rm
Var_{A|B}}(P_{A},P_{B})$ are plotted in Fig. 4b. The calculated Reid criterion
for the EPR paradox, given by ${\rm Var_{A|B}}(X_{A},X_{B})\times{\rm
Var_{A|B}}(P_{A},P_{B})$, surpasses the classical limit of 1, thus confirming
the entanglement between the two modes. More recently, the multimode squeezed
state has found applications in entangled sensor networks, addressing various
problems related to optical phase [87, 17, 88, 89], radiofrequency [18],
optomechanical [90], and data processing [20]. For detailed information, we
refer the readers to Sec. 7 and Sec. 8.

####  2.2.3 Polarization Entangled-photon Sources

One of the most widely studied and utilized forms of discrete variable
photonic entanglement is polarization entanglement. Photons entangled in their
polarization degree of freedom have played a central role in numerous key
experiments, ranging from the early stages of quantum information science [42,
43, 44, 45] to recent ground-satellite quantum communication testbeds [91].
Polarization states can be easily manipulated, transmitted, and detected using
readily available optical components with minimal disturbances, making them
highly attractive for long-distance quantum communication.

The pioneering experiments to test Bell’s inequalities [42, 43, 44, 45]
utilized polarization entangled photons generated from atomic systems and
measured using photomultiplier tubes. Subsequently, the use of SPDC in solid-
state quantum optics platforms offered a promising approach to generate high-
flux, wavelength-tailored, and high-fidelity polarization entangled photons.
The non-collinear type-II SPDC process in a BBO crystal, for instance, enabled
the construction of the first polarization entangled-photon source without the
need for post-selection [92]. Various geometries utilizing type-I SPDC
processes in BBO crystals were later developed for polarization entanglement,
leading to the generation of entangled states with more than 10 photons [93,
94, 95]. By employing quasi-phase-matching techniques, such as using LiNbO3 or
KTP crystals, polarization entangled photons spanning a wide spectral range
can now be generated and then detected using state-of-the-art single-photon
detectors.

A particularly robust configuration for stable polarization entangled-photon
sources is based on the Sagnac interferometer [50, 96]. Such sources have been
extensively used in quantum optics and quantum information experiments.
Notably, the polarization entangled-photon source deployed in the Micius
quantum-communication satellite (Fig. 5a) follows this scheme [91]: a type-II
PPKTP crystal is pumped bidirectionally within a Sagnac loop, generating
counter-propagating signal-idler photon pairs in the state
$|H\rangle|V\rangle$. A half-wave plate rotates the polarization of the
counterclockwise photon pairs by 90∘, transforming the state to
$|V\rangle|H\rangle$. The counterclockwise and clockwise photons are then
combined using a polarizing beam splitter. When a pair of photons is detected
at the two output arms, it is impossible to distinguish whether they were
generated by the clockwise or counterclockwise pump. As a result, the two
quantum states must be coherently added, yielding

$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle|V\rangle+e^{i\varphi}|V\rangle|H\rangle\right),$
(18)

where $\varphi$ represents the overall phase difference between the clockwise
and counterclockwise paths, which remains stable over time. The photon
correlations of the polarization entangled-photon source were characterized
on-satellite, and the measurement data is presented in Fig. 5b. To generate
each curve, the polarization of the signal photons was fixed while sweeping
the polarization of the idler photon during detection. The recorded normalized
coincidence rates clearly exhibit a strong dependence on the polarizations of
the signal and idler photons.

Figure 5:  Polarization entanglement sources. (a) Schematic of the
polarization entanglement source in a Sagnac configuration. Such a source was
deployed in the Micius quantum-communication satellite. HWP: half-wave plate;
QWP: quarter-wave plate; PL: pump laser; PBS: polarizing beam splitter; DM:
dichroic mirror; LP: long-pass filter. (b) Measurement data illustrating the
correlation of polarization entangled photons on-satellite. (c) Schematic of
the polarization entanglement source in a one-way configuration. (a, b)
reprinted from Ref. [91]. (c) reprinted from Ref. [54].

Apart from the Sagnac structure, an alternative configuration for generating
polarization entangled photons is the one-way configuration first introduced
in Ref.[51]. As shown in Fig. 5c, a PPKTP crystal is pumped by two parallel
beams from the same laser, which are then separated by a beam displacer (BD1).
A second beam displacer (BD2) diverts the different polarization states from
the upper and lower arms into four beams. A set of wave plates processes the
polarizations, followed by a third beam displacer (BD3) that merges the four
beams into the entangled signal and idler. By carefully designing the optical
paths for the two combining beams in the upper or lower arm to be identical,
the which-path information is effectively erased, resulting in high-quality
polarization entangled photons. The one-way configuration offers robustness
against environmental disturbances, leading to a measured visibility of
coincidence detections exceeding 99.5% for both the horizontal-vertical and
diagonal-antidiagonal bases. The high quality of this source has recently been
leveraged in the loophole-free test of Bell’s inequalities [54], along with
another demonstration that relies on a Sagnac-like implementation [55].

####  2.2.4 High-Dimensional Entangled Photons in Space and Time

In addition to the polarization degree of freedom, SPDC processes have been
utilized to generate photons that exhibit entanglement in the spatial and
temporal domains [65] that arises due to the conservation of energy and
momentum during the SPDC process.

In the regime of low pump power and paraxial approximation, where the pump
laser is treated as a classical light field and the SPDC crystal is thin, the
resulting biphoton state can be described as [97, 98, 99]

$\displaystyle\ket{\psi}_{\rm SPDC}$ $\displaystyle=$ $\displaystyle\iint
d\mathbf{q_{S}}d\mathbf{q_{I}}d\omega_{S}d\omega_{I}\Phi(\mathbf{q_{S}},\mathbf{q_{I}},\omega_{S},\omega_{I})\hat{a}_{S}^{\dagger}(\mathbf{q_{S}},\omega_{S})\hat{a}_{I}^{\dagger}(\mathbf{q_{I}},\omega_{I})\ket{0}_{S}\ket{0}_{I}$
(19) $\displaystyle=$ $\displaystyle\iint
d\mathbf{q_{S}}d\mathbf{q_{I}}d\omega_{S}d\omega_{I}\Phi(\mathbf{q_{S}},\mathbf{q_{I}},\omega_{S},\omega_{I})\ket{\mathbf{q_{S}},\omega_{S}}\ket{\mathbf{q_{I}},\omega_{I}}.$

Here, the signal and idler photons are characterized by their energies
$\omega_{S,I}$ and transverse momenta $\mathbf{q}_{S,I}$. Unlike the Fock
notation, which represents the state of the entire system, here we use
separate ket-vectors to describe each photon in the generated pair. Each ket-
vector specifies the spatio-temporal mode in which the photon can be measured
or the mode that the single photon excites.

The joint bi-photon amplitude
$\Phi(\mathbf{q_{S}},\mathbf{q_{I}},\omega_{S},\omega_{I})$ is determined by
the phase matching of the SPDC process, which involves the interplay between
the frequencies and wave vectors of the pump, signal, and idler photons. When
the parameters of the SPDC process are appropriately chosen, the function
$\Phi(\mathbf{q_{S}},\mathbf{q_{I}},\omega_{S},\omega_{I})$ becomes non-
separable in momentum or frequency (or both), indicating the entangled nature
of the generated photons.

This feature can also be intuitively understood in both domains. On one hand,
the photon pair is naturally generated at the same time and in the same
transverse spatial location, which is often referred to as the “common birth
zone” [100], resulting in perfect time and position correlations. On the other
hand, due to energy and momentum conservation in the SPDC process governed by
the phase matching conditions, one also observes frequency and momentum anti-
correlations, given by $\omega_{P}=\omega_{S}+\omega_{I}$ and
$\mathbf{q}_{P}=\mathbf{q}_{S}+\mathbf{q}_{I}$, respectively.

Perfect correlations in the complementary bases of time and energy, as well as
position and momentum, indicate the generation of entanglement in either
domain. Exemplary correlation measurements and the corresponding joint
spectral, temporal, spatial, and momentum amplitudes are shown in Fig. 6a and
7a. In both cases, the correlations were strong enough to verify the presence
of entanglement.

However, it is important to note that in both domains, the coherence of the
pump field plays a crucial role in entanglement generation. The coherence of
the pump field is transferred to the generated bi-photon state and,
consequently, to the entanglement between the two photons [101, 102].
Additionally, it is worth mentioning that in the ideal case of a single-
frequency, plane-wave pump, along with an infinitely thin and infinitely
extended crystal in the transverse direction, the down-converted photons would
exhibit perfect correlations in frequencies and momenta. However, in realistic
scenarios where the pump field has a non-zero frequency bandwidth and momentum
spread, the correlations become less well-defined, leading to wider joint
amplitudes. The limited frequency and momentum spread of the photon pair,
determined by the phase matching conditions, defines the strength of the
correlations and, consequently, the achievable dimensionality of the
entanglement.

To generate discrete high-dimensional entangled states, the continuous spatio-
temporal domain needs to be discretized into a specific number of orthogonal
modes or discrete units, known as bins. The number of modes or bins that can
be used depends on the resolution of the employed devices as well as the
strength of the correlations discussed earlier.

These different modes or bins are commonly labeled by the letter $d$,
representing a system that allows for $d$ orthogonal states. This terminology
gives rise to the term “qudits”, as compared qubits in two-level systems such
as polarization. By utilizing the temporal or spatial domain to address photon
pairs generated in the SPDC process, it becomes possible to generate high-
dimensional entangled quantum states. These states offer many advantages in
various aspects of quantum photonics, spanning from fundamental research to
information technologies. One of the primary benefits is the ability to encode
more than just a single bit of quantum information. In fact, high-dimensional
entangled states can encode log${}_{2}d$ qubits. Furthermore, they exhibit
increased resilience to noise [103, 104, 105, 106] and can simplify operations
in quantum information processing [107]. From a fundamental perspective, such
states are also valuable in tests of quantum versus classical theories [108,
109]. For more detailed information, interested readers can refer to recent
review articles [110, 111, 65], which provide further insights into the field
of high-dimensional entanglement and its applications in various research
areas.

