advantages. One evident quantum feature is the possibility of achieving
perfectly correlated images in the near and far fields, resulting in inverted
images [303]. Moreover, entanglement-based ghost imaging enables signal-to-
noise ratios beyond what is attainable with classical correlation imaging,
thanks to the perfect correlations of SPDC and its quantum statistics [314,
315]. In fact, by employing a similar protocol to the aforementioned setup but
imaging both entangled photons with the same camera, sub-SNL imaging has been
demonstrated [316]. Overall, the advantage of entanglement-based ghost imaging
is particularly significant at very low light levels, as quantum statistics
offer a means to surpass the limitations of classical Poissonian statistics,
which exhibit increasing uncertainty in the few-photon regime.

An interesting question regarding the spatial resolution limits arises in
ghost imaging protocols: which of the three wavelengths for pump, signal, and
idler defines the diffraction limit? It has been found that the so-called
effective de Broglie wavelength, which is half the wavelength of the photon
pair, sets a fundamental limit [317]. Additionally, it has been demonstrated
that the resolution is not only limited by the point spread function of the
camera system but also by the quality of the spatial correlations [318].

In recent decades, the concept of utilizing quantum correlations for imaging
applications has been extended to various regimes and applications. Here are a
few noteworthy examples:

1\. Quantum Secure Imaging: Quantum correlations, in combination with
polarization and the BB84 quantum cryptography protocol, have been employed to
develop a quantum secure imaging protocol. This protocol enables the detection
of any unwanted changes to the image, such as those caused by eavesdroppers
[319].

2\. Combination with Interaction-Free Imaging: Correlation-based quantum
imaging has been successfully combined with interaction-free imaging, another
popular quantum imaging technique. By incorporating spatially-resolved
interference effects and correlation-based ghost imaging, it becomes possible
to obtain both intensity and phase images of an object. This counter-intuitive
approach allows imaging through the detection of photons that cannot have
interacted with the object [320].

3\. Image Recognition: Recent research has shown that by structuring the pump
beam and employing special pattern projection techniques during the detection
process, an image recognition protocol based on ghost imaging principles can
be realized. This approach offers promising possibilities for image
recognition tasks [321].

4\. Four-Photon Entanglement: Entanglement-based imaging has been extended to
four-photon states using entanglement swapping. By projecting two photons onto
an anti-symmetric state, it has been demonstrated that the image placed into
one arm of a photon pair can be swapped to the other pair while being
contrast-inversed, showcasing the complexity of achieved quantum imaging
operations [322].

5\. Holographic Tasks: Quantum correlation-based imaging concepts have been
adapted for holographic tasks. Fig. 18d presents some of the results. In such
a scheme, spatial correlations derived from SPDC are utilized to perform
holographic reconstructions of birefringent images encoded in polarization
entangled states. This approach offers insensitivity to spatial phase
disturbances, enhanced spatial resolution due to reduced de Broglie wavelength
of the bi-photon state, and strong resistance to classical noise [304].

6\. Quantum Image Distillation: Quantum correlation-based imaging techniques
have been applied to quantum image distillation. This refers to the retrieval
of images based on quantum correlations (not necessarily entanglement) from a
strong uncorrelated noisy background or when overlaid with a strong classical
light field carrying its own imaging information. Quantum image distillation
experiments have demonstrated the capability to extract quantum images from
such challenging scenarios [305] (see Fig. 18e for experimental results).

These examples illustrate the diverse range of applications and advancements
in quantum imaging based on correlation and entanglement principles.

###  5.3 Quantum Imaging Using Nonlinear Interference

While correlation-based quantum-imaging protocols offer numerous advantages,
they also pose challenges, particularly associated with the camera technology,
as many of these protocols require cameras to be single-photon sensitive with
high timing resolution or necessitate special post-processing for correlation
extraction. To combat these challenges, a new quantum-imaging approach based
on nonlinear interferometry has emerged that leverages entanglement without
the need for correlation measurements. In nonlinear interferometry, nonlinear
crystals are used in lieu of beamsplitters in an interferometer [323],
yielding nonlinear interference that has found various applications in quantum
optics, most notably in quantum-enhanced phase sensing [324].

Quantum imaging based on nonlinear interference builds on the demonstration by
Zou, Wang, and Mandel to showcase the counterintuitive effect of induced
coherence [325, 326]. The method involves coherently pumping two SPDC crystals
in a way that only one pair is generated. The signal photon from the first
crystal is then directed into the second crystal, aligned to perfectly overlap
with the path of the signal photon from the second crystal, as sketched in
Fig. 19a. In measuring the signal photon, it is impossible to distinguish
which SPDC process has occurred, thereby also inducing coherence in the idler
photon. As a result, the idler photons’ paths from both crystals can be
brought to interference using a beamsplitter. Importantly, the interference
occurs regardless of whether the signal photon is detected, eliminating the
need for correlations. Remarkably, an object placed in the signal path between
the two crystals manifest as a reduction in the interference visibility in the
detection of the idler photons. Therefore, the image of the object can be
directly retrieved without correlation measurements, a technique known as
“quantum imaging with undetected photons” [326]. Since the setup acts as an
interferometer, phase images, in addition to amplitude images, can be captured
by modifying the interference pattern (Fig. 19c-e). Similar to correlation-
based imaging, the photon interacting with the object does not need to have
the same wavelength as the photon recorded by the camera. Additionally, it
does not require detecting the photons directly interacting with the object,
offering the potential to image objects at wavelengths where cameras and
detectors may not exist.

Figure 19:  Quantum imaging based on nonlinear interference. (a) Schematic of
a quantum imaging protocol based on nonlinear interference, where the object
is placed in the path of the signal photon (red) between the two crystals. The
image is obtained by recording the interference of the idler photon (green)
without the need for correlation measurements. (b) Experimental setup for
quantum imaging using nonlinear interference implemented in the commonly used
Michelson geometry to perform mid-infrared microscopy. Figure reprinted from
Ref. [327]. (c) Captured interference images at both output ports of the
beamsplitter in the idler arm when an intensity mask is placed into the arm of
the signal photon. Figure reprinted from Ref. [326]. (d) Mid-infrared
absorption (left) and phase (right) microscopy images recorded with the idler
at around 800 nm. Scale bar, 200 $\mu$m. Figure reprinted from Ref. [327]. (e)
Hyperspectral absorption microscopy images in the mid-infrared for signal
(idler) wavelengths of 3.18 $\mu$m (653.2 nm) on the left and 3.32 $\mu$m
(638.8 nm) on the right showing the sensitivity of the sample to different
probe wavelengths. Figure reprinted from Ref. [328].

Since the introduction of quantum imaging with undetected photons in 2014, the
concept has undergone significant developments and expansions in various
settings. These advancements include achieving video rate recording speed
[329], exploring classical analogs [330], extending its application to sensing
in the THz frequency regime [331], utilizing it for mid-infrared microscopy
[327, 328] (see Fig. 19b), and employing it in holography tasks [332].
Furthermore, it has been applied to spectroscopy [333, 334] and integrated
into frequency-domain optical coherence tomography [335]. In the imaging
domain, recent advancements have combined it with single pixel imaging and
interaction-free measurement techniques, leading to remarkable outcomes [336].
Leveraging nonlinear imaging enables spatially resolved imaging without the
need for cameras or raster scanning, without direct interaction with the
object, and in wavelength regions where good detectors do not exist.

A current area of extensive study in quantum imaging with undetected photons
revolves around understanding the resolution limits of the protocol. Research
has shown that resolution strongly depends on the quality of momentum
correlations, specifically the position-momentum entanglement of the SPDC
process [337]. In the paraxial regime, resolution is determined by the
wavelength of the undetected photon. However, in tightly focused SPDC
processes with significant non-paraxial effects, quantum imaging with
undetected photons becomes fundamentally diffraction-limited by the longer
wavelength between the signal and idler [338]. Nonetheless, proposals for
achieving subdiffraction imaging with undetected photons have been put forward
[339].

Finally, it is worth mentioning that while the concept of nonlinear
interference can be connected to different protocols for entanglement
generation [164], and the setup has been employed to quantify momentum
correlations [340], entanglement may not be indispensable for such protocols
as position correlations alone appear sufficient for image retrieval [341].
Consequently, the role of entanglement and its advantages in the domain of
quantum imaging with undetected photons requires further investigation [342].

###  5.4 Imaging with Entangled Photon-Number States

In the realm of quantum imaging, it is worth exploring the imaging-related
advantages offered by entangled photon-number states, commonly referred to as
N00N states (see also Sec. 3). These states have been shown to surpass the
standard diffraction limit of light wavelength $\lambda$ by a factor of $N$.

One early example of utilizing N00N states in imaging applications is quantum
lithography sketched in Fig. 20a, where an interference pattern formed by $N$
photons is generated through the creation of N00N states in different paths,
resulting in $N$ photons in a superposition between the two paths [343]. By
overlapping these paths with a slight tilt and measuring two-photon
correlations in space, the resulting interference fringe pattern becomes $N$
times denser than the classical interference pattern produced by light of the
same wavelength [344]. This phenomenon is the spatial equivalent of the phase
super-sensitivity exhibited by N00N states [239]. Notably, this phase super-
sensitivity has been combined with polarization microscopy to image
birefringent materials with enhanced sensitivity [252].

More recently, the concept of N00N states has been extended to transverse
spatial modes instead of paths, enabling super-resolution measurements shown
in Fig. 20b [345]. This extension has led to improvements in angular
resolution for orbital angular momentum modes [246] and enhancements in
longitudinal resolution for radial modes [346]. Interestingly, the latter
study also revealed that the commonly used intuitive explanation for the
$N$-fold enhancement using the photonic de Broglie wavelength of the
$N$-photon states, i.e., $\lambda/N$ [347], was only valid for plane waves.
When N00N states implemented through spatial modes are considered, the simple
notion of the photonic de Broglie wavelength needs to be reevaluated [346].

Figure 20:  Quantum imaging using entangled photon-number states. (a)
Schematic for quantum lithography based on 2-photon path-entangled N00N states
generated through parallel pumping of two crystals (left). When brought to
interference on a beam splitter (BS2), two-photon correlations together with
optical centroid measurements result in a high-visibility fringe pattern with
double the frequency (bottom right) compared to single-photon interference
fringes (top right). Figure reprinted from [348]. (b) Schematic for super-
resolution imaging of an object illuminated with 405 nm light and imaged using
810 nm photons with the optical centroid measurement technique (top). The
bottom images show recordings using spatially coherent light at 810 nm,
spatially incoherent light at 810 nm, two-photon states at 810 nm, and
coherent light at 405 nm, respectively (from left to right). Figures reprinted
from [345].