As discussed earlier, in an ideal scenario with perfect momentum and frequency
anti-correlations in the SPDC process, the joint bi-photon amplitude
$\Phi(\mathbf{q_{S}},\mathbf{q_{I}},\omega_{S},\omega_{I})$ can be
approximated by delta functions, allowing for the discretization of continuous
space into a large set of correlated spatio-temporal modes or bins
$|n\rangle$. This discretization enables the generation of a state that
closely resembles the maximally entangled state given by

$|\psi\rangle_{\rm
qudit}=\frac{1}{\sqrt{d}}\sum_{n=0}^{d-1}|n\rangle|n\rangle.$ (20)

In the laboratory, various sets of orthogonal light modes or non-overlapping
bins in space and time can be utilized for this purpose. The exact
discretization and the resulting dimensionality of entanglement depend on
several factors, including the specific SPDC parameters, the chosen
discretization scheme, and the resolution of the devices used in the
experiment.

In the following sections, we will provide a brief overview on important
recent advances in both the spatial and temporal domains, highlighting
advancements in generating high-dimensional entanglement.

####  2.2.5 Time-Energy Entanglement

For the case of time-energy entanglement, let us consider a simplified
scenario with a single spatial mode for both the pump and the down-converted
photon pair. This allows for a further simplification of Eq. (19) to

$\ket{\psi}_{\omega}=\int
d\omega_{S}d\omega_{I}\Phi(\omega_{S},\omega_{I})\ket{\omega_{S}}\ket{\omega_{I}},$
(21)

In experiments, the correlations observed in frequency and their corresponding
Fourier-related time domain are often expressed and measured using the joint
spectral amplitude [112] and the joint temporal amplitude [113] (see Fig. 6a).
The discretization in either basis is set by the spectral or timing resolution
of the detectors, and the resulting states are commonly referred to as
spectral or time bins.

In Fig. 6b, a schematic of such a discretization scheme is depicted,
illustrating how the continuous time and frequency domains are divided into
discrete bins or modes, with each bin corresponding to a specific range of
frequencies or time intervals. The correlations between these bins capture the
entanglement between the photons.

In the time domain, achieving high-dimensional entanglement requires the
timing resolution of the detectors to be much finer than the coherence time of
the pump laser. Moreover, a high level of stability is necessary for the pump
laser to enable the measurement of correlations not only in time but also in
superpositions of time bins. To accomplish the latter, which is essential for
entanglement certification, a commonly used measurement technique is an
unbalanced Mach-Zehnder interferometer, also known as a Franson interferometer
[114]. A schematic of an exemplary setup is depicted in Fig. 6e.

These approaches have facilitated the observation of high-dimensional
entanglement, surpassing hundreds of time bins and even higher
dimensionalities [115, 106]. Additionally, the utilization of pulse shaping
techniques [116] in conjunction with nonlinear interactions has enabled the
measurement of timing correlations at resolutions beyond the capabilities of
conventional detectors [117, 113]. This breakthrough overcomes the existing
technological limitations of detectors and achieves significantly enhanced
precision in temporal measurements.

Another strategy to achieve high-dimensional entanglement in the temporal
domain involves employing a coherent mode-locked laser that emits a sequence
of coherent pump pulses. Each pulse presents an opportunity for generating a
photon pair, effectively defining distinct time bins as sketched in Fig. 6c.
The utilization of coherent mode-locked lasers with a high repetition rate is
essential for this purpose [118]. The repetition rate plays a crucial role as
it determines the time interval between two successive pulses and,
consequently, the delay between the individual time bins. It is important to
note that the high repetition rate poses challenges for the detection system
due to the substantial imbalance between the two arms of the Franson
interferometer.

Despite these challenges, experimental studies have successfully demonstrated
entanglement dimensions of up to 18 using a setup illustrated in Fig. 6f
[119]. This approach highlights the potential of coherent mode-locked lasers
in generating high-dimensional temporal entanglement.

Figure 6:  Time-energy entanglement sources. (a) The SPDC process leads to
correlation in time and anti-correlation in frequency, due to the conservation
laws. To obtain high-dimensional entanglement, the correlations can be
discretized into temporal/spectral bins. Figure reprinted from Ref. [113]. (b)
Using a discretization scheme or (c) frequency comb as a pump can directly
lead to high-dimensional entanglement in $d$ possible photon pulses. (d) A
temporal quantum state can also be discretized into orthogonal temporal modes,
which correspond to discrete orthogonal spectral modes in frequency. (e) A
typical setup to generate and measure temporal correlations using a
continuous-wave laser as the pump and Franson interferometers to verify the
entanglement. Figure reprinted from Ref. [106]. (f) A setup generating high-
dimensional entanglement using a pulse train. Figure reprinted from Ref.
[119]. (g) A setup to generate frequency bin entangled states using an
integrated frequency comb. Figure reprinted from Ref. [120].

Another highly promising approach is the study of temporal modes, which
constitute an orthogonal set of Hermite-Gauss shaped wave packets as depicted
in Fig. 6d. These modes can be considered the most natural basis in the
temporal domain [121, 122]. In recent years, significant progress has been
made in utilizing advanced detection and manipulation tools [123] as well as
direct entanglement generation through nonlinearity engineering [124].

In addition to exploring the temporal domain, it is also possible to leverage
the generated frequency correlations. In this context, frequency bins serve as
an analogous discretization to time bins [125], and techniques similar to
pulse shaping methods [126] can be employed to address them. Integrated
frequency combs have emerged as a particularly promising approach, as
evidenced by recent advancements [120, 127, 128, 129, 130, 131]. Integrated
frequency combs can be seen as the analog counterpart to the aforementioned
pulse trains, as depicted in Fig. 6b and c. Utilizing frequency combs to
generate frequency bin entanglement offers a well-controlled and potentially
scalable approach. Fig. 6g provides a sketch of such a setup. With
entanglement dimensions already demonstrated in the tens to hundreds range,
this approach is particularly attractive due to its integrated nature and
compatibility with telecom equipment. Additionally, by employing a continuous
pump laser and discretizing the biphoton state using a Fabry-Pérot cavity to
form a biphoton frequency comb, it has been shown that high dimension can be
achieved without post-selection [132, 130].

By combining the technology to shape single photons in time [116] and
frequency [129], both approaches for generating high-dimensional entanglement
in the energy-time domain hold great promise for expanding the size of the
underlying Hilbert space. Moreover, their compatibility with integrated
systems and optical fibers positions time-energy entanglement as a vital
player in future high-dimensional QIT.

####  2.2.6 Position-Momentum Entanglement

Another widely exploited degree of freedom to generate high-dimensional
entanglement through SPDC is the spatial domain, where non-classical
correlations in position and momentum can be observed. Similar to the
discussion in the time-energy domain, we can simplify the general SPDC state
introduced in Eq. (19) by considering correlations only in space. This
simplification involves assuming a single pair of frequencies for the two
photons, allowing us to write the state as

$\ket{\psi}_{\mathbf{q}}=\int
d\mathbf{q}_{S}d\mathbf{q}_{I}\Phi(\mathbf{q}_{S},\mathbf{q}_{I})\ket{\mathbf{q}_{S}}\ket{\mathbf{q}_{I}}.$
(22)

The conservation of transverse momentum in the SPDC process, i.e.,
$\mathbf{q}_{P}=\mathbf{q}_{S}+\mathbf{q}_{I}$, leads to highly correlated
joint transverse momentum amplitudes. These anti-correlations have been
leveraged to couple the photons into multiple fibers in a bundle. In doing so,
the continuous space is discretized to generate high-dimensional path
entanglement [133]. In addition to momentum anti-correlations, correlations in
the complementary space, i.e., the transverse spatial locations, have been
utilized to verify entanglement using the EPR criteria over the last few
decades [134, 135, 136]. Fig. 7a shows the measurements of the joint spatial
and momentum amplitude on an electron-multiplied CCD camera.

As conceptually depicted in Fig. 7b, the use of single-photon sensitive
cameras, particularly in the pixel basis, has made it appealing to discretize
the continuous transverse space, similar to time bins in the temporal domain.
This approach to discrete high-dimensional entanglement has greatly benefited
from the latest advancements in camera technologies, which have enabled the
demonstrations of entanglement in more than 100 dimensions [137]. In
conjunction with advanced compressed sensing methods, a channel capacity of
8.4 bits per photon has been achieved, corresponding to over 300 dimensions
[138].

In another approach, spatial light modulators (SLMs) are inserted into the
beam path of each down-converted photon to introduce filtering masks that
define macro-pixels of customized shapes and sizes. This configuration offers
great flexibility in adjusting the macro-pixels to match the experimental
setup that comprises crystals and imaging optics. Fig. 7e shows a schematic
for such a setup in which SLMs enabled the generation of a nearly ideal
maximally entangled state described in Eq. (20) [139, 140]. Additionally, the
use of SLMs has allowed for measurements in superposition states of the macro-
pixels to bolster in-depth characterization of the generated state, achieving
entanglement dimensions of more than 50 [139, 141].

Figure 7:  Position-momentum entanglement sources. (a) The SPDC process leads
to correlation in space and anti-correlation in momentum, due to its phase
matching conditions. Figure reprinted from Ref. [135]. (b) To obtain high-
dimensional entanglement, the correlations can be discretized into spatial
bins, i.e., pixels. (c) Generating a superposition of multiple pump paths in
one (or many) SPDC crystal generates high-dimensional entanglement entangled
in $d$ different paths. (d) Another popular approach to discretize space into
$d$-dimensional quantum states is by using different spatial mode families,
three of which are shown here with the color depicting their phase structure
and the brightness representing their intensity. (e) A sketch of a typical
setup to study spatial correlations using spatial light modulators to project
onto pixels or spatial modes. Figure reprinted from Ref. [141]. (f) By using a
metalens array, tens of parallel SPDC processes can be pumped, thereby
generating high-dimensional path entanglement. Figure reprinted from [142].
(g) Integrated waveguide chips can be used to generate, manipulate, and
analyze high-dimensional path-entangled photon pairs (inset shows a photo of
the chip). Figure reprinted from Ref. [143].

The discretization of the transverse space into specific areas can also be
achieved within the pump beam itself. In this configuration, a single pump is
sent into the SPDC crystal with coherence across locations, as schematically
sketched in Fig. 7c. Recently, this approach was demonstrated using a lens
array implemented through a nano-structured metasurface, enabling 100
parallel, coherently driven SPDC processes that generated up to six photons in
a compact arrangement, as depicted in Fig. 7f. The main challenge with such a
scheme lies in the verification of high-dimensional entanglement, as it
requires a method to project onto superposition states. To date, only three-
dimensional entanglement in a subset of the whole array has been demonstrated
[142].

Following a similar concept, splitting the pump into multiple paths, each
passing through an SPDC crystal, leads to what is known as path entanglement.
This approach was recently extended to achieve an entanglement dimension of 32
in an efficient bulk-optics setup [144]. Encoding the high-dimensional quantum
state in different paths is particularly appealing not only because it can be
directly implemented using single-mode fibers compatible with fiber networks
[145], but also because it can be realized with integrated photonic waveguide
chips (see Sec. 2.2.8). These chips hold great promise for scaling up quantum
states to very large dimensions. An experiment using this technology
demonstrated 14-dimensional entanglement, where the generation, manipulation,
and analysis of the state were performed on the chip, as illustrated in Fig.
7g [143].

Notably, the spatial domain of photons can be discretized into complex
orthogonal transverse structures of light known as the transverse spatial
modes. Fig. 7d illustrates various mode families that exist subject to the
underlying symmetry and coordinate system used to describe them. While
entanglement has been demonstrated for all common mode families, such as
Hermite-Gauss modes [146], Laguerre-Gauss (LG) modes [147], Ince-Gauss modes
[148], as well as modes with special properties like Bessel modes that are
self-healing [149] and Airy modes that self-accelerate [150], the most popular
ones are the LG modes in cylindrical coordinates. LG modes possess a special
property known as orbital angular momentum (OAM), which is based on a twisted
phase structure. In fact, the first demonstration of spatial mode entanglement
was achieved using OAM-carrying photons generated through the SPDC process,
which conserves angular momentum [151]. Entanglement in OAM has been further
verified by correlating OAM with its complementary variable, i.e., angle
[152]. The popularity of high-dimensional OAM entanglement is also driven by
virtue of the technical simplicity. OAM-carrying photons can be relatively
easily measured using phase modulations on SLMs and single-mode fibers [153,
154]. By removing the helical phase structure of OAM modes through phase
modulations, photons with flat phase profiles can be efficiently coupled into
single-mode fibers, which is not the case for other mode types [155].
Additionally, OAM quanta are conserved in cylindrically symmetric systems,
making quantum states encoded in OAM-carrying spatial modes particularly
robust when transmitted through cylindrical-shaped optical systems. By
exploiting the full-field correlations of LG mode entanglement involving the
azimuthal OAM degree of freedom and the radial degree of freedom, entanglement
in dimensions up to 100 has been demonstrated [156]. However, it is important
to note that the physical size of the modes increases when higher-order modes
are utilized, and the apertures of the optical system impose a practical
limitation on high-dimensional entanglement. Nevertheless, LG modes are known
to be the maximally dense solution in cylindrical systems in terms of the
information capacity [157], with entanglement of OAM quanta up to 10,000
already achieved [158]. Generating high-dimensional entanglement using spatial
modes of photon pairs from an SPDC process poses a challenge due to the
decreasing likelihood of higher-order modes appearing in the down-converted
pair, resulting in a non-maximally entangled state. Various methods have been
developed to address this drawback. One approach is to carefully tune the
phase-matching conditions [159] to increase the probability of higher-order
mode generation and, consequently, enhance the entanglement dimensionality.
Another strategy is to shape the spatial mode of the pump light in the SPDC
process to tailor the generated mode content and improve the quality of
entanglement. However, this approach has thus far been limited to up to
5-dimensional entanglement [160, 161]. More recently, there have been
proposals to enhance the dimensionality by not only shaping the pump, but also
engineering the nonlinear interaction through crystal design [162] to offer
greater control over the entanglement properties.

It is worth noting that a novel approach building upon the previously
discussed concepts has been introduced to generate high-dimensional
entanglement through path identity, leading to successful generation of three-
dimensional entangled states [163]. By exploiting the indistinguishability of
multiple SPDC processes, each contributing additional modes to the entangled
photon pair, this approach offers unprecedented flexibility and customization
in realizing the desired entangled state. While the concept of path identity
draws inspiration from well-known schemes used for generating polarization
entanglement [93], it can also be extended to entangle more than two photons
and can be linked to quantum imaging schemes [164]. This innovative approach
opens up new possibilities for exploring and harnessing high-dimensional
entanglement in various quantum information processing applications.

In the future, it is conceivable that transversely entangled photons in their
spatial modes could serve as flying qudits, which are high-dimensional quantum
information carriers, for long-distance communication in both fiber-based
[165] and free-space [166] scenarios, as they exhibit robust propagation
characteristics. Following transmission, these qudits could be utilized
directly for quantum operations [167] or interfaced with waveguide chips by
converting them into path qudits through simple transformations [168]. On
quantum photonic chips, the entangled state can be further manipulated and
converted into polarization encoding or multi-mode waveguide structures [169].
This integration of transverse mode entanglement with various platforms and
encoding schemes holds great promise for advancing QIT.

####  2.2.7 Entanglement in Multiple Degrees of Freedom

Instead of generating high-dimensional entangled states for each degree of
freedom (DOF) separately, significant progress has been made in utilizing
hyper-entanglement, which involves simultaneous entanglement in multiple DOFs,
such as polarization, space, and time. By treating each DOF as an additional
resource for encoding quantum information, the overall Hilbert-space dimension
is multiplied, resulting in a significantly enlarged state space. Various
combinations of hybrid entanglement have been explored, including polarization
and momentum [170, 171, 172], polarization and spatial modes [173],
polarization and time [174, 175], and hyper-entanglement involving all DOFs
[176, 177].

The expanded state space provided by hyper-entanglement has enabled notable
achievements, such as efficient entanglement distillation [178], the
generation of an 18-qudit entangled state using six photons and their
polarization, path, and spatial mode DOFs [179], and the demonstration of an
entanglement-based quantum network [180]. These advancements highlight the
potential of hyper-entanglement for various QIT applications.

####  2.2.8 Integrated Entanglement Sources

The remarkable progress in understanding and implementing quantum information
processing techniques has sparked great enthusiasm for the development of
large-scale quantum systems. As these systems continue to increase in
complexity and fulfill real-world requirements, there is an urgent need to
transition from traditional, bulky optical entanglement sources to compact
photonic integrated chips (PICs). This shift is made possible by the rapid
advancements in classical PICs, which not only enable superior generation of
entanglement states but also offer improved manipulation and detection
capabilities. Moreover, PICs offer several advantages, including reduced
footprint and cost, enhanced integration of functionalities, improved
fidelity, and increased phase stability. Furthermore, the well-established
fabrication techniques and process design kits (PDKs) associated with PICs
allow for the integration of a wide range of optical functional components
onto a single chip, including beam splitters, waveguides, delay lines,
filters, cavities, detectors, among others. Additionally, PICs provide a
diverse selection of material platforms, allowing researchers to choose the
most suitable platform for their specific goals and targets.

Figure 8:  Integrated entanglement sources. (a) Schematic of fully on-chip
generation of frequency-bin entangled photons on an InP-SiNOI hybrid
integration configuration. Figure reprinted from Ref. [181]. (b) Programmable
integrated lithium niobate photonic circuits for the generation, manipulation,
and detection of squeezed-vacuum and TMSV states through the SPDC process.
Figure reprinted from Ref. [182]. (c) Sketch of dual-pump degenerate SFWM for
squeezed-vacuum generation on the SiOI platform. Measurement data show the
quadrature variance in relative to the shot noise level. Figure reprinted from
Ref. [183].

In the discrete-variable regime, integrated entanglement sources can
probabilistically generate entangled photons through SPDC in materials with
second-order nonlinearity or spontaneous four-wave mixing (SFWM) through
third-order nonlinearity. SPDC typically offers higher conversion efficiency,
while SFWM platforms provide more mature and scalable low-cost fabrication
possibilities.

The initial breakthrough in generating entangled states in PICs is traced back
to 2008 when Alberto et al. achieved this milestone in a silica-on-silicon
waveguide platform [184]. They utilized single-mode waveguide interferometers
to replace bulky beam splitters, enabling the realization of high-fidelity
path-entangled states, two-photon quantum interference, and controlled-NOT
gates. Since then, significant progress has been made in various PIC platforms
over the past two decades, benefiting from the rapid development of micro- and
nanofabrication technologies. These platforms include silicon-on-insulator
(SOI) [185, 186], silicon nitride-on-insulator (SiNOI) [187], thin film
lithium niobate-on-insulator (TFLNOI) [188], gallium arsenide-on-insulator
(GaAsOI) [189], and others.

Akin to the bulk-optics systems, PICs can generate entangled photons with
multiple degrees of freedom (DOFs), such as time-energy [190], polarization
[191], optical path [192], transverse mode [193], time bin [194], and
frequency bin [120]. In particular, PICs provide a pathway for controlling and
scaling up the dimensionality of entanglement using transverse modes supported
by multi-mode waveguide structures [193]. Harnessing the capabilities of PICs,
this method offers a compact and efficient solution to produce high-
dimensional entanglement. The use of multi-mode waveguide structures also
facilitates the manipulation and control on entangled states, paving the way
for scalable entanglement-based QIT.