In general, quantum-light sources with spatial correlations are known to
enable resolution enhancement in imaging protocols, achieving the Heisenberg
limit and providing an $N$-fold resolution improvement over the diffraction
limit [349]. One can intuitively consider the spatially correlated state
obtained from ideal SPDC, as described in Eq. (37), as a multidimensional bi-
photon N00N state where the two photons are in a superposition of all possible
locations. This concept holds the potential for super-resolution imaging that
surpasses the classical limit by a factor of two. However, realizing this
enhancement is challenging due to the limited availability of photon-number-
resolving detectors and the exponential decrease in efficiency in an attempt
to localize $N$ photons within a sub-diffraction-limited area. To overcome
this limitation, an optical centroid measurement was proposed [350] and
subsequently implemented for photon numbers up to 4 [348, 351]. In this
protocol, the detection does not rely on registering $N$-photon coincidence
events at a specific location, but rather evaluates all events, particularly
their centroid, which represents the mean position of all detected photons.
Through appropriate post-processing, sub-wavelength fringes and quantum-
enhanced resolution can be achieved. Similar to other quantum-imaging
techniques, recent advancements in the camera technology and data processing
approaches have enabled successful super-resolution measurements [345, 352,
247, 353].

The aforementioned results represent promising initial strides toward the
development of quantum imaging protocols that benefit from entanglement.
However, for these techniques to be applicable in real-world scenarios,
various technological challenges must be overcome, including improved spatio-
temporal resolution and enhanced single-photon measurements.

##  6 Entanglement-Enhanced Light-Matter Interactions and Spectroscopy

Spectroscopy techniques play a pivotal role in extracting valuable information
about the energy dynamics and chemical structure of various substances,
ranging from small molecules to large photosensitive complexes [354, 355, 356,
357]. Accurate identification of unknown molecular samples has been
fundamental in advancing technologies used in modern society, including water
and air pollution monitoring, homeland security, and healthcare applications
[358]. Traditionally, these techniques rely on the use of laser light in the
optical regime. However, recent investigations have revealed the potential of
nonclassical light, such as entangled photon pairs, to open new and exciting
avenues in experimental nonlinear spectroscopy [359, 360, 11, 361, 362, 363,
364].

Figure 21: Schematic of an ETPA experiment. Entangled photons are produced by
non-degenerate SPDC. In this process, a nonlinear crystal is pumped with light
of central frequency $\omega_{P}$ to produce frequency-correlated photons with
central frequencies $\omega_{S}$ and $\omega_{I}$. A tunable delay $\tau$ is
introduced into the path of one of the photons. Finally, both photons impinge
on the sample under investigation, producing a ETPA signal either as a ration
between the generated and the transmitted photon pairs (transmission-based
configuration) or as the fluorescence produced by the sample after the two-
photon excitation (fluorescence-based configuration). Figure reprinted from
Ref. [365].

Remarkably, the time- and frequency-correlations exhibited by entangled photon
pairs have facilitated the observation of intriguing phenomena known as
entangled two-photon absorption (ETPA), as schematically sketched in Fig. 21.
These correlations have played an important role in predicting and exploring
various fascinating effects, including the linear dependence of the two-photon
absorption rate on the photon flux [366, 367, 368]. Such observations have led
to the discovery of phenomena such as two-photon-induced transparency [369,
370], virtual-state spectroscopy [371, 372, 373, 374, 375, 376, 377, 378, 379,
380, 381, 382, 383, 384, 365], the induction of usually forbidden atomic
transitions [385], the manipulation of matter’s quantum pathways [386, 387,
388, 389, 390, 391], the analysis of many-body states [392], and the control
of molecular processes [393, 394]. Notably, one of the most appealing features
of ETPA is the linear relationship between two-photon absorption and the
incident photon flux. This implies that multi-photon processes can be
effectively excited using low-power, continuous-wave, single-frequency laser
sources [32]. Such findings hold significant promise for the development of
novel compact and cost-effective quantum-enhanced spectrometers.

While the theoretical foundation for the linearization of ETPA has been
established [366, 367], there is an ongoing debate regarding the actual
quantum enhancement that such a process can offer to spectroscopy [395, 396,
397]. This discussion has sparked significant experimental work to investigate
ETPA and its predicted linear behavior [398, 399, 400, 401, 402, 403].
Interestingly, some of these studies have confirmed the predictions made in
theory papers from the 1980s-1990s, while others suggest that previous results
may have overestimated the potential enhancement that ETPA could provide for
future quantum-enhanced spectroscopy. In essence, the question of whether ETPA
can be observed using standard entangled-photon sources, such as those based
on SPDC (see Sec. 2.2 for an introduction), and common fluorophores like
Rhodamine and Tetraporphyrins, remains open. There is a need to establish a
series of tests to probe genuine ETPA phenomena [404]. This is particularly
important considering recent arguments that single-photon-loss mechanisms,
such as scattering [405] and hot-band absorption [406], may mimic the expected
linear absorption behavior of entangled photons.

In this section, we aim to equip the reader with fundamental tools to delve
into the research field of ETPA. We begin by presenting a simple theoretical
model that describes the interaction between entangled light and matter. This
model serves as a basis for designing practical schemes for experimental
implementation of entangled-photon absorption spectroscopy. Subsequently, we
delve into current endeavors focused on the development of innovative
applications enabled by ETPA. By providing this foundation, we hope to
facilitate further exploration and advancement in the field of ETPA research.

###  6.1 Theoretical Description of Entangled Two-Photon Absorption

We begin by examining the interaction between an absorbing medium and a two-
photon optical field, denoted as $\ket{\Psi}$. The interaction is described by
the interaction Hamiltonian
$\hat{H}_{\text{int}}(t)=\hat{d}(t)\hat{E}^{(+)}(t)$, where $\hat{d}$ is the
dipole moment operator and $\hat{E}^{(+)}(t)$ represents the positive-
frequency part of the electric field operator. Notably, the total field
comprises both the signal ($S$) and idler ($I$) fields. Each electric field
operator is given by:

$\hat{E}^{(+)}_{S,I}(t)=\int
d\omega_{S,I}\sqrt{\frac{\hbar\omega_{S,I}}{4\pi\epsilon_{0}cA}}\hat{a}_{S,I}(\omega_{S,I})e^{-i\omega_{S,I}t},$
(38)

where $c$ represents the speed of light, $\epsilon_{0}$ is the vacuum
permittivity, $A$ is the effective area of the field interacting with the
sample, and $\hat{a}_{S,I}(\omega_{S,I})$ corresponds to the annihilation
operator of a photonic mode with frequency $\omega_{S,I}$.

We consider the initial state of the absorbing medium to be its ground state
$\ket{g}$, with energy $\varepsilon_{g}$. Using time-dependent second-order
perturbation theory, we can calculate the probability of the medium being
excited to a doubly-excited final state $\ket{f}$, with energy
$\varepsilon_{f}$, through a two-photon absorption process. This probability
is given by:

$P_{g\rightarrow
f}=\left|\frac{1}{\hbar^{2}}\int_{-\infty}^{\infty}dt_{2}\int_{-\infty}^{t_{2}}dt_{1}\mathcal{D}(t_{1},t_{2})\mathcal{E}(t_{1},t_{2})\right|^{2},$
(39)

where

$\displaystyle\mathcal{D}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\sum_{j=1}d_{fj}d_{jg}e^{-i(\varepsilon_{j}-\varepsilon_{f})t_{2}}e^{-i(\varepsilon_{g}-\varepsilon_{j})t_{1}},$
(40) $\displaystyle\mathcal{E}(t_{1},t_{2})$ $\displaystyle=$
$\displaystyle\bra{0}\hat{E}_{S}^{+}(t_{2})\hat{E}_{I}^{+}(t_{1})\ket{\Psi}+\bra{0}\hat{E}_{S}^{+}(t_{2})\hat{E}_{I}^{+}(t_{1})\ket{\Psi},$
(41)

Here, $d_{fj}=\bra{f}\hat{d}\ket{j}$ and $d_{jg}=\bra{j}\hat{d}\ket{g}$ are
the transition matrix elements of the dipole moment operator, representing the
coupling between the states $\ket{j}$ and $\ket{f}$, as well as $\ket{j}$ and
$\ket{g}$. The sum over $j$ in Eq. (40) indicates that the two-photon
excitation of the medium occurs through intermediate states $\ket{j}$ with
energy eigenvalues $\varepsilon_{j}$. Furthermore, since we are interested in
the absorption of the two-photon states, the final state of the field is
assumed to be the vacuum state.

We extend the model by assuming that the two-photon state is generated through
type-II SPDC [407]. In this process (see Fig. 21), a second-order nonlinear
crystal of length $L$ is pumped by a Gaussian pulse with temporal duration
$T_{p}$, producing two photons with orthogonal polarizations: the signal
photon and the idler photon. To ensure time indistinguishability of the paired
photons, the signal and idler photons undergo polarization interchange and
pass through a similar crystal of length $L/2$. Finally, an external time-
delay is introduced between the photons. The resulting two-photon state is
given by [374]:

$\displaystyle\ket{\Psi}$ $\displaystyle=$
$\displaystyle\left(\frac{T_{p}T_{e}}{2\pi\sqrt{\pi}}\right)^{1/2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega_{S}d\omega_{I}\exp\left[-T_{P}^{2}(\omega_{P}-\omega_{S}-\omega_{I})^{2}\right]$
(42)
$\displaystyle\times\text{sinc}\left[T_{e}(\omega_{I}-\omega_{S})\right]e^{i\omega_{I}\tau}\hat{a}^{\dagger}_{S}(\omega_{S})\hat{a}^{\dagger}_{I}(\omega_{I})\ket{0},$

where $\omega_{j}$ ($j=P,S,I$) represents the frequencies of the pump, signal,
and idler fields, respectively. The term
$\text{sinc}[T_{e}(\omega_{I}-\omega_{S})]$ ensures the temporal overlap
between the signal and idler photons. The entanglement or correlation time
between the photon pairs is given by:

$T_{e}=\frac{(N_{S}-N_{I})L}{4},$ (43)

where $N_{S,I}$ are the inverse group velocities of the signal and idler
photons, respectively.