To integrate more quantum components in SOI PICs, foundry-compatible
complementary metal-oxide-semiconductor (CMOS) fabrication processes have been
leveraged to achieve the generation of multipartite multidimensional
entanglement via monolithic integration of over 2,400 quantum components on a
single chip [195]. Recently, Hatam et al. realized fully on-chip turnkey
entanglement sources on a SiNOI platform, as depicted in Fig. 8a. They
followed a hybrid integration approach, combining an electrically pumped
indium phosphide (InP) reflective semiconductor optical amplifier (RSOA) with
an efficient programmable SiN vernier filter as the low-noise pump source. As
a result, frequency-bin entangled states were generated with up to 99%
fidelity without the need for an external bulky laser [181]. Since SiNOI is a
foundry-compatible PIC platform, their approach allows for seamless
integration with other quantum processing circuits, paving the way for
commercially available, cost-effective, scalable, and stable quantum PICs for
QIT applications.

In contrast to the probabilistic SPDC process, semiconductor quantum dots
(QDs) offer the capability of on-demand generation of single and entangled
photons. While other platforms, such as color centers in crystals [196, 197],
rare earth atoms [198], and two-dimensional materials [199], have been
investigated for on-chip quantum-light generation, QDs exhibit advantages
including high brightness and flexible operation wavelength through bandgap
engineering. Efforts to surpass the fine structure splitting have led to
several recent experiments that successfully demonstrated on-demand generation
of high-fidelity entangled photons via biexciton-exciton cascade in an
individual QD [200, 201, 202]. Building upon this approach, Frejia et al.
embedded QDs in a nanophotonic waveguide and transferred the intrinsic
polarization entanglement into photon path entanglement [203] to enable
interfacing with other quantum components in PICs. However, integrating QDs
with waveguides or cavities requires precise QD positioning and alignment
lithography. This is particularly important for epitaxial QDs that are
typically randomly distributed on the substrate. Despite the development of
the wafer bonding and “pick-and-place” techniques, which often come with
tradeoffs in fidelity and uniformity [204, 205], achieving large-scale
integration of QDs in PICs remains challenging. For more comprehensive
discussions on QDs-based integrated quantum platforms, readers should refer to
the review paper by Hepp et al. [206].

In contrast to the discrete-variable counterpart, continuous-variable
entanglement offers advantages of deterministic generation and efficient
detection at room temperature, while its fidelity is more susceptible to loss.
In the pioneering experiments, researchers employed single-pass quasi-phase-
matched waveguides in lieu of bulk crystals to generate on-chip squeezed
light. Squeezing levels of 12% and 14% were measured in the KTP [207] and
lithium niobate (LiNbO3)[208] platforms respectively. In these experiments,
however, the experimental components apart from the SPDC processes remained
off-chip and required precise beam shaping, alignment, and phase locking. More
recently, Genta et al. demonstrated the on-chip generation and
characterization of EPR entanglements by employing optical waveguide
interferometers to interfere two off-chip single-mode squeezing states [209],
an approach discussed in Sec. 2.2.1 and Sec. 2.2.2. The use of optical
waveguides inherently ensures near-perfect spatial mode matching between the
squeezed light and the local oscillator (LO), significantly reducing the
complexity associated with homodyne measurements. Later, Lenzini et al.
integrated two individual single-pass periodically poled LiNbO3 (PPLN)
squeezed-light sources with waveguides, phase shifters, and interferometers on
the same PIC to enable the reconfigurable generation, manipulation, and
characterization of squeezed vacuum states and EPR entangled states [182], as
shown in Fig. 8b. For QIT based on continuous variables, high levels of
squeezing are crucial. For example, entanglement swapping typically requires 3
dB of squeezing [210] to surpass the classical limit; 2D cluster state
generation needs 4.5 dB of squeezing [61]; and fault-tolerant quantum
computing requires 10 dB of squeezing [211]. To date, the highest detected on-
chip squeezed vacuum is 6 dB over a 2.5 THz bandwidth, demonstrated by
Takahiro et al. using a single-mode ZnO-doped PPLN waveguide[212], showing
great potential for large-scale QIT on PIC platforms.

In addition to the above low-index-contrast platforms, recent research has
focused on generating squeezed states in more compact nanophotonic platforms.
In 2015, Avik et al. made significant strides in this direction by
demonstrating the use of a single silicon nitride (SiN) microring resonator
for squeezed-light generation. Utilizing the non-degenerate SFWM process above
the parametric oscillation threshold, they measured amplitude squeezing of 1.7
dB between the generated signal and idler beams [213]. Subsequently, Zhao et
al. further advanced this field by demonstrating the generation of quadrature
squeezed-vacuum states using a degenerate SFWM scheme, as illustrated in Fig.
8c. In their approach, both the squeezed light and LO were simultaneously
generated on the same chip [183]. They utilized two identical ring resonators
operating in the over-coupling and critical coupling conditions for squeezing
generation and LO generation. The large separation of the two pump signals
helped minimize the noise caused by thermal-refractive effects. However, the
individual parametric processes of each pump introduced unwanted noise
fluorescence to the squeezing states. As a result, a 1.34 dB quadrature
squeezed vacuum was detected, with an inferred on-chip squeezing level of 3.09
dB accounting for coupling losses. To address this issue, Zhao et al. proposed
a novel approach by leveraging a photonic molecule design, which involved an
auxiliary resonator strongly coupled with the squeezing resonator [214]. By
effectively detuning the resonances supporting parametric processes, the noise
from the pump were significantly suppressed, while the squeezing resonator
remained unaffected. The directly measured squeezing level was 1.65 dB,
corresponding to 8 dB on-chip squeezing, with the measured squeezing limited
by the out-chip collection and detection efficiencies. Although non-degenerate
SFWM suffers from higher thermal-refractive noise in squeezed-light
generation, it inherently supports a large number of spectral modes, making it
appealing for large-scale QIT applications. Based on non-degenerate SFWM,
Vaidya et al. demonstrated 1 dB quadrature squeezed vacuum in a SiN ring
resonator[215]. Later, Yang et al. achieved a 1.6 dB squeezing in an ultra-
high-Q silica microdisk with 20 operational pairs of TMSV states [216]. In
their experiment, the LOs were generated by means of an electro-optic
modulator to circumvent the need for complex phase locking of multiple lasers.
Despite the recent tremendous advances, the generation of squeezed states in
PICs remains nascent. Assuming optimal performance for each individual
component in PIC platforms, it would become feasible to achieve squeezing
levels in excess of 10 dB and high-quality 3D cluster states for one-way
quantum computing [66]. Remarkably, recent advancements in programmable
photonic processors based on continuous-variable entanglement have shown
quantum computational advantages at room temperature [217, 218].

##  3 Quantum Metrology Using Entangled States

Figure 9: Schematic of quantum metrology using entangled states, which
consists of three steps. The first step prepares an entangled state of signal
and idler. The second step involves the interaction between the signal and the
target. In the third step, the signal and infer are measured to infer the
properties of the target.

Metrology is one of the important topics of optical sciences. The use of
photons to measure small quantities is one of the most important applications
of light [219, 13, 220, 221, 222, 223, 224]. Unfortunately, with classical
states of light produced by, e.g., the laser, one cannot surpass the
fundamental shot-noise limit [225, 226, 227]. Quantum metrology builds on
nonclassical states of light such as entanglement and squeezing to beat the
conventional limit. With the recent development in experimental capabilities,
quantum metrology has become tangible in laboratory settings and is moving
quickly toward real-world applications [219, 13, 220, 228, 221]. Here, we will
introduce the theory of quantum metrology followed by a review on its
realizations.

###  3.1 Theoretical Description of Quantum Metrology

A metrology protocol aims to measure a parameter $\varphi$ of a physical
system. As shown in Fig. 9, quantum metrology consists of three steps: Step 1
generates the probe state. In Step 2, the probe state is sent to interact with
the system. In the last step, the signal is measured to estimate the parameter
of interest for the system. In classical metrology, one has a data set
consisting of $N$ copies of independent identically distributed random
variables drawn from a probability density function (PDF), $p_{\varphi}(X)$,
where the PDF is solely depending on an unknown parameter $\varphi$ being
estimated. The goal of the estimation is to construct an estimator
$\tilde{\varphi}_{N}(\mathbf{x})$ to minimize the uncertainty. To this end,
one can use the Fisher information (FI) to derive an optimal estimator [229].
In this case, one could construct the classical _Cramér-Rao bound_ (CRB) that
lower bounds the mean square error (MSE) of any unbiased estimator
$\tilde{\varphi}_{N}$:

$\Delta^{2}{\tilde{\varphi}}_{N}\geq\frac{1}{N\,F\\!\left[p_{\varphi}\right]}\,,$
(23)

where $F\\!\left[p_{\varphi}\right]$ is the FI defined as follows:

$F\left[p_{\varphi}\right]=\int\\!\\!\textrm{d}x\,\left(\frac{\partial}{\partial\varphi}\log
p_{\varphi}(x)\right)^{2}p_{\varphi}(x)=\int\\!\\!\textrm{d}x\,\frac{1}{p_{\varphi}(x)}\left[\frac{\partial\,p_{\varphi}(x)}{\partial\varphi}\right]^{2}.$
(24)

It should be noted that the classical CRB is dependent on the measurement
performed on the signal. As such, one would ask: what is the best achievable
performance for estimating the parameter $\varphi$ irrespective of the
measurement? The _quantum Cramér-Rao bound_ (QCRB) provides the answer. QCRB
is a generalization of the classical CRB, setting the lower bound for the
estimation variance over all locally unbiased estimators subject to
measurements allowed by quantum mechanics [230, 231, 232]. The QCRB is derived
from the quantum FI (QFI) defined as

$F_{\mathrm{Q}}\left(\rho_{\varphi}\right)=\max_{\left\\{p_{\varphi}\right\\}}F(p_{\varphi}).$
(25)