By substituting Eqs. (40)–(42) into Eq. (39), we can derive the expression for
the two-photon absorption probability as follows:

$\displaystyle P_{g\rightarrow f}(T_{e},T_{p},\tau)$ $\displaystyle=$
$\displaystyle\frac{4T_{p}}{T_{e}}\frac{\sqrt{2\pi}\omega_{0}^{2}}{\hbar^{2}\epsilon_{0}^{2}c^{2}A^{2}}\exp\left[-2T_{p}^{2}(\omega_{p}-\varepsilon_{f})\right]$
(44) $\displaystyle\times$
$\displaystyle\left|\sum_{j}d_{fj}d_{jg}\left(\frac{1-e^{-i(\varepsilon_{j}-\omega_{0})(2T_{e}-\tau)}}{\varepsilon_{j}-\omega_{0}}+\frac{1-e^{-i(\varepsilon_{j}-\omega_{0})(2T_{e}+\tau)}}{\varepsilon_{j}-\omega_{0}}\right)\right|^{2},$

where we have assumed that the photons are degenerate with central wave-packet
frequencies $\omega_{S}^{0}=\omega_{I}^{0}=\omega_{0}=\omega_{P}/2$. In this
expression, $\omega_{P}$ represents the central frequency of the pump field.
We have also set $\varepsilon_{g}=0$ by displacing the energy levels for
simplicity. Notably, Eq. (44) reveals two important features: (1) The
probability of entangled two-photon absorption is determined by the coherent
superposition (interference) of all possible pathways involving intermediate
states $j$ for two-photon excitation, and (2) the ETPA signal can be
controlled by adjusting the external delay between the paired photons, denoted
by $\tau$.

###  6.2 Entangled-Photon Absorption Spectroscopy: First Approach

In 1997, Saleh _et al._ introduced a technique called “entangled-photon
virtual-state spectroscopy” based on the ETPA probability described by Eq.
(44) [371]. This technique utilizes the interference pattern observed in the
ETPA signal as a function of the external delay $\tau$ to extract
spectroscopic information about the sample. Figure 22a illustrates an example
of the predicted ETPA signal for the 1s $\rightarrow$ 2s transition of atomic
hydrogen, showcasing its non-monotonic behavior that encodes the sample’s
spectral information.

To extract this spectroscopic information, Saleh and colleagues proposed
performing an average of Eq. (44) over a range of values of $T_{e}$, yielding
the weighted-and-averaged ETPA transition probability [371]:

$\overline{P}(\tau)=\frac{1}{T}\int_{T_{e}^{\rm min}}^{T_{e}^{\rm
max}}P_{g\rightarrow f}(T_{e},T_{p},\tau)T_{e}dT_{e},$ (45)

where $T=T_{e}^{\rm max}-T_{e}^{\rm min}$. To perform this average
experimentally, a set of measurements with different values of $T_{e}$ is
required. While this can be technically challenging, the parameter $T_{e}$ can
be adjusted by various methods depending on the type of entangled-photon
source used. For example, in type-I SPDC (parallel-polarized photons), the
width of the pump beam can be modified to change $T_{e}$ [408], whereas in
type-II SPDC, the crystal length can be varied as $T_{e}$ is linearly
proportional to it [409]. As such, a set of wedge-shaped nonlinear crystals
can be used to achieve the desired range of $T_{e}$.

Figure 22: Entangled two-photon absorption spectroscopy. (a) Calculated non-
monotonic absorption probability as a function of the delay between paired
photons. (b) Fourier transform of the normalized weighted-and-averaged ETPA
transition probability as a function of the energy mismatch
$\Delta^{(j)}=\varepsilon_{j}-\omega_{0}$. The $y$-axis is shown on a
logarithmic scale. Figures reprinted from Ref. [374].

Figure 22b shows the Fourier transform of the normalized weighted-and-averaged
ETPA transition probability for atomic hydrogen. Note the presence of
different peaks, whose locations signal the frequencies of the intermediate
$p$-transitions through which the 1s $\rightarrow$ 2s two-photon excitation
takes place. These results demonstrate the potential of ETPA spectroscopy for
retrieving relevant information about the electronic structure of arbitrary
samples. Remarkably, even in the case when the SPDC crystal is pumped by a
continuous-wave, monochromatic laser, i.e., $T_{p}\rightarrow\infty$,
information regarding the various intermediate levels is still accessible.
However, there are challenges associated with the original proposal for ETPA
spectroscopy. First, it requires performing multiple experiments with two-
photon states bearing different temporal correlations, which necessitates
having hundreds of entangled-photon sources available. Second, as stated by
the authors of Ref. [371], it requires _a priori_ knowledge of the lowest-
lying intermediate electronic energy level of the system under study, which
limits the applicability of ETPA spectroscopy for probing unknown samples.

In view of the aforementioned challenges, new approaches to ETPA spectroscopy
have been proposed over the years. In the following sections, we will discuss
some alternative schemes for obtaining information about the energy dynamics
and electronic structure of unknown samples.

###  6.3 ETPA Spectroscopy with Varying Entanglement Time and Pump Wavelengths

Entangled photons possess unique nonclassical spectral and temporal features,
which make them excellent probes for monitoring the excitation dynamics of
molecular complexes. In striking contrast to classical light, entangled
photons can circumvent the time-frequency uncertainty principle, meaning they
exhibit independent temporal and spectral characteristics.

As described earlier, in SPDC, a photon from the pump pulse with frequency
$\omega_{P}$ is down-converted in a birefringent crystal into a pair of
entangled photons with frequencies $\omega_{S}$ and $\omega_{I}$. Energy
conservation ($\omega_{P}=\omega_{S}+\omega_{I}$) induces a time-energy
correlation between the pair of photons, where the bandwidth of the pump pulse
limits the width of the sum of the down-converted photons’ frequencies.
However, different group velocities within the SPDC crystal can result in a
broad bandwidth of the individual photons, which can exceed the bandwidth of
their sum. This group velocity difference determines the entanglement time
between the photons [see Eq. (43)], and it sets a limit on the time delay
between the absorption events of the correlated photons. Remarkably, this
property has been successfully utilized by Schlawin _et al._ [390] to reveal
exciton dynamics in molecular complexes.

Figure 23: Control of exciton dynamics via ETPA. (a) Bacterial RC comprising
12 single-exciton and 41 two-exciton states, labeled as $\ket{e}$ and
$\ket{f}$, respectively. (b) Two-photon absorption probability of state
$f_{11}$ as a function of the entanglement time $T_{e}$ and the central pump
frequency $\omega_{p}$. (c) The same for state $f_{23}$. Note that the
entanglement time between photons defines whether two-photon absorption takes
place via transport or direct excitation. Figures reprinted from Ref. [390].

Consider the system shown in Fig. 23a, which depicts the bacterial reaction
center (RC) of _Blastochloris viridis_. The electronic structure of the RC can
be modeled by including 12 single-exciton and 41 two-exciton states, labeled
as $\ket{e}$ and $\ket{f}$, respectively. By varying the entanglement time and
the pump frequency, one can reveal the influence of exciton dynamics in the
single-exciton manifold of the RC.

If a two-photon absorption peak becomes stronger with increasing $T_{e}$, it
indicates that the intermediate state is populated by transport rather than by
direct excitation. This can be understood by monitoring the population (or,
equivalently, the absorption probability) of the two-exciton state $f_{11}$,
with frequency $\omega_{f_{11}}=22,160\;\text{cm}^{-1}$. This state is mostly
excited through the single-exciton state $e_{5}$, which can be populated by
photon absorption or exciton transport.

As shown in Fig. 23b, when $T_{e}\rightarrow 0$, only direct excitation
$g\rightarrow e_{5}\rightarrow f_{11}$ survives. However, as $T_{e}$ is
increased, a number of resonances between $22,000$ and
$24,000\;\text{cm}^{-1}$ appear. This is due to transport processes within the
single-exciton manifold. It is particularly interesting to look at the
$24,000\;\text{cm}^{-1}$ resonance, which corresponds to the excitation energy
of the $f_{23}$ level. This state is solely excited through the short-lived
single-exciton state $e_{6}$, which decays within $100$ fs to the state
$e_{5}$. Figure 23c shows that by increasing the entanglement time, direct
excitation of the state $f_{23}$ is lost to the $f_{11}$ level due to the
transport process between the $e_{6}$ and $e_{5}$ single-exciton levels,
sketched in the inset in Fig. 23c.

The aforementioned results demonstrate the potential of entangled photon
pulses for resolving specific single-exciton pathways through precise
manipulation of two-photon states. By varying the entanglement time and the
pump frequency, it becomes possible to selectively probe and characterize the
dynamics of single-exciton states in complex systems. This level of control
and resolution offered by entangled photon pulses opens up new avenues for
studying and understanding the intricate processes and pathways involved in
exciton dynamics. It provides a powerful tool for investigating and unraveling
the detailed electronic structure and energy flow in various materials and
molecular complexes.

###  6.4 ETPA Spectroscopy with Temperature-Controlled SPDC

Sec. 6.3 described how varying the entanglement time and the pump wavelength
can bolster ETPA spectroscopy. There is, however, another feature of entangled
photons that can be exploited to design realistic ETPA spectroscopy schemes,
namely the non-degeneracy of entangled photons. Along this line, in 2019 León-
Montiel _et al._ [383], proposed a rather simple technique for performing
ETPA-based molecular spectroscopy with continuous-pumped SPDC . The main
contribution of this proposed protocol is that it solves the two major
drawbacks of the existing ETPA spectroscopy protocols, namely the need to
perform multiple experiments with two-photon states bearing different temporal
correlations, which translates into the necessity to have at the
experimenter’s disposal hundreds of entangled-photon sources, and the need to
have a priori knowledge of the absorbing medium’s lowest-lying intermediate
energy level.