Like the classical CRB and FI, the QCRB is linked to the QFI by

$\Delta^{2}\varphi_{N}\geq\frac{1}{NF_{\mathrm{Q}}\left[\rho_{\varphi}\right]}\geq\frac{1}{NF\left[p_{\varphi}\right]}.$
(26)

The QFI $F_{\mathrm{Q}}$ can be calculated by

$F_{\mathrm{Q}}\left[\rho_{\varphi}\right]=\operatorname{Tr}\left\\{\rho_{\varphi}L\left[\rho_{\varphi}\right]^{2}\right\\},$
(27)

where $L\left[\cdot\right]$ is called the symmetric logarithmic derivative
operator. In simple scenario of a pure state
$\rho_{\varphi}\\!=\\!\left|\psi_{\varphi}\right\rangle\\!\left\langle\psi_{\varphi}\right|$,
the QFI is derived as

$F_{Q}=4(\langle\dot{\psi}_{\varphi}|\dot{\psi}_{\varphi}\rangle-|\langle\dot{\psi}_{\varphi}|\psi_{\varphi}\rangle|^{2}),\quad|\dot{\psi}_{\varphi}\rangle=\frac{d|\psi_{\varphi}\rangle}{d\varphi}.$
(28)

It is interesting to note that, in the context of optical interferometry, when
the estimated parameter is encoded on the state by a unitary operation
described as $\rho_{\varphi}\\!=\\!U_{\varphi}\rho U_{\varphi}^{\dagger}$ with
$U_{\varphi}\\!=\\!\textrm{e}^{-\textrm{i}\hat{H}\varphi}$, QFI on a pure
state $\rho=\ket{\psi}\bra{\psi}$ is proportional to the variance of $\hat{H}$
[13]:

$F_{Q}(\ket{\psi_{\varphi}})=4\Delta^{2}H=4(\bra{\psi}\hat{H}^{2}\ket{\psi}-\bra{\psi}\hat{H}\ket{\psi}).$
(29)

We should note that care needs to be taken in calculating the QFI [220, 224,
227, 233, 221]. Specifically, the QFI is deriving by optimizing over all
possible measurement schemes, which may include cases where one already has
prior knowledge of the parameter being estimated. To ensure a valid QFI
calculation, it is necessary to identify a measurement that can attain the
QCRB. For a more comprehensive review on the quantum estimation theory, the
readers should refer to Refs. [234, 220, 221]. Finally, performance metrics
need to be defined to benchmark metrology protocols. For optical-metrology
protocols using, on average, $N$ photons to perform the estimation,
$1/\sqrt{N}$ is defined as the shot-noise limit (SNL) while $1/N$ is referred
to as the Heisenberg limit (HL) [12]. The SNL is considered the fundamental
limit for a classical metrology protocol, dictating its maximal attainable
sensitivity. However, quantum metrology protocols based on nonclassical states
are not subject to the SNL. In the following sections, we will discuss
experiments that utilized entanglement to surpass the SNL.

###  3.2 Quantum Metrology Using N00N States

The N00N state is maximally entangled with multiple particles distributed over
two modes, presenting one of the most significant and paradigmatic classes of
quantum states with a fixed number of particles [235, 236]. Formally, the N00N
state is described as

$\left|\psi_{\mathrm{N00N}}\right\rangle=\frac{|N0\rangle+e^{iN\varphi}|0N\rangle}{\sqrt{2}},$
(30)

where $N$ is the number of particles in the system, and $\varphi$ is the
relative phase between the two modes. Using the N00N state in phase
estimation, the ultimate sensitivity without loss is given by

$\Delta\varphi_{\mathrm{N00N}}\geq\frac{1}{N},$ (31)

showing performance at the HL.

This distinct feature of the N00N state makes it an attractive resource to
address phase-estimation problems [236]. Nevertheless, one should note that
creating a N00N state is daunting and typically probabilistic, rendering
N00N-state-based sensing impractical in real-world scenarios. To date, only
the N00N state with $N=2$ can be deterministically generated via the Hong-Ou-
Mandel effect [237]. The generation of N00N states with $N>2$ has to resort to
probabilistic pre-selection or post-selection with the success rate decay
exponentially with respect to $N$. Moreover, N00N state is susceptible to
loss, causing the QFI to decrease exponentially with the photon number $N$.

Figure 10: Experimental quantum metrology using N00N states. (a) Pure four-
photon interference due to path-entangled four-photon state. Figure reprinted
from Ref. [238]. (b) Super-resolution phase measurement with two and three
photons. Figure reprinted from Ref. [239]. (c) Experimental results of
coincidence measurements, demonstrating 4- and 5-fold super-resolution. Figure
reprinted from Ref. [240]. (d) Two-photon and four-photon count rate with a
high-visibility fringe that beats the SNL. Figure reprinted from Ref. [241].
(e)Experimental data for Fisher information and phase uncertainty, with
unconditional violation of SNL achieved. Figure reprinted from Ref. [242].

The very first few experiments with N00N states were indeed based on post-
selection on the detection events. For example, Walther et al. demonstrated a
interferometer using four-photon N00N states generated by an SPDC source
[238]. As shown in Fig. 10a, the reduction of the oscillation wavelength
derived from post-selected four-photon coincidence events confirmed the super-
resolution enabled by the N00N state. At the same time, Mitchell et al.
experimentally demonstrated a technique to produce three-photon N00N states
[239] and demonstrated super-resolving phase measurements with post-selected
multiphoton events, as shown Fig. 10b. These two experiments proved the
experimental plausibility to generate N00N states using linear optics in
conjunction with post-selection. Another important experiment by Afek et al.
generated N00N-like states by mixing a coherent state with the state from SPDC
[240]. As illustrated in Fig. 10c, $N$-fold super-resolution with up to 5
photons was experimentally observed. These prominent experiments demonstrated
the feasibility of producing N00N states and their use in quantum metrology
protocols, sparking a series of follow-on experiments that demonstrated super-
resolution using N00N states [243, 244, 245, 94, 246, 247]. Despite these
encouraging results, it is important to note that these experiments only
showed super-resolution in estimating an optical phase inside the
interferometer, manifesting as the interference fringes oscillating faster
than the these obtained with classical states. In this regard, achieving
super-resolution does not require surpassing the SNL [220, 221]. In fact, one
can achieve super-resolution using carefully prepared classical states without
beating the SNL [248, 224, 249, 250].

More recently, researchers explored using N00N states to surpass the SNL [241,
171, 251, 252, 242, 89, 253]. Fig. 10d shows one of the earliest attempts in
this direction made by Nagata et al., demonstrating that entangled four-photon
interference can outperform the SNL [241]. However, it should be emphasized
that Nagata et al.’s study employed conditional measurements based on post-
selection and therefore did not constitute an unconditional violation of the
SNL. The probabilistic nature of the protocol, as well as the loss incurred in
the experiment and the low detection efficiency made the task of unconditional
surpassing of the SNL arduous. Recently, Slussarenko et al. demonstrated the
first unconditional violation of SNL in quantum metrology protocol based on a
high-fidelity N00N state with $N=2$ photons [242], with experimental data
depicted in Fig. 10e. The high generation efficiency of the N00N state and the
low overall loss in the experiment paved the way for such a breakthrough.

###  3.3 Quantum Metrology Using Squeezed States

As discussed in Sec. 2.2, the squeezed state has reduced noise in one field
quadrature at the expense of increased noise in the other quadrature. The
squeezed state can be leveraged to improve the sensing precision if the
information of interest resides in the less noisy quadrature and is captured
by homodyne measurement on that quadrature. In 1987, Xiao et al. utilized
squeezed states to estimate the optical phase at a measurement precision
exceeding the SNL [254]. Almost simultaneously, Grangier et al. experimentally
demonstrated an squeezed-light-based interferometer beating the SNL [255]. The
two experiments spurred many subsequent quantum-metrology studies using
squeezed states. At present, the most well-known application of squeezed-state
quantum metrology is its application to gravitational wave detection [256,
257, 258]. The original Laser Interferometer Gravitational-Wave Observatory
(LIGO) was designed to measure gravitational waves that induce distance
changes on the order of $10^{-18}$ m [256, 257, 258]. However, the
photodetection shot noise fundamentally limited the original LIGO’s frequency
of gravitational observations. To mitigate this impediment, the advanced LIGO
enhanced by squeezed light was developed and employed, now operating below the
SNL [259, 260, 261, 262, 263]. Readers should refer to Refs. [256, 264, 257,
258, 265] for more comprehensive reviews on for gravitational-wave detection
enhanced by squeezed light.

Figure 11: Experimental quantum metrology using squeezed states with photon
counting. (a) The experimental setup of an unconventional nonlinear
interferometer that achieves an unconditional and robust quantum metrological
advantage. The Fisher information per photon exceeds the predicted value using
an ideal five-photon N00N state and perfect detectors. Figure reprinted from
Ref. [266]. (b) Schematic of the experimental setup for quantum metrology with
photon counting. The Fisher information extracted from multiphoton
interference fringes shows advantage over the SNL. Figure reprinted from Ref.
[267]. (c) An interferometric scheme using TMSV states and photon-number-
resolving detectors. The calculated Fisher information of the post-selected
events beats the SNL. Figure reprinted from Ref. [268] (d) Experimental setup
to demonstrate multiphoton quantum-enhanced phase estimation. The Fisher
information for detection scheme with photon-number resolution outperforms the
SNL and beats the detection scheme without photon-number resolution. Figure
reprinted from Ref. [269].

Apart from detecting the field quadratures, one can also measure the photon
statistics of squeezed states to achieve quantum enhancements in metrology. A
common approach is to exploit the TMSV state generated through the SPDC
process, as previously discussed in Sec. 2.2, in phase sensing. In fact, as
shown in Sec. 3.2, many quantum metrology protocols create the desired N00N
state from a TMSV-state source [238, 239]. As discussed in Sec. 2.2.1, at weak
squeezing levels, i.e., low gain of the parametric process, the generated TMSV
state is effectively a photon-pair source with the multi-pair probability
negligible. In this case, photon pairs from the source in tandem with photon
statistics measurements are leveraged to achieve measurement precision
superior to the SNL [242, 88].