Figure 24: Temperature-controlled entangled two-photon absorption
spectroscopy. (a) The proposed schematic. (b) Central wavelength of the SPDC
photons as a function of the temperature of the nonlinear crystal, pumped by a
continuous-wave laser at 400 nm. (c) Normalized ETPA signal as a function of
the crystal temperature $T$ and the external delay between photons $\tau$. (d)
Normalized Fourier transform of the ETPA signal with respect to the delay.
Figures reprinted from Ref. [383]

The proposed protocol, depicted in Fig. 24a, makes use of a collinear, type-II
SPDC source, where a PPKTP crystal is pumped by a continuous-wave laser
centered at 400 nm. Interestingly, this configuration provides frequency
anticorrelation of the down-converted photons, which guarantees the strongest
ETPA signal [374]. The wavelengths of the paired photons can be tuned around
the degenerate wavelength ($\omega_{0}=800$ nm) by controlling the crystal
temperature [96, 410]. The state produced by this temperature-dependent source
is equivalent to the one depicted in Eq. (42), with $T_{p}\rightarrow\infty$
and $\omega_{s,i}^{0}(t)=\omega_{0}\pm\Delta\left(T\right)$, where
$\Delta\left(T\right)$ stands for the temperature-dependent, linear frequency
shift of the signal and idler photons, as drawn in Fig. 24b. Using this setup,
we can thus monitor the ETPA signal as a function of the temperature of the
crystal and the delay between photons. Figure 24c illustrates an example of a
predicted ETPA signal for a model sample comprising two intermediate-state
levels, whose wavelengths are arbitrarily chosen to be 967 and 1063 nm. Note
the non-monotonic behavior of the signal results from the interference between
different pathways through which two-photon excitation of the medium occurs.
To reveal information about these pathways or equivalently, the intermediate-
state energy locations, a Fourier transform is performed on the ETPA signal
with respect to the delay. Figure 24d shows the normalized Fourier transform
of the ETPA signal. Note that two characteristic patterns of $X$-shaped and
straight lines appear. Remarkably, the $X$-shaped lines indicate
$\varepsilon_{j}$, the energy location of intermediate states, whereas the
straight lines appear at the combined frequencies
$\pm\left[\varepsilon_{j}\pm\varepsilon_{k}\right]$. This contrasting behavior
between line-signals appear because the former contain frequency components
that are temperature dependent, while the latter are constant with the
temperature. Readers should refer to the Supplemental Material of Ref. [383]
for details.

In contrast to previous schemes for ETPA spectroscopy, the proposed protocol
makes use of a single temperature-controlled SPDC crystal, a routinely-used
technology in quantum optics laboratories. Moreover, the spectroscopic
information about the probed samples can be obtained directly from the
experimental data, without requiring complex post-processing. These features
constituted a major simplification of the ETPA spectroscopy and established a
new route toward its first experimental demonstration.

In summary, the investigation of entangled-light-matter interactions
represents a current and captivating research domain at the forefront of
physics and technology. It holds significant potential to revolutionize
various fields of science and engineering, including material science,
sensing, imaging, and spectroscopy. With the growing community dedicated to
the ETPA spectroscopy, our objective through this concise review is to equip
readers with the fundamental knowledge required to explore the ETPA research
field. By advancing our comprehension of quantum light and its interactions
with matter, we strive to develop innovative QIT that can positively transform
our daily lives.

##  7 Distributed Quantum Sensing

The quantum metrology, sensing, imaging, and spectroscopy protocols reviewed
in the previous sections focus on measurement problems at a single sensor,
while a multitude of real-world sensing tasks rely on a network or an array of
sensors working collectively. Examples of sensor networks range from telescope
arrays for astronomical observation to MEMS-based seismic networks[411] for
earthquake detection.

In a sensor network, the impinging signals are recorded by independent sensors
and extracted with joint signal processing [412]. In general, a classical
sensor network improves the signal-to-noise ratio by coherently adding the
signals while the uncorrelated noise adds up incoherently. As a rule of thumb,
collective measurements with $M$ independent sensors can improve the
measurement sensitivity by $1/\sqrt{M}$, known as the standard quantum limit
(SQL). Substantial progress has been made in the study of quantum
multiparameter estimation [413, 414, 415], where the measurement sensitivity
can potentially scale as 1/$M$, known as the Heisenberg limit, by utilizing
entangled probes. Distributed quantum sensing (DQS) [19, 14, 15], an emergent
paradigm of quantum multiparameter estimation, aims to leverage entanglement
to enhance the measurement sensitivity in estimating global properties shared
by distributed sensors.

Figure 25: Concept of distributed quantum sensing. Multipartite entangled
probes are prepared and distributed to spatially distant sensor nodes to
tackle measurement problems in the sensor network.

Figure 25 provides a schematic diagram of DQS. Multipartite entangled probes
are prepared and delivered to distant sensors. Each sensor measures a local
physical property of the object of interest. The measurement data from all
sensors are then jointly post-processed to derive a global property of the
sensor network, such as the average amplitude or a phase gradient seen by all
sensors. To quantify the quantum advantage of a DQS protocol over its
corresponding classical sensing (DCS) protocol based on separable probes, it
is necessary define a resource constraint applicable to both protocols.
Theoretical studies [15, 14, 413] have shown that a DQS protocol can approach
a measurement sensitivity at the Heisenberg limit of $1/NM$, surpassing the
optimal DCS protocol’s $1/N\sqrt{M}$ and the SQL’s $1/\sqrt{NM}$ scaling when
the protocols are subject to a resource constraint of consuming on average $N$
photons in sensing. Intuitively, the quantum advantage stems from the
correlated measurement noise arising from photon-number fluctuations among
different sensors. Hence, the measurement noise from different sensors in DQS
is canceled out during post-processing when combining the acquired data, in
sharp contrast to the independent measurement noise in DCS. A detailed review
of DQS protocols and their quantum advantage is given in Ref.[19]. We
hereafter focus on the recent experimental advances of DQS in the optical
domain.

###  7.1 Distributed Quantum Sensing for Optical Phase Estimation

Figure 26: DQS for optical phase sensing with CV multipartite entangled
states. (a) Experimental setup. Entangled states are prepared by splitting a
displaced squeezed state with a beam splitter network. The phase quadratures
of the four beams are measured by four homodyne detectors, followed by
classical processing to derive the average phase. OPO: optical parametric
oscillator; EOM: electro-optic modulator; FFT: fast Fourier transform. (b) The
power spectral densities (PSDs) of the phase quadratures at four individual
sensors for 6 different phase shifts. (c) Average PSDs from four detectors for
average phase estimation show higher signal-to-noise ratios than that from a
single sensor. (d) Comparison of measurement sensitivities $\sigma$ between
DQS (red), DCS (blue), and SQL (dashed black) at different average number of
photons $N$. Figures reprinted from Ref. [17].

Following the theoretical proposal in Ref. [15], a proof-of-concept DQS
experiment based on CV multipartite entanglement was reported by Guo et al.
[17]. The schematic of the experiment is shown in Fig. 26a. A displaced phase
squeezed state is generated by injecting a phase-modulated seed laser to an
OPO cavity pumped below the threshold. It is then split by a balanced beam
splitter network consisting of three 50:50 beam splitters into 4 modes to
create entangled probes for DQS at 4 sensor nodes. At the $j$th sensor, a
half-wave plate introduces a phase shift $\phi_{j}$. The objective of DQS is
to estimate a global parameter, e.g., the averaged phase shift $P_{\rm
ave}=(\sum_{j=1}^{M}\phi_{j})/M$ across the sensor network. To this end, the
phase quadratures at the 3 MHz sideband frequency of the entangled probes are
measured by homodyne detectors at four sensor nodes followed by joint post-
processing on the measurement data from all nodes. The power spectral
densities at individual single sensors are depicted in Fig. 26b, showing that
signal peaks at 3 MHz increase with the optical phase shifts. The noise floor
lies $0.8$ dB below the SNL due to the vacuum noise entered through the three
spare ports of the beam splitter network. The power spectral densities of
averaged signals are depicted in Fig. 26c, showing that the noise floor is
significantly lower than those of the individual sensors. The reduced noise
floor corresponds to the single-mode squeezing level $5$ dB prior to the beam-
splitter network [15]. To carry out the same sensing task, DCS employs
separable single-mode displaced phase squeezed states as probes while the SNL
is defined as the performance of coherent-state probes. Figure 26d compares
the measurement sensitivities at various mean photon numbers between DQS, DCS,
and the SNL. It should be noted that loss on the entangled probes diminishes
the advantage of DQS. Quantum repeaters [416] and quantum error correction
[417] will be necessary to reinstate DQS’s advantage over DCS in the presence
of loss.

Figure 27: DQS for optical phase sensing with DV multipartite entangled
states. (a) Experiment setup. SC-YVO4 and TC-YVO4: spatial compensation (SC)
and temporal compensation (TC) yttrium orthovanadate crystals (YVO4); HWP:
half-wave plate; QWP: quarter-wave plate; PBS: polarizing beam splitters.
(b-d) The interferometer configurations to produce the mode-entangled
particle-entangled (MePe), mode-entangled particle separable (MePs), and mode-
separable particle-entangled (MsPe) quantum states. (e) (top) The average
outcome probabilities in the measurement basis $\sigma_{x}^{\otimes 6}$,
$\sigma_{x}^{\otimes 3}$, $\sigma_{x}^{\otimes 2}$ and (bottom) Fisher
information (FI) for the MePe, MePs and MsPe states. The theoretical limit of
FI is shown as the red dot-dashed line (MePs), blue dashed line (MsPe), and
black dotted line (MsPs). Figures reprinted from Ref. [87].

Complementary to CV-DQS, Liu et al. demonstrated a proof-of-concept DQS
experiment based on DV entangled states [87]. Figure 27a shows their
experimental setup: three BBO crystals are pumped by ultrafast ultraviolet
lasers to produce three entangled photon pairs each in the state
$\ket{\phi^{+}}=(\ket{HH}+\ket{VV})/\sqrt{2}$, where $H(V)$ denotes the
horizontal (vertical) polarization. Mode-entangled and particle-entangled
(MePe) probes are then prepared by passing three photon pairs through a
tunable interferometer shown in Fig. 27b, with the form of $\ket{\phi_{\rm
MePe}}=(\ket{HH}_{M_{1}}\ket{HH}_{M_{2}}\ket{HH}_{M_{3}}+\ket{VV}_{M_{1}}\ket{VV}_{M_{2}}\ket{VV}_{M_{3}})/\sqrt{2}$.
This three-mode entangled state consists of two entangled photons in each mode
and is used to probe optical phases $\theta_{1,2,3}$ configured by a
combination of a quarter-wave plate and a half-wave plate at three nodes. The
goal is to estimate the average phase shifts
$\hat{\theta}=\sum_{k=1}^{3}\theta_{k}/3$. By performing projective
measurements in $\sigma_{x}^{\otimes 6}$ basis, an interference fringe is
obtained: $P_{\pm 1}^{\rm MePe}=[1\pm V_{\pm}\cos{(NM\hat{\theta})}]/{2}$,
where $V_{\pm}$ is the interference visibility, $N=2$ is the number of
entangled photon in each mode and $M=3$ is the number of entangled modes. The
Heisenberg limit of estimation sensitivity $\delta\hat{\theta}\sim 1/NM$ is
achieved when the visibility $V=1$. The quantum advantage of DQS based on the
MePe state is benchmarked against the performance of DCS using mode-separable
and particle-entangled (MsPe) probes in a local optimal state $\ket{\phi_{\rm
MsPe}}=(\ket{HH}+\ket{VV})_{M_{1}}(\ket{HH}+\ket{VV})_{M_{2}}(\ket{HH}+\ket{VV})_{M_{3}}/2\sqrt{2}$,
generated directly from three SPDC processes without passing through the
interferometer as shown in Fig. 27d. At each sensor node, the probability of
projection outcome in the $\sigma_{x}^{\otimes 2}$ basis is given by $P_{\pm
1}^{\rm MsPe}=[1\pm V_{\pm}\cos{(N\hat{\theta}_{i})}]/{2}$. The estimation
sensitivity of individual phase shifts with unity visibility is $1/N$,
resulting in an overall estimation sensitivity for averaged phase shifts
$1/\sqrt{M}N$, similar to the separable squeezed states in the CV-DCS. Liu et
al. also prepared mode-entangled particle-separable (MePs) states
$\ket{\phi_{\rm
MePs}}=(\ket{H}_{M_{1}}\ket{H}_{M_{2}}\ket{H}_{M_{3}}+\ket{V}_{M_{1}}\ket{V}_{M_{2}}\ket{V}_{M_{3}})^{\otimes
2}/2$ by replacing one of the combinations of BBO crystals with a single piece
of crystal shown in Fig. 27c. This scheme is equivalent to repeating twice the
measurements with mode entangled probes where only one photon is in each mode.
The measurement outcome probability in $\sigma_{x}^{\otimes 3}$ basis is given
by $P_{\pm 1}^{\rm MePs}=[1\pm V_{\pm}\cos{(M\hat{\theta})}]/{2}$, enabling a
measurement sensitivity scaling $1/M$. The estimation uncertainty of $N$
measurements with one photon in each mode at a time follows $1/\sqrt{N}M$,
also surpassing the standard quantum limit of $1/\sqrt{NM}$. The average
probabilities for obtaining different phases in the measurement basis of
$\sigma_{x}^{\otimes 6},\sigma_{x}^{\otimes 3},\sigma_{x}^{\otimes 2}$ for
MePe, MePs, MsPe probe states are plotted in Fig. 27e top. The corresponding
Fisher information fitted from the measurement probabilities is shown in Fig.
27e bottom. The blue (black) dashed line represents the theoretical limit of
the FI for DCS (SQL). The red dot-dashed line represents the theoretical limit
of the FI for MePs. The DQS (MePe) and DCS (MsPe) achieve 2.7 dB and 1.43 dB
reductions in the estimation variance as compared to the SQL. In addition,
MePs probes outperform the performance of MsPe when the mode number is larger
than the photon number in each mode, i.e., $M>N$. Overall, the DQS protocol
provides the best performance subject to a fixed total mean photon number.
Combining the DQS protocol with the sequential scheme, where probes coherently
interact with the sample for multiple times, could further improve the
measurement sensitivity. A total number of photon passes at $n=21$ across 6
modes has been demonstrated. The joint detection probability for 6 modes is
$P_{\pm 1}^{\rm MePs}=[1\pm V_{\pm}\cos{(21\hat{\theta})}]/2$, and a 4.7 dB
noise reduction compared to the SNL has been achieved, opening a new path to
improving the measurement sensitivity of DQS.

Recently, Zhao et al. demonstrate a field test of DV-DQS [88] based on a
loophole-free Bell test setup where entangled photon pairs are distributed to
distant locations separated by 240 m. A global phase estimation precision of
0.916 dB below the SNL has been achieved unconditionally by virtue of a state-
of-the-art average heralding efficiency of $73.88\%$.

###  7.2 Distributed Quantum Sensing for Radio-Frequency Measurements

DQS has been rapidly growing and has demonstrated great potential for
practical quantum advantages in various sensing applications.

Figure 28: Entangled RF-photonic sensor network. (a) Concept of entangled RF-
photonic sensor network. EOT: electro-optic transducer; HD: homodyne detector.
(b) experiment diagram. HWP: half-wave plate; PBS: polarizing beam splitter;
VBS: variable beam splitter; LO: local oscillator. (c-e) Optimization of
multipartite entangled state for (c) RF-field average amplitude estimation,
(d) phase-difference estimation at a central node, and (e) phase-difference
estimation at an edge node. Insets show the tuning range of VBSs in preparing
entangled states. (b-e) reprinted from Ref. [18].

Beyond optical phase sensing, DQS can be harnessed to tackle measurement
problems in a variety of physical domains by exploiting transducers or other
types of quantum sensors, such as spin sensors [418]. Xia et al. demonstrated
CV-DQS of RF signals using RF-photonic transducers [18], as sketched in Fig.
28 the concept and experimental setup. The entangled probes were generated by
splitting squeezed light on two variable beam splitters each composed of a HWP
and a PBS. The entangled probes were distributed to three distant sensor
nodes. At each sensor node, an electro-optical modulator driven by an input RF
signal created quadrature displacement at the sideband of the probe. Three
balanced homodyne detectors characterized the quadrature displacement
$\alpha_{i}\propto E_{i}\phi_{i}$, where $E_{i}(\phi_{i})$ represents the
amplitude (phase) of RF signal at $i_{\rm th}$ sensor. A global property of
the RF fields was derived $\alpha_{g}=\sum_{i}a_{i}\alpha_{i}$, for example,
$a_{i}=1/3$ for mean amplitude estimation at a constant RF phase. An essential
contribution of this work is the investigation of the optimal entanglement
structure for different DQS tasks. Specifically, Xia et al. optimized the
entangled states for three different sensing cases shown in Fig. 28: (c)
averaged RF amplitude, (d) RF phase gradient at the center node, (e) RF phase
gradient at the edge node. The estimation variance normalized to the SQL is
plotted as a function of the beam-splitting ratios for different entangled
states for the probes. The performance of the DQS is compared to that of DCS
with laser light carrying the same optical power. It is important to note that
the optimal performance is only achieved with the specific entangled state for
a defined sensing task. The divergence of the estimation variance, as the
splitting ratio approaches 0 or 1, indicates no information about the global
property can be gained as no photon is sent to one of the sensors. The minus
sign of the splitting ratio in Fig. 28d, e represents a $\pi$ phase delay to
the signal $\phi_{i}\xrightarrow{}-\phi_{i}$ along with a sign flip
$a_{i}\xrightarrow{}-a_{i}$ in post-processing to ensure an unbiased
estimator. In these two cases, the performance curves of DQS display an
asymmetric behavior in contrast to the symmetric DCS curves, which manifests
the correlations in the measurement noise across different sensors.

###  7.3 Distributed Quantum Sensing for Optomechanical Measurements

Optomechanical sensors leverage the parametric coupling between a mechanical
oscillator and a light field to achieve highly precise measurements of force,
acceleration, mass, and magnetic fields [419]. Recently, Xia et al.
demonstrated entanglement-enhanced joint force sensing using two
optomechanical sensors, as depicted in Fig. 29a [90]. The experiment employed
entangled probe light at 1550 nm, similar to that used in continuous-variable
distributed quantum sensing (CV-DQS) for RF measurements [18]. This entangled
light was coupled into two optomechanical sensors, each comprising a
mechanical membrane placed on top of a highly reflective mirror. The
mechanical displacement at the $i$th sensor, encoded in the phase quadrature
of the probe beam $Y^{(i)}_{\rm out}=Y^{(i)}_{\rm
in}+\alpha_{i}\beta_{i}\chi_{i}F_{i}$, was measured using a balanced homodyne
detector. Here, $|\alpha_{i}|^{2}$ represents the mean photon number of the
coherent state at the carrier wavelength, $\beta_{i}$ denotes the
optomechanical transduction efficiency, $\chi_{i}$ stands for the mechanical
susceptibility, and $F_{i}$ encompasses both thermal force and signal forces.
The entangled light exhibits quantum correlations between the input probe
quadratures over a wide range around the mechanical resonant frequencies,
thereby enhancing the performance of optomechanical sensors.

Figure 29b presents the normalized power spectral densities (PSDs) of
individual sensors, revealing two broad thermal peaks and two delta peaks
originating from radiation pressure test forces. The noise floor is slightly
below the SNL depicted by the gray line. In Fig. 29c, the normalized joint PSD
of the two sensors demonstrates a significantly improved signal-to-noise ratio
(SNR), with the noise floor positioned 2 dB below the joint SNL. To assess the
performance of averaged force sensing $(F_{1}+F_{2})/2$ with entangled probes,
two scenarios are compared: the imprecision-noise-dominated regime in Fig. 29d
and the thermal-noise-dominated regime in Fig. 29e. When a substantial
difference in mechanical resonant frequency exists, the minimum joint force
noise $\sqrt{S_{FF}^{\rm min}}$ is achieved near the averaged mechanical
resonant frequency, constrained by the laser shot noise. However, with
entangled probes, the shot noise can be reduced by a factor of $\sqrt{V}$, as
shown in Fig. 29d (left), where $V=\left<(Y_{\rm in}^{(1)}+Y_{\rm
in}^{(2)})^{2}\right>/2$ represents the squeezing factor offered by the
entangled probes. The 3-dB bandwidth $\mathcal{B}_{3\text{dB}}$ with entangled
probes is slightly inferior to that with classical probes due to the
frequency-independent nature of the entangled states in the experiment. To
fully exploit the quantum correlations, frequency-dependent entanglement akin
to frequency-dependent squeezing[420] is required. The minimum force noise and
3-dB bandwidth at different probe powers are depicted in Fig. 29d (right).
Increasing the mean photon number $\alpha_{c}^{2}$ at the carrier wavelength
leads to a reduction in force noise at a rate of $1/\alpha_{c}$. The bandwidth
is primarily determined by the resonant frequency difference and exhibits a
slight increase with the probe power.