Very recently, Qin et al. demonstrated a 5.8-fold enhancement above the SNL in
terms of the FI extracted per photon, without discarding photons due to loss
and imperfections [266]. This was achieved using a stimulated squeezing
nonlinear interferometer consisting of a pair of two-mode squeezing operations
together with two superconducting nanowire single-photon detectors (SNSPDs).
The FI per photon as a function of the phase clearly surpassed the SNL and
outperformed an ideal 5-N00N state in some regions of the phase space, as
shown in Fig. 11a.

Although an idealized model ignores the multi-pair contributions in a TMSV
state, in practice the multi-pair events in SPDC, albeit occurring with small
probabilities, induce errors in protocols without accounting for these
contributions using photon-number-resolving detectors. To address this issue,
Matthews et al. first used photon-number-resolving detectors based on
multiplexing 16 avalanche photodiodes (APDs) [267] to measure a four-mode
squeezed state (see Fig. 11b). They were able to detect four-photon
coincidences and beat the SNL with post-selection, despite the presence of
significant loss. Indeed, the ability to detect multi-photon events opens the
door to achieving further quantum advantages with squeezed states [270]. In
particular, the fast development of highly efficient photon-number-resolving
detectors such as the superconducting transition edge sensors (TES) has
greatly facilitated surpassing the SNL by retaining the photon-number
information [271, 272, 273]. In this regard, Thekkadath et al. utilized high-
gain SPDC sources and photon-number-resolving detectors to perform
interferometry with heralded quantum probes of up to $N=8$ [268], as
illustrated Fig. 11c. They employed two type-II parametric down-conversion
sources to herald photon pairs and injected the heralded state into an
interferometer. At the output, photon-number-resolving measurements were taken
to estimate an unknown phase difference. Despite the protocol being
intrinsically probabilistic, quantum advantages in phase estimation were
observed. In 2021, You et al. experimentally demonstrated, for the first time,
a scalable protocol for quantum-enhanced optical phase estimation across a
broad range of phases, without any pre- or post-selected measurements [269].
The efficient design of an SPDC source combined with photon-number-resolving
detection enabled the utilization of all detected photons in an unconditional
quantum-enhanced phase estimation. As shown in Fig. 11d, the ability to probe
all possible multi-photon interference events facilitated the identification
of complex multipartite interactions occurring in the interferometer.
Consequently, this scheme enabled the estimation of a broad range of phases
with sensitivities unconditionally surpassing the SNL. Finally, we note that
these experiments represent only a subset of the extensive body of research
conducted on quantum metrology using squeezed states. Interested readers are
encouraged to refer to Refs. [274, 275, 221] for greater detail.

##  4 Quantum Illumination

Quantum illumination (QI) is an entanglement-enhanced sensing scheme that aims
to improve the precision and sensitivity of target detection in the microwave
frequency region, as illustrated in Fig. 12. By entangling the signal probe
with locally stored idler reference, QI seeks to enhance the performance of
target detection compared to classical approaches [1, 2, 4].

In the original QI proposal, which focused on determining the absence or
presence of a target, the error-probability exponent exhibited a 6 dB
advantage resulting from the utilization of entanglement [2]. This advantage
was achieved while consuming the same probe power as the classical target
detection scheme. Recently, QI has been extended to enhance target ranging and
angle detection [9, 6]. These extensions show the potential for even greater
advantages over classical methods in the intermediate signal-to-noise ratio
(SNR) region, thanks to the threshold phenomena of nonlinear parameter
estimation [6]. Furthermore, the same mechanism that enables the quantum
advantage in target detection has also been applied to phase-sensing scenarios
[276]. This demonstrates the versatility and potential applicability of QI
beyond its original context.

Figure 12: Schematic of quantum illumination for target detection. An
entanglement source delivers signal and idler photons to the transmitter and
receiver. The transmitter then sends the signal to interrogate a target
residing in a lossy and noisy environment. The receiver takes a joint
measurement on the signal received from the environment and the idler to infer
the presence or absence of the target.

###  4.1 Theory for Quantum Illumination

In the QI protocol, a total of $M$ signal-idler pairs are utilized for the
sensing task. Each pair consists of a signal mode $\hat{a}_{S}$ and an
entangled idler mode $\hat{a}_{I}$, generated through the SPDC process
described in Eq. (2.2.1). The signal mode is sent to probe a potential target,
while the returned mode
$\hat{a}_{R}=\sqrt{\kappa}\hat{a}_{S}+\sqrt{1-\kappa}\hat{e}$ is collected at
the quantum receiver. Here, $\kappa$ represents the transmissivity, and
$\braket{\hat{e}^{\dagger}\hat{e}}=N_{B}/(1-\kappa)$ denotes the mean photon
number of the background noise. The idler mode is stored locally, awaiting a
joint measurement with the returned signal at the quantum receiver.

In the original QI target-detection protocol proposed by Tan et al. [2], where
the target’s presence or absence is equally probable, the asymptotic error
probability derived from the quantum Chernoff bound [277, 278] is given by

$\displaystyle P_{E}^{\rm QCB}\sim\frac{1}{2}\exp\left(-\frac{M\kappa
N_{S}}{N_{B}}\right),$ (32)

where $N_{S}\equiv\braket{\hat{a}_{S}^{\dagger}\hat{a}_{S}}$ represents the
signal brightness. QI operates in the regime of weak signal and strong
background noise, characterized by $N_{S}\ll 1$ and $N_{B}\gg 1$. On the other
hand, with the same set of parameters, the optimal classical illumination (CI)
scheme, employing a coherent-state transmitter and a homodyne-detection
receiver, exhibits an error probability given by

$\displaystyle P_{C}\sim\frac{1}{2}\exp\left(-\frac{M\kappa
N_{S}}{4N_{B}}\right),$ (33)

where the exponent is four times inferior to that of QI. Notably, the
advantage of QI over CI persists even in the presence of entanglement-breaking
noise and loss, which render the correlation between the returned signal and
retained idler fully classical. However, the quantum Chernoff bound alone does
not provide guidance on how to design a structured quantum receiver to fully
exploit QI’s advantage.

Before delving into the design problem of the quantum receiver, it is
important to highlight that the original QI proposal utilizes a quantum
transmitter that emits a TMSV state, which has been demonstrated to be optimal
for target-detection tasks in various scenarios [279, 280, 281]. Reference
[279] showed that the TMSV state is optimal in the asymmetric scenario, where
the objective is to minimize the decay rate (maximize the error-probability
exponent) of the probability of a false positive given a certain probability
of a false negative. Conversely, the coherent state is the optimal input in
the absence of an ancilla, such as the idler mode. Reference [280]
demonstrated that in the symmetric scenario of minimizing the overall error
probability, the TMSV state is optimal in the regime of low signal brightness
and bright thermal noise. Furthermore, Ref. [281] showed that in the limit of
zero reflectivity, the TMSV state minimizes the error probability, while the
coherent state is optimal among unentangled sources. However, for scenarios
with non-zero reflectivity and finite signal and noise brightness, other
quantum states for the probe may slightly outperform the TMSV state.

The real challenge in QI lies in the design of the quantum receiver, which
aims to implement a joint measurement on the returned signal and idler modes
while harnessing the advantage of QI over CI. The optimal measurement for a
specific quantum-state discrimination or parameter estimation problem can
generally be solved mathematically by formulating a set of positive operator-
valued measure (POVM) elements. However, the practical implementation of such
a measurement requires a structured design comprising experimental modules
that can realize the desired POVM elements while considering practical
technological constraints. For example, in the microwave domain, quantum-
limited detection requires extensive cooling due to ambient noise backgrounds,
and basic measurement apparatuses such as photon-counting modules are less
developed compared to the optical domain [282, 283].

The first generation of quantum receivers for QI was based on parametric
amplification [284]. Two specific receiver designs emerged from this approach:
the optical parametric amplifier receiver (OPAR) and the phase-conjugate
receiver (PCR), both achieving half of the six decibel error-probability
exponent advantage offered by QI.

The OPAR operates by applying a two-mode squeezing operation, as described by
Eq. (2.2.1), to create the mode
$\hat{c}=\sqrt{g}\hat{a}_{I}+\sqrt{g-1}\hat{a}_{R}^{\dagger}$ from each pair
of signal and idler modes, $\hat{a}_{I}$ and $\hat{a}_{R}$. Photodetection is
then performed on all the $\hat{c}$ modes to determine the presence or absence
of the target. The optimal gain for the two-mode squeezing operation is
derived as $G\sim 1+\sqrt{N_{S}}/N_{B}$. Similarly, the PCR follows a similar
principle by first generating a phase-conjugate mode using
$\hat{c}=\sqrt{g}\hat{a}_{v}+\sqrt{g-1}\hat{a}_{R}^{\dagger}$, where
$\hat{a}_{v}$ represents a vacuum mode. The phase-conjugate mode is then
interfered with the idler mode, followed by balanced photodetection. The PCR
offers similar performance to the OPAR but provides practical benefits in
high-noise regimes [10]. In the regime of weak signal and strong noise, where
$N_{S}\ll 1$ and $N_{B}\gg 1$, the error probability for both the OPAR and PCR
can be approximated as:

$\displaystyle P_{E}^{\rm OPAR}\sim P_{E}^{\rm
PCR}\sim\frac{1}{2}\exp\left(-\frac{M\kappa N_{S}}{2N_{B}}\right).$ (34)

Although the OPAR and PCR are sub-optimal compared to the ultimate quantum
limit, they offer the advantage of using off-the-shelf components and have
been successfully implemented.