Figure 29: Entanglement-enhanced optomechanical sensing. (a) Experimental
diagram. BS: beam splitter; HWP: half-wave plate; QWP: quarter-wave plate.
PBS: polarizing beam splitter. (b) Normalized power-spectral density (PSD) for
individual sensors. (c) Normalized joint PSD for both sensors. The circles
mark the delta peaks yielded by test radiation pressure forces applied on the
sensors. The SNL is shown as grey lines. (d, e) Joint force noise at 50 $\mu$W
probe power (left); minimum force noise and 3-dB bandwidth at different probe
power (right). The resonant frequency difference is (d) 1422 Hz (e) and 262
Hz. (f) Sensitivity-bandwidth product (SBP) versus frequency difference at 50
$\mu$W probe power. SBP versus different probe power at resonant frequency
difference of 262 Hz (g) and 1422 Hz (h). Figures reprinted from Ref. [90].

When the mechanical resonant frequencies are nearly identical, as illustrated
in Fig. 29e, both classical and entangled probes experience a minimum force
noise limited by thermal noise. However, entangled probes offer a larger
bandwidth by a factor of $1/\sqrt{V}$, as demonstrated by the force noise
curves in Fig. 29e (left). The minimum force noise and 3-dB bandwidth at
different probe powers are plotted in Fig. 29e (right). As the probe power
increases, the minimum force noise converges to the thermal noise limit, while
the bandwidth scales with $\alpha_{c}$.

The performance of the sensor array for broadband sensing tasks is
characterized by the sensitivity-bandwidth product (SBP)[421], given by
$\mathcal{S}\times\mathcal{B}_{3\text{dB}}\propto 1/\sqrt{VS_{FF}^{\rm min}}$,
where $\mathcal{S}=1/\sqrt{S_{FF}^{\rm min}}$ and is similar to integrated
sensitivity. Fig. 29f illustrates the SBP of classical, entangled, and optimal
entangled probes at different resonant frequency differences, with the probe
power fixed at 50 $\mu$W for each sensor. In the experiment, the SBP of
entangled probes was suboptimal, but it approached the performance of optimal
entangled probes at small resonant frequencies, surpassing the SBP of the
classical probe by approximately $1/\sqrt{V}\approx 1.25$. The SBPs increase
with respect to the square root of the probe power in the thermal noise
dominant regime, as shown in Fig. 29g, and with respect to the probe power in
the imprecision noise dominant regime, as shown in Fig. 29h.

Before concluding this section, we would like to make a few remarks. It is
worth noting that DQS is not limited to the photonic platform and has rapidly
expanded to encompass various physical systems, including atomic spins[418]
and microwave cavities[422]. These advancements have opened up a wide range of
applications, from inertial navigation to dark matter search[421, 423].
Furthermore, we would like to highlight a recent milestone in the field,
namely the successful distribution of squeezed light over a 40-km fiber
link[424]. In this experiment, the transmitted squeezed light was detected
using a real local oscillator, demonstrating a promising pathway for
implementing CV-DQS in large-scale quantum networks. These developments
underscore the growing significance and potential of DQS in various physical
systems, as well as its applicability in enabling advanced QIT applications.

##  8 Quantum Machine Learning

Figure 30: Schematic for quantum machine learning with photons. The quantum
state of light from a source is processed by a variational quantum circuit
configured by a classical algorithm to generate the quantum state for a
defined classical or quantum processing task. Measurements on the quantum
state produce data used by a classical algorithm to optimize the parameters of
the variational quantum circuit and seek solutions of the data-processing
task.

Machine learning has rapidly grown over the last two decades and continues to
have a significant impact on everyday life, from speech recognition to
autonomous vehicles. In the meantime, quantum hardware is advancing at a great
pace, leading to quantum computation advantage in solving certain computation
problems. It is thus expected that quantum computers can also significantly
enhance the performance of machine learning tasks in several aspects,
including quantum speedups in optimization, improved classification accuracy,
and sampling of classically intractable systems [425]. Many existing quantum
machine-learning algorithms require a large-scale fault-tolerant quantum
computer, which is not yet available in the current era of noisy intermediate-
scale quantum (NISQ) technology. Instead, hybrid approaches combining
variational quantum circuits and classical processing, as schematically
illustrated in Fig. 30, have emerged as a leading strategy for quantum machine
learning tasks[426] such as variation quantum eigensolver[427, 428, 429] and
quantum kernel method[430, 431]. Along this line of research, researchers have
been exploring quantum photonic platforms in conjunction with classical
processing to improve classical processing [432, 20] as well as quantum
processing [433]. In the same vein as quantum machine learning in photonic
platforms, a more recent experiment in the trapped-ion platform exploited a
classical algorithm in tandem with variational quantum circuits [434] to
produce entangled spin states for a Ramsey interferometer to approach the
ultimate measurement precision at the Heisenberg limit. We will next review
the recent advances in quantum machine learning for both classical and quantum
processing.

###  8.1 Quantum Machine Learning for Classical Processing

Figure 31: Entanglement enhanced data classification. (a) Schematic of
supervised learning assisted by an entangled sensor network (SLAEN). (b)
Classical classifiers built on separable states and pure classical processing.
(c) Experiment diagram for RF signals classification using SLAEN. Figure
reprinted from Ref. [20].

Quantum machine learning algorithms, such as quantum support vector machine
[435] and quantum principal analysis [436], often require quantum memories to
load classical data, which could hinder their near-term quantum advantage for
big-data problems. Supervised learning assisted by an entangled sensor network
(SLAEN) [31] is a different paradigm that takes advantage of quantum metrology
to acquire data at the physical layer instead of using classical data given a
priori. SALEN can improve the accuracy of classical data classification and
compressing. The schematics of SLAEN and classical classifiers are compared in
Fig. 31 a, b. SLAEN consists of a variational quantum circuit (VQC)
parameterized by $\boldsymbol{v}$ to generate an entangled probe state shared
by $M$ sensors. The objects, modeled by quantum channels $\Phi(\alpha_{m})$,
are probed by the entangled states, which are subsequently measured to produce
results $\tilde{\alpha}_{m}$. The measurement data are processed in a
classical computer running a support vector machine (SVM) algorithm. The goal
of the training stage of SLAEN is to find the optimum entangled probes and the
hyperplane used to generate labels for the data. This is done by using
training data to optimize the hyperplane $\\{\boldsymbol{w},b\\}$ stored in
the classical computer with the measurement data $\tilde{\alpha}_{m}$ and
known labels $y$. The VQC is reconfigured in real-time to generate entangled
probes that minimize measurement noise by mapping
$\boldsymbol{w}\rightarrow\boldsymbol{v}$ at each optimization step. At the
end of the training stage, the obtained optimal entangled state and hyperplane
would facilitate the data classification with reduced error probability. In
contrast, the classical classifier acquires measurement data with separable
probes and conducts training in post-processing in the classical computing
without invoking physical hardware.

Figure 32: Convergence of error probability during training for (a) RF phase
gradient classification with two sensors and (c) RF mean amplitude
classification with three sensors. (b, d) Evolution of the hyperplane
parameters during training, corresponding to (a, c). Colorbar: error
probability. Figure reprinted from Ref. [20].

Xia et al. demonstrated improved data classification accuracy of RF signals
using a variational entangled sensor network on a continuous-variable quantum
optics platform[20]. The experiment setup is shown in Fig. 31c. A source
generated squeezed light, which was then processed by a variational quantum
circuit composing two variable beam splitters (VBSs) and three phase shifters.
The splitting ratio of each VBS was controlled by an external DC voltage. The
output entangled probes were delivered to three sensors equipped with electro-
optic modulators that converted RF signals into displacements on the phase
quadrature of the $m_{th}$ mode. The optical phase quadratures were
subsequently measured by three homodyne detectors and the measurement results
$\tilde{\alpha}_{m}$ were fed into a classical computer for training,
classification, and VQC optimization. Fig. 32 shows the experimental training
results of SLEAN and classical classifiers using laser light aiming to
classify (a,b) RF-phase gradient with two sensors and (c,d) RF mean amplitude
with three sensors. SLAEN was provided with $N$ sets of training data and
labels for two training tasks and the hyperplane was optimized iteratively by
minimizing a cost function. The setting for the VQC was updated at each step.
Fig. 32b, d show the trajectories of the hyperplanes $\\{\boldsymbol{w},b\\}$
evolving toward the optimum hyperplanes (Hexagrams) during the two training
processes. The error probabilities measured at different training steps of
SLAEN and classical classifier are plotted in Fig. 32a, c, which demonstrate
two advantages of SLAEN over the classical classifier. First, SLAEN takes less
step to converge to the optimal configuration. Second, once trained SLAEN
enjoys reduced error probabilities as compared to the classical classifier.
The reduction in classification error probability with SLAEN stems from the
quantum correlation between measurement noise at different sensors, akin to
the estimation of global properties in DQS tasks. The experiment Xia et al.
verified SLAEN in RF sensing. As a versatile architecture, SLAEN can also be
readily adapted to cope with other types of quantum sensors[437] such as NV
centers, nanomechanical oscillators, etc.

Figure 33: Quantum speedup in reinforcement learning. (a) Schematic of
reinforcement learning. (b) Experimental diagram. Configurations of the
nanophotonic chip for (c) classical and (d) quantum epoch. (e) Averaged reward
for quantum (blue) and classical (orange) strategy. (f) Average reward for the
combined strategy (green), where the agent switches to a classical strategy
after reaching the optimal point of the quantum strategy, as compared to the
classical-only strategy (orange). Figures reprinted from Ref. [432].

Reinforcement learning (RL)[438], a powerful paradigm in artificial
intelligence with a wide range of applications including robotics and
autonomous driving, has recently been incorporated into quantum photonic
platforms. The RL process involves iterative interactions between an agent and
the environment through a classical channel, as shown in Fig. 33a, referred to
as an epoch. In general, the agent’s learning process can be divided into two
critical steps within each epoch and could potentially benefit from quantum
speedups through multiple pathways. First, the agent takes actions based on
received rewards from the environment. Quantum computers could potentially
reduce the running time of agent’s internal program for selecting actions that
lead to higher rewards. Second, quantum communication could enable a quadratic
reduction in environment queries, similar to Grover’s search algorithm. Along
the second pathway, Saggio et al. experimentally demonstrated quantum speed-up
in reinforcement learning [432] on an integrated nanophotonics platform. They
achieved this by amplifying the amplitudes of states corresponding to high
reward actions, as shown in Fig. 33b. The configurations of the nanophotonic
processor for classical RL and quantum-enhanced RL are shown in Fig. 33c, d
respectively. In their experiment, the wining (losing) action state
$\ket{w}_{A}$ ($\ket{l}_{A}$) of the agent was represented by a single photon
qubit via $\ket{1}_{A}$ ($\ket{0}_{A}$), and the reward state by the
environment was denoted by another qubit state $\ket{1}_{R}$ or $\ket{0}_{R}$.
This four-level Hilbert space was encoded in the path degree of freedom of a
single photon passing through four waveguides. The behavior of the environment
was modeled by a controlled-NOT gate $U_{E}$ such that the reward state was
flipped only in the case of the winning state $\ket{w}_{A}$. A single photon
was initially coupled into the mode $\ket{0_{A}0_{R}}$ and then interfered at
a Mach-Zehnder interferometer (MZI), resulting in the state
$\ket{\psi}_{A}\ket{\psi}_{R}=(\cos(\xi)\ket{0}_{A}+\sin(\xi)\ket{1}_{A})\ket{0}_{R}$
where $\varepsilon=\sin^{2}(\xi)$ is the winning probability of winning action
state $\ket{1}_{A}$. During the classical RL, the environment applied a
controlled-NOT gate on the two qubits, by implementing another MZI between the
third and fourth waveguide. The photon was then coupled out and detected by
two single-photon detectors D1 and D2 at the output of two waveguides,
corresponding to states $\ket{0_{A}0_{R}}$ and $\ket{1_{A}1_{R}}$ with success
probability $\cos^{2}(\xi)$ and $\sin^{2}(\xi)$. The winning probability
$\varepsilon_{j}$ after $j_{th}$ photon arriving at D2 was updated according
to $\varepsilon_{j}={(1+2j)}/{(100+2j)}$, where the initial winning
probability was set to $\varepsilon_{0}=1/100$. In quantum-enhanced RL, the
initial state $\ket{\psi}_{A}\ket{\psi}_{R}$ was first transformed to
$(\cos(\xi)\ket{0}_{A}+\sin(\xi)\ket{1}_{A})\ket{-}_{R}$ via two unitary
operators $U_{H0}$ and $U_{H1}$, where
$\ket{-}_{R}=(\ket{0}_{R}-\ket{1}_{R})/\sqrt{2}$. Then, the environment
applied $U_{E}$, which flipped the sign of the winning state,
$(\cos(\xi)\ket{0}_{A}-\sin(\xi)\ket{1}_{A})\ket{-}_{R}$. After interaction
with the environment, the quantum agent reversed operations $U_{H0}$ and
$U_{H1}$, and performed a reflection $U_{R}=2\ket{\psi}\bra{\psi}_{A}-1_{A}$,
amplifying winning probability $\varepsilon=\sin^{2}(3\xi)$ at detector D3.
This quantum-enhanced winning probability, in contrast to classical winning
probability $\sin^{2}(\xi)$, enabled a speed-up in RL. Since the reward was
not revealed in each quantum epoch, an additional classical test epoch was
required and the update rule of winning probability remained the same as in
the classical case. Fig. 33e shows the average reward of classical (orange)
and quantum (blue) RL strategies. To ensure a fair comparison with classical
RL, the reward after each classical test epoch in quantum RL was distributed
(averaged) over both the quantum and classical test epochs. The quantum
strategy reached its optimum point when the averaged reward
$\eta_{Q}=\sin^{2}(3\xi)/2$ equaled that of classical strategy
$\eta_{C}=\sin^{2}(\xi)$ with $\eta_{Q}=\eta_{C}=0.396$, then started to
decrease, which is common to any Grover-like algorithm. After reaching the
optimum point, the agent switched back to the classical strategy while
maintaining the quantum advantage as shown in Fig. 33f, where a combined
strategy (green) is compared to the classical case.

Figure 34: Quantum unsampling with a photonic processor. (a) Illustration of
quantum unsampling. (b) Layer-wised approach for quantum unsampling. (c)
Experiment diagram and layout of the programmable nanophotonic processor.
Minimizing the loss function of the (d) first layer to find a photon in Mode 1
and (e) second layer to find a photon in Mode 2. The four red points show the
probabilities for all six twofold coincidence events in (f). Figures reprinted
from Ref. [433].

###  8.2 Quantum Machine Learning for Quantum Processing

In addition to improving the performance of classical processing, one of the
most promising applications of quantum machine learning is to validate and
calibrate quantum states and quantum dynamics[439], which are of great
importance to NISQ devices. Carolan et al. introduced a novel variational
quantum unsampling protocol to unravel quantum dynamics on the known input
states via a layer-wised approach and demonstrated it on a quantum photonic
processor[433]. Quantum unsampling is a process aimed at learning the inverse
of the quantum dynamics of a black box. Given the known input state
$\ket{\psi_{\rm in}}$ and unknown quantum operation $\hat{U}$, a quantum
unsampling protocol seeks an appropriately parametrized circuit
$\hat{V}(\boldsymbol{\phi})$ such that
$\hat{V}(\boldsymbol{\phi})\approx\hat{U}^{\dagger}$ by minimizing a loss
function: $L(\boldsymbol{\phi})=1-|\langle\psi_{\rm
in}|\hat{V}(\boldsymbol{\phi})|\psi_{\rm out}\rangle|^{2}$, as illustrated in
Fig. 34a. In general, the probability of an individual event is exponentially
unlikely and the gradient-based optimization becomes exponentially
inefficient[440]. To address this challenge, researchers have developed a
layer-by-layer training approach shown in Fig. 34b, where each layer optimizes
over only a polynomially sized subset of the full Hilbert space by
disentangling one qubit at a time. Consider an $n$-qubit input state
$\ket{\psi_{\rm in}}=\ket{\alpha_{1},\alpha_{2},...\alpha_{n}}$, the output
state $\hat{U}\ket{\psi_{\rm in}}$ is fed into the first layer
$\hat{V}_{n}(\boldsymbol{\phi}_{n})$ acting on all $n$ qubits
$\rho_{1}=\hat{V}_{n}(\boldsymbol{\phi}_{n})\ket{\psi_{\rm out}}\bra{\psi_{\rm
out}}\hat{V}_{n}^{\dagger}(\boldsymbol{\phi}_{n})$ and an optimization
algorithm varies the circuit parameter $\boldsymbol{\phi}_{n}$ to minimize the
loss function
$L_{1}(\boldsymbol{\phi}_{n})=1-|\langle\alpha_{1}|\text{tr}_{2...n}(\rho_{1})|\alpha_{1}\rangle|^{2}$,
aiming to find the first qubit in state $\ket{\alpha_{1}}$. The reminder $n-1$
qubit state is then fed to the next circuit layer
$\hat{V}_{n-1}(\boldsymbol{\phi}_{n-1})$, which maximizes the overlap between
the second qubit state $\ket{\alpha_{2}}$. The optimization continues until
all qubits are processed. The overall unsampling quantum operation is thus
$\hat{V}=\Pi_{i=1}^{n}\hat{I}_{n-i}\oplus\hat{V}_{i}(\boldsymbol{\phi}_{i})$,
and the probability at each stage is exponentially enhanced. The unsampling
quantum circuits are over-parameterized and only partially characterize the
unknown quantum operations over certain input states. Additional information
of $\hat{U}$ can be obtained by training with more input basis states.

A proof of concept variational quantum unsampling experiment of boson sampling
process was carried out on a state-of-the-art programmable nanophotonic
processor (PNP) drawn in Fig. 34c. Pairs of photons generated via SPDC from a
PPKTP crystal were coupled into the two optical modes of PNP which comprised
reconfigurable MZIs arranged in a mesh. Two photons initially passed through
the four-mode sampling circuit (orange block) in Fig. 34c. The output state
was then fed into a two-layer upsampling circuit (green and blue blocks) and
detected by an array of single photon detectors. During the training of first
unsampling layer (green block), a classical optimizer running the local
gradient-free BOBYQA algorithm [441] varied the phase shifters of MZIs to
minimize the loss function $L_{1}(\boldsymbol{\phi}_{4})=1-\tilde{P}_{1}^{1}$,
which maximized the probability of a single photon in optical mode 1
$\tilde{P}_{1}^{1}$. The loss function $L_{1}(\boldsymbol{\phi}_{4})$ was
initialized to 0.55 and converged to around 0.2 after 28 iterations to the
noise floor, as shown in Fig. 34d. Then the output state was fed into the
second unsampling layer (blue block) while leaving first unsampling layer
untouched. The classical optimizer now aimed to find a single photon in mode 2
by minimizing the loss function $L_{2}(\boldsymbol{\phi}_{2})$. The loss
function was initialized to 0.97 and converged to 0.31 after 20 steps, as
plotted in Fig. 34. The probabilities of six twofold coincidence events at
different training steps (red dots in Fig. 34d, e) are depicted in Fig. 34f.
At the end of the training, the coincidence count probability in optical mode
(1,2) was found to be $P=0.695\pm 0.053$, equivalent to the overlap between
the initialization state and the output unsampling state $|\bra{\psi_{\rm
in}}\hat{V}(\boldsymbol{\phi})\ket{\psi_{\rm out}}|^{2}$.

Quantum machine learning is still in its early stages and is rapidly evolving
with the potential to solve complex problems and advance the capabilities of
quantum computing. PIC platforms hold the advantage of providing scalable
architectures for quantum machine learning tasks where quantum-light sources,
beam splitters, and single-photon detectors can be packed onto small chips,
facilitating the development of compact and efficient quantum photonic
circuits[195, 442, 443, 444].