###  4.2 Quantum Illumination Experiments

A proof-of-concept demonstration of QI for target detection was carried out in
the optical domain [285]. As schematically depicted in Fig. 13, the QI sensor
comprised an entanglement transmitter and a quantum receiver. The entanglement
transmitter entailed a periodically poled lithium niobate (PPLN) crystal
pumped by a laser to generate entangled signal and idler, serving respectively
as the probe and reference, through a non-degenerate SPDC process as
illustrated in Fig. 2a. The probe was transmitted to a lossy and noisy
environment where a target may reside. The environmental noise was emulated by
injecting optical thermal noise to mimic the microwave scenario. The probe
returned from the environment was first combined with the reference retained
in a quantum memory composed of a fiber spool and then with the pump prior to
being processed at an OPAR composed of a second PPLN crystal and other optics
for filtering and collection. The OPAR converted the phase information carried
on the returned signal into amplitude modulation on the reference. By
detecting the reference at the OPAR output, one can infer the properties of
interest for the target such as its presence or absence. In this experiment,
the entanglement transmitter and OPAR enabled a $\sim 20\%$ SNR advantage
despite a 14 dB overall environmental loss and a noise background 75 dB more
intense than the signal.

Figure 13: Diagram of the proof-of-concept quantum illumination target
detection experiment reported in Ref. [285]. The setup entails an non-
degenerate entanglement transmitter and a quantum receiver based on OPAR. The
environmental noise is emulated by injecting optical thermal noise.

The proof-of-concept QI experiment in the optical domain has verified the
benefit of entanglement in an entanglement-breaking sensing environment.
However, to demonstrate the entanglement-enabled advantage, the optical QI
experiment had to artificially mixed thermal noise with the probe signal as
the ambient noise at the optical wavelengths is negligible. In contrast,
blackbody radiation noise at the microwave frequencies is appreciable,
rendering QI a useful tool to address microwave sensing problems. To implement
QI in the microwave domain, a proposal based on optical-microwave transduction
was conceived [286]. The proposed QI protocol exploits entangled microwave
signal and optical reference. The storage of the optical reference does not
require cooling as one can simply store it in an optical fiber loop with
insignificant loss. The returned signal is up-converted from the microwave to
the optical domain to enable quantum-limited joint measurements with the
reference. Notably, the state-of-the-art optical-microwave transduction
efficiencies [287, 288, 289, 290] still fall short of what is needed by the
proposed microwave QI protocol.

On the experimental front of microwave QI, Ref. [291] reported a setup based
on Josephson parametric converter (JPC) as the entanglement transmitter and a
digital PCR as the quantum receiver, as shown in Fig. 14a. JPC generated
entangled microwave signal photons serving as the probe and idler photons
serving as the reference. The physical process of entanglement generation from
a JPC resembles that of the SPDC in the optical domain as they can both be
described mathematically by the Hamiltonian in Eq. (2.2.1). Unlike the OPAR
and PCR that perform joint measurements on the returned probe and the
reference at the receiver, the digital PCR took a measurement on the reference
microwave photons at the transmitter and recorded the digitized measurement
outcomes. A second set of measurements were taken on the returned probe to
construct the covariance matrix of the probe and the reference. Shown in Fig.
14b, the SNR of the microwave QI plotted alongside these of the CI based on
correlated probe and reference as well as with the coherent-state transmitter
in tandem with a homodyne receiver. Due to the limitation of the digital PCR,
its associated SNR could not surpass that of the optimal CI scheme based on a
coherent-state transmitter and a homodyne receiver. Nonetheless, the
experiment did show that the calibrated SNR for the microwave QI, assuming the
availability of a physical PCR and perfect detectors, would beat that of the
optimal CI, thereby constituting a significant step in experimental microwave
QI. In a more recent microwave QI experiment, two microwave resonators coupled
by a Josephson ring modulator were employed to produce entangled microwave
photons and to serve as the OPAR [283]. The microwave OPAR exploited a
superconducting qubit dispersively coupled to a microwave resonator, in lieu
of a photodetector as used in the optical QI experiment, to read out the state
of the microwave resonator that stored the reference photons. This microwave
QI experiment achieved a $\sim 20\%$ advantage over the optimal CI scheme in
the error-probability exponent, on par with the demonstrated quantum advantage
in the optical QI experiment [285].

Figure 14: Microwave quantum illumination experiment. (a) Schematic for
microwave QI based on a Josephson parametric converter (JPC) to generate
entangled microwave photons and a digital PCR. Both the probe (signal) and
reference (idler) photons are first amplified and subsequently detected at
room temperature. (b) Measured SNR at different source brightness $N_{S}$.
Solid blue: microwave QI based on a JPC entanglement transmitter and a digital
PCR; solid orange: CI based on correlated probe and reference; solid green: CI
based on a coherent-state transmitter and homodyne receiver; solid yellow: CI
based on a coherent-state transmitter and heterodyne receiver; dashed blue:
calibrated microwave QI. Figures reprinted from Ref. [291].

The above experiments based on OPAR and PCR are limited in their quantum
advantage, due to the experimental constraints as well as the receivers being
sub-optimal. To design an optimal receiver, one needs to construct a
measurement to perform the optimal quantum state discrimination between
multiple copies of identical quantum states $\hat{\rho}_{RI,0}^{\otimes M}$
versus $\hat{\rho}_{RI,1}^{\otimes M}$, where the indices $0,1$ represent the
target being present or absent. In general, for mixed states, local operations
on each copy and collectively processing the data from $M$ copies cannot solve
such an optimal state discrimination problem [292]. A joint measurement on $M$
copies is in general needed by an optimal QI receiver.

The first optimal receiver design was proposed in Ref. [3]. The design
utilizes a sum-frequency generation process to jointly interact the $M$
return-idler pairs and produce a sum mode in a coherent state embedded in weak
thermal noise. The optimal receiver for QI adopts a feedforward mechanism, in
analogy to the Dolinar receiver [293] designed for optimal coherent-state
discrimination. However, the optimal receiver requires unit-efficiency sum-
frequency generation at the single photon level, therefore challenging
experimental realizations.

A recent proposal of the optimal receiver design based on correlation-to-
displacement (‘$\rm C\hskip
0.56905pt\scalebox{0.8}{\begin{picture}(0.0,0.0)\roundcap\put(0.0,0.0){\leavevmode\hbox{\set@color\char
41\relax}} \put(0.0,0.0){\line(1,0){missing}} \end{picture}}\hskip
0.56905ptD$’) conversion surprisingly shows that an optimal receiver design
can be achieved by separately heterodyne-detecting the return mode, and
consequently processing the idler field [294]. Conditioned on the results of
the heterodyne detection of the return modes, the associated idler modes are
in coherent states embedded in weakly thermal noise. A beamsplitter array will
then be combining the idlers—coherently add up the displacements while
maintaining the same level of noise. By well-established coherent-state
discrimination protocols [293], one can achieve the optimal error probability
of quantum illumination in Eq. (33). In fact, it achieves a tighter lower
bound of the error probability of QI with an arbitrary entanglement source
[280].

Before we close on the topics of QI for target detection, we would like to
introduce extensions of QI to more complex target detection scenarios. While
the original QI protocol considers detecting the presence or absence of a
single target at a specific spatial spot and at a certain time, real world
target detection scenarios are more complex, involving detecting target’s
range, angle, and speed [295]. Therefore, it is important to extend the
quantum advantages in the original QI to general scenarios. Refs. [9, 6]
showed that quantum advantages in ranging—the estimation of the distance
between the observer and the target—is possible. As shown in Fig. 15, due to
the nonlinear nature of the parameter estimation problem, the quantum range
accuracy is much better than the classical one at the intermediate signal-to-
noise region. The caveat in such advantage is that the advantage still assumes
the weak brightness of the signal field. Therefore, with practical modeling of
target ranging, it is found that quantum advantages are limited to detecting
targets at hundreds of meters away such that a reasonable integration time of
seconds is adequate to provide useful information about target.

Figure 15: Quantum advantage in ranging at the long integration time limit.
The plot assumes a bandwidth of $10^{6}$ HZ and an initial range uncertainty
of $5$ kilometers. Figure reprinted from Ref. [6].

###  4.3 Covert Sensing

Figure 16: Schematic for entanglement-enhanced covert sensing. The
entanglement transmitter generates signal and idler photons to execute the
sensing task. The signal photons are used to probe a target residing in a very
noisy environment while the idler photons are delivered to the quantum
receiver. The opponent, Willie, attempts to detect the sensing operation by
measuring the intensity of his captured photons. The quantum receiver performs
a joint measurement on the signal photons received from the environment and
the retained idler photons to estimate the probed physical properties of the
target. By concealing the signal in a large noise background, one can ensure
that Willie is unable to detect the sensing operation due to the statistical
fluctuation in his measurement. Entanglement enhances the estimation precision
and the covertness of the protocol. Figure reprinted from Ref. [276].

A variant of the QI protocols is entanglement-enhanced covert sensing. In
covert sensing, the probe signal is concealed in a bright noise background
such that the opponent, Willie, who is granted access to the probe power
scattered in the environment, is unable to detect the sensing attempt.
Entanglement can be leveraged to enhance the performance and security of
covert sensing, as schematically sketched in Fig. 16. Akin to QI, in covert
sensing the transmitter generates entangled signal and idler modes, transmits
the signal to probe an object, and retains the idler for the quantum receiver.
Willie aims to detect the signal embedded in a bright noise background but the
statistical fluctuation of the noise impedes his capability to observe the
sensing attempt. Willie’s error probability $\mathbb{P}_{e}^{\rm(w)}\geq
1/2-\epsilon$, where $\epsilon$ is referred to as the covertness parameter.
Within the relevant parameter region, one can show that

$\epsilon\propto\frac{\sqrt{M}N_{S}}{N_{B}},$ (35)

where $M$ is the number of probe modes proportional to the integration time,
$N_{S}$ is the per-mode mean photon number for the probe, and $N_{B}\gg 1$ is
the per-mode mean photon number for the noise background. Eq. (35) suggests
that $\sqrt{M}N_{S}={\rm constant}$ is needed to fix the covertness parameter
at a given background noise level, leading to ${\rm SNR}\propto\sqrt{M}$,
known as the square-root law of covert communication and sensing [296]. In
analogy to QI, the quantum receiver takes a joint measurement on the received
signal and the idler to infer the physical properties of the object under
investigation. The estimation precision is derived by the quantum Cramér-Rao
bound:

$\delta\theta^{2}_{Q}\geq\frac{1}{MF},$ (36)

where $F$ is the quantum Fisher information [297].

An entanglement-enhanced covert sensing experiment built on a tabletop
platform was recently reported [276], as depicted in Fig. 17a. The performance
of entanglement-enhanced covert sensing was benchmarked against that of covert
sensing based on classical light sketched in Fig. 17b. In the entanglement-
based experiment, the transmitter consisted of a PPLN crystal that produced
non-degenerate signal and idler modes. The signal served as the probe to
measure the phase shift imparted by an object. The environmental noise was
emulated by mixing thermal light with the signal. Willie captured a portion of
the signal and noise power that did not reach the quantum receiver and aimed
to measure its power to infer the sensing attempt. Different from the optical
QI target detection experiment with an OPAR [285], a PCR was implemented in
the covert sensing experiment. The PCR comprised a second PPLN that generatd a
phase-conjugate beam of the signal returned from the environment. The phase-
conjugate beam interfered with the retained idler on a 50:50 beam splitter
followed by two photodetectors in a balanced configuration. The classical
covert sensing scheme employed a thermal light source and a balanced homodyne
receiver while Willie’s setup remained unchanged. Figure 17c plots Willie’s
detection error probability as a function of the interrogation time when the
square-root law is obeyed (black) or violated (red), with the corresponding
estimation mean squared error shown in the inset. One can observe that the
mean squared error drops at a slower rate when the square-root law is obeyed,
unveiling a tradeoff between performance and security.

Figure 17: Entanglement-enhanced covert sensing experiment. (a) Diagram of
entanglement-enhanced covert sensing experiment based on a non-degenerate
entanglement source and a PCR. (b) Diagram of classical covert sensing
experiment based on a thermal light source and a balanced homodyne receiver.
(c) Experimental data of Willie’s detection error probability when the square-
root law is obeyed (black) or violated (red). Dots: experimental data. Red and
black curves: theory. Blue curves: lower bound for Willie’s detection error
probability. Inset: corresponding mean squared errors for classical (solid)
and entanglement-enhanced (dotted) covert sensing, highlighting different
slopes when the square-root law is obeyed and violated. Figures reprinted from
Ref. [276].

##  5 Quantum Imaging with Entangled Photons

Another fast-growing field of entanglement-based QIT is quantum imaging, which
promises enhanced resolutions at ultra-low light intensities and measurements
in spectral regions where cameras do not exist. With the development of
efficient entanglement sources and advanced measurement devices, a multitude
of encouraging demonstrations of quantum imaging have been conducted in recent
years. In the following, we will provide a brief introduction to the most
important features required for a quantum-light source to be used in nearly
all quantum-imaging protocols. We will then highlight a few significant recent
developments in the field of quantum imaging. For more extensive background
information, we encourage interested readers to refer to recent review
articles on quantum imaging, such as Refs. [298, 299, 300].

###  5.1 Entangled-Photon Sources for Quantum Imaging

Nearly all entanglement-based quantum-imaging protocols hinge on the position
and momentum degrees of freedom, as introduced in Sec. 2.2.6. The most common
method to generate a two-photon position-momentum entangled state is through
the SPDC process, in which a pump photon is down-converted into an entangled
pair of photons.

A vital property for imaging is the spatial correlation in the position-
momentum entanglement. As described in Sec. 2.2.6, a paraxial Gaussian pump
beam and a specific combination of pump, signal, and idler frequencies (i.e.,
$\omega_{P}=\omega_{S}+\omega_{I}$) generate entangled states expressed as

$\displaystyle\ket{\psi}_{\mathbf{q}}=\int
d\mathbf{q}_{S}d\mathbf{q}_{I}\Phi(\mathbf{q}_{S},\mathbf{q}_{I})\ket{\mathbf{q}_{S}}\ket{\mathbf{q}_{I}}.$
(37)

Under ideal conditions that lead to perfect anti-correlations in transverse
momenta, i.e.,
$\Phi(\mathbf{q}_{S},\mathbf{q}_{I})\approx\delta(\mathbf{q}_{S}+\mathbf{q}_{I})$,
one can detect, in the far-field of the crystal, a photon in precisely the
opposite transverse position of its entangled partner.

In the near-field, which corresponds mathematically to the Fourier transform
and experimentally to imaging the crystal onto the observation plane, the
joint spectral amplitude can be approximated as
$\Phi(\mathbf{x}_{S},\mathbf{x}_{I})\approx\delta(\mathbf{x}_{S}-\mathbf{x}_{I})$,
where $\mathbf{x}_{S}$ and $\mathbf{x}_{I}$ are the transverse position
vectors. This approximation implies that detecting one photon at a particular
transverse position requires the entangled partner photon to be at the same
position, which can be utilized for imaging purposes.

For many quantum imaging tasks, the achievable resolution is directly linked
to the strength of correlations, making it critical to obtain a state that
closely resembles this ideal situation. To achieve strong correlations, the
pump should be collimated as much as possible, indicating that larger
apertures are advantageous. Additionally, the crystal should be as thin as the
required detection rate allows. A highly collimated pump field minimizes the
transverse momentum spread, while a thin crystal allows for a larger bandwidth
of transverse momenta in the down-converted photons. When both conditions are
met, the correlations are maximized, resulting in enhanced imaging resolution.
Optimizing quantum imaging parameters shares many similarities with preparing
high-dimensional entangled states, and research findings in either field often
synergistically advance the other.

In a broad sense, entanglement-based quantum imaging protocols can be divided
into three categories: correlation-based imaging, nonlinear interference-based
imaging, and imaging using photon-number entangled states. The following
sections will delve into each category.

###  5.2 Correlation-Based Quantum Imaging

In correlation-based quantum imaging protocols, the image is obtained by
analyzing correlation between measurements, specifically coincidences between
two or more involved photons. While these protocols impose high technological
requirements on the imaging devices, they offer the advantage of improved
signal-to-noise ratios, as discussed below.

One of the most popular imaging methods in the field of quantum imaging is
known as ghost imaging falling into the category of correlation-based imaging
protocols. In a ghost imaging protocol, the information on the imaged object
is not obtained from the detection of individual photons but rather from the
correlations between different measurements.

In ghost imaging, a spatially entangled photon pair is split into two beams of
single photons: a signal beam and an idler beam. The signal photon is used to
illuminate the object to be imaged, and after transmission or reflection, it
is detected by a single-pixel detector, known as a bucket detector, that does
not provide any spatial information. The idler photons, on the other hand, do
not interact with the object and are recorded by a spatially resolving
measurement, such as raster scanning with a single-pixel detector or using a
single-photon sensitive camera. Fig. 18a depicts a schematic of a ghost
imaging protocol.

The camera recordings alone do not directly produce an image but rather show
the incoherent mixture of all possible detection data, representing the cross-
section of the SPDC emission. However, when the camera recordings of the idler
photons are correlated with the bucket detector events from the signal photons
(i.e., the camera only records the idler photons when their corresponding
signal photons are measured), the image is reconstructed based on the spatial
correlations.

An intuitive way to visualize such a ghost imaging protocol, and many other
protocols based on photon pairs, is through the Klyshko picture [301], in
which the SPDC crystal generating entangled photons is replaced by a mirror,
and the bucket detector in the signal-photon arm is replaced by a light
source. Using this arrangement, the same image recorded by the camera in the
idler arm can be observed [302].

Figure 18:  Correlation-based quantum imaging. (a) Schematic of a correlation-
based imaging protocol, where the object is placed in the path of the signal
photon (red), which are not directly imaged. The image is obtained by
recording the idler photon (green) and correlating the camera exposures to the
detections of its entangled partner. (b) Sketch of an experimental setup for
quantum ghost imaging with a triggered ICCD camera. Results obtained with this
setup are displayed in (c), showing that by the virtue of the entanglement
images can be obtained in the image plane (left) and momentum plane (right)
depending on the location of the object and camera after the crystal. (d)
Quantum holography using four phase-shifted correlation images (left), from
which the phase pattern can be retrieved (right). (e) An image encoded into
entangled photons can be distilled through correlation measurements, even when
superposed with an image encoded in a strong classical light field. Results
show the mixed images (left, A), the distilled quantum image (middle, B), and
the classical image obtained through subtracting the two (right, C). (b, c)
reprinted from Ref. [303]. (d) reprinted from Ref. [304]. (e) reprinted from
Ref. [305].

The first demonstration of ghost imaging in 1995 [306] utilized a scanning
approach to record the idler photons due to the limited camera technology at
that time. Thanks to the advancement of modern cameras capable of nanosecond
triggering and single-photon sensitivity, subsequent demonstrations have
showcased the power of spatial correlations in ghost imaging [303, 307, 308,
309]. Figure 18b illustrates a sketch of a ghost imaging experiment.

One intriguing feature of entanglement-based ghost imaging is the ability to
obtain either the image or its inverted counterpart depending on whether the
object and camera are placed in the image or Fourier plane of the crystal,
which corresponds to utilizing position correlations or momentum anti-
correlations, respectively, as shown in Figure 18c. Another significant
advantage of ghost imaging is the flexibility to employ two different
wavelengths for the signal and idler photons, known as two-color ghost
imaging. This allows for wavelength adjustment based on detector and camera
efficiencies or noise characteristics [310].

While entanglement was initially considered a critical ingredient for ghost
imaging, subsequent theoretical and experimental studies raised questions
about its necessity. It was shown that thermal light sources and properly
utilized coherent light can also be employed in correlation imaging protocols,
exhibiting similar features (for further details on the debate and classical
ghost imaging results, interested readers can refer to [311, 312, 313, 308]
and references therein). These debates helped to identify specific quantum