##  9 Entanglement-Assisted Communication

Entanglement is a critical ingredient to enable a multitude of communication
capabilities beyond the reach of classical communication. This section
introduce the fundamentals and recent advances in entanglement-assisted
communication (EACOMM), a paradigm that harnesses the pre-shared entanglement
between communicating parties to increase the rate of transmitting classical
information. A representative EACOMM regime is the well-known quantum
superdense coding protocol [445]. This section will review the recent
advancements of EACOMM, with a particular focus on experimental
demonstrations. For a comprehensive review of quantum superdense coding,
readers can refer to a recent review [446]. This review article will not cover
entanglement-based quantum key distribution. Readers interested in this topic
may refer to recent reviews [34, 447].

EACOMM enables the transmission of classical information at rates surpassing
those achievable by classical communication alone, given the same channel
parameters and power constraints. The configuration of EACOMM is illustrated
in Fig. 35: prior to transmitting classical information, two communicating
parties, Alice and Bob, establish shared entangled states, the signal and the
reference, through entanglement distribution channels and store this pre-
shared entanglement in their quantum memories. To initiate communication,
Alice retrieves her portion of the entangled state from her quantum memory and
encodes the classical information through modulations. The encoded signal is
then transmitted to Bob over a noisy and lossy channel. Upon receiving the
signal, Bob retrieves his share of the entangled state, i.e., the reference,
from his quantum memory. Both the signal and reference are forwarded to a
quantum receiver, which performs a joint measurement on them to decode the
classical information originally encoded by Alice.

The concept of EACOMM originated from the development of a protocol known as
quantum superdense coding [445], where two classical bits of information can
be transmitted using one qubit of an entangled pair. The discovery of the
enhanced classical communication rates achievable with pre-shared entanglement
led to the exploration of the entanglement-assisted (EA) classical capacity.
Early seminal results from the early 2000s demonstrated that the ratio between
the EA classical capacity and the ordinary classical capacity could become
infinite under certain circumstances [7]. However, the derived EA capacity did
not provide specific information regarding the encoding format and quantum
measurements required to fully exploit the advantage over classical
communication. In recent years, structured designs and implementations of
encoders and quantum receivers for EACOMM have emerged, utilizing both DV and
CV states.

Figure 35: Schematic for entanglement-assisted communication. Entanglement is
distributed to two communicating users, Alice and Bob, prior to classical
communication. To communicate, Alice retrieves her share of the entangled
state, signal, from a quantum memory and encodes on the state. The encoded
signal is transmitted to Bob through a lossy and noisy quantum channel. At
Bob’s terminal, he exploits a quantum receiver to take a joint measurement on
the received signal from the channel and the idler. In doing so, Bob infers
Alice’s encoded classical information.

###  9.1 Superdense Coding with DV Entanglement

The principles of superdense coding, also known as dense coding, were
initially introduced by Bennett and Wiesner in the context of classical
communication using EPR states [445]. In this scenario, Alice and Bob each
possess one particle from an entangled pair, prepared in a known entangled
state. For simplicity, let us consider the particles as qubits, and the
prepared state as the Bell state

$\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}\left(|0\rangle_{A}|0\rangle_{B}+|1\rangle_{A}|1\rangle_{B}\right).$
(46)

Alice chooses one of the operations $\\{I,Z,X,Y\\}$ from the set of Pauli
matrices to apply to her particle $A$. Each operation maps one of the four
possible bit pairs $\\{00,01,10,11\\}$ to one of the four orthogonal Bell
states:
$|\Phi^{\pm}\rangle=\left(|0\rangle_{A}|0\rangle_{B}\pm|1\rangle_{A}|1\rangle_{B}\right)/\sqrt{2}$
and
$|\Psi^{\pm}\rangle=\left(|0\rangle_{A}|1\rangle_{B}\pm|1\rangle_{A}|0\rangle_{B}\right)/\sqrt{2}$.
After applying the chosen operation, Alice sends her particle to Bob. Bob, who
now controls both particles, performs a complete Bell-state measurement to
determine the identity of the joint state. Bob then extracts the two bits
representing the state created by Alice, allowing her to transmit two bits of
information to Bob using only a single qubit and shared entanglement. It is
important to note that without entanglement, Alice’s transmission of a qubit
can convey only one classical bit of information. This can be proven by the
communication channel’s capacity, defined as the supremum of the mutual
information between the input and output of the channel, which quantifies the
maximum rate of information transmission per qubit.

Several significant experimental demonstrations of superdense coding have
showcased the achievable capacity using different transmission and measurement
techniques. In 1996, Mattel et al. demonstrated a capacity of 1.58 bits per
photon using polarization-encoded states generated at 702 nm with a type-II
degenerate SPDC source [448]. Their experiment employed a joint measurement
device that interfered the two photons to reveal the encoded Bell state
through polarization detection. However, this setup based on linear optics
allowed only partial Bell-state measurement, limiting the observed
communication capacity below the quantum limit of 2 bits per channel use.
Barriero et al. later demonstrated a method to surpass the restriction of
detecting only three out of the four Bell states by employing
hyperentanglement [449]. By utilizing the additional orbital angular degree of
freedom in the generated photons, they achieved a channel capacity of
$1.630\pm 0.006$ bits per channel. By first distributing a separable joint
state entangled in polarization and orbital angular momentum, the resulting
measurement could directly detect any of the four polarization-entangled Bell
states.

In a recent experiment, Williams et al. used similar methods to demonstrate
superdense coding from complete Bell-state measurements on time-polarization
hyper-entanglement that takes advantage of an extra degree of freedom in the
time domain [450]. The quantum states of the hyper-entangled photon pairs can
be described as

$\displaystyle|\Phi^{\pm}(t)\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|H\rangle_{A}|H\rangle_{B}\pm|V\rangle_{A}|V\rangle_{B}\right)\otimes|\phi(t)\rangle$
(47) $\displaystyle|\Psi^{\pm}(t)\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|H\rangle_{A}|V\rangle_{B}\pm|V\rangle_{A}|H\rangle_{B}\right)\otimes|\phi(t)\rangle,$
(48)

where $|H\rangle$ and $|V\rangle$ are, respectively, the horizontal and
vertical polarization states, and $\phi(t)$ is the temporal quantum state that
exhibits two-photon coherence within the timescale relevant to the
measurements. The complete linear-optics Bell-state measurement apparatus
comprised 50:50 beam splitters, polarizing beam splitters, delay lines, half-
wave plates for polarization rotation, and single-photon detectors, as
sketched in Fig. 36a.

Figure 36: Experiment of superdense coding with complete Bell-state
measurements. (a) Schematic for complete Bell-state measurement based on
linear optics and time-polarization hyper-entanglement. (b) Experimental
diagram. (c) Experimental data for Bell-state discrimination. Figures
reprinted from Ref. [450].

The temporal and polarization coherence of the hyper-entangled photon pairs
allowed for the following transformations on the input Bell states:

$\displaystyle|\Phi^{+}\rangle$
$\displaystyle\rightarrow\frac{1}{\sqrt{2}}\left(|H\rangle_{A}|V\rangle_{A}+|V\rangle_{B}|H\rangle_{B}\right)$
(49) $\displaystyle|\Phi^{-}\rangle$
$\displaystyle\rightarrow\frac{1}{\sqrt{2}}\left(|H\rangle_{A}|H\rangle_{B}-|V\rangle_{A}|V\rangle_{B}\right)$
(50) $\displaystyle|\Psi^{+}\rangle$
$\displaystyle\rightarrow\frac{1}{\sqrt{8}}(|H\rangle_{A}|H^{\prime}\rangle_{B}+|H^{\prime}\rangle_{A}|H\rangle_{B}+|V\rangle_{A}|V^{\prime}\rangle_{B}+|V^{\prime}\rangle_{A}|V\rangle_{B}$
$\displaystyle+|H\rangle_{A}|V^{\prime}\rangle_{B}-|H^{\prime}\rangle_{A}|V\rangle_{B}+|V\rangle_{A}|H^{\prime}\rangle_{B}+|V^{\prime}\rangle_{A}|H\rangle_{B})$
(51) $\displaystyle|\Psi^{-}\rangle$
$\displaystyle\rightarrow\frac{1}{\sqrt{8}}(|H\rangle_{A}|V^{\prime\prime}\rangle_{A}+|H^{\prime\prime}\rangle_{A}|V\rangle_{A}-|H\rangle_{B}|V^{\prime\prime}\rangle_{B}-|H^{\prime\prime}\rangle_{B}|V\rangle_{B}$
$\displaystyle+i|H\rangle_{A}|V^{\prime\prime}\rangle_{B}-i|H^{\prime\prime}\rangle_{A}|V\rangle_{B}+i|V\rangle_{A}|H^{\prime\prime}\rangle_{B}-i|V^{\prime\prime}\rangle_{A}|H\rangle_{B}),$
(52)

where the subscripts $A$ and $B$ denote the two output ports of the second
50:50 beam splitter, and the prime (double prime) suggests that the photon
carries a $\Delta T_{0}$ ($\Delta T_{1}$) delay introduced by the first
(second) delay line. As such, a time and polarization resolving measurement on
both photons discriminates the four Bell states. The complete Bell-state
measurement apparatus was integrated in the experimental setup depicted in
Fig. 36b. The hyper-entanglement source entailed a PPKTP crystal embedded in a
Sagnac interferometer introduced in Sec. 2.2.3. The crystal was pumped by a
continuous-wave 405 nm diode laser to produce more than 200,000 hyper-
entangled photon pairs per second. Alice executed the encoding operations to
map the photon pairs to one of the four Bell states while Bob leveraged the
complete Bell-state measurement apparatus to infer Alice’s encoded classical
bits. The measurement data plotted in Fig. 36c show that the success
probability in discriminating all four Bell states exceeded 90%. This quantum
superdense experiment demonstrated a capacity of $1.665\pm 0.018$ per detected
photon pair, approaching the limit of 2 bits derived from qubit transmission.

In 2018, Hu et al. demonstrated the idea of superdense coding using photon
pairs that were encoded in path and polarization [451] degrees of freedom. As
illustrated in Fig. 37a, the experiment hinged on a polarization-path
entanglement source comprising a PPKTP crystal in a Sagnac interferometer. The
source was pumped bidirectionally by two beams to create entangled photon
pairs encoded in both polarization and path degrees of freedom. This joint
path-polarization encoding was used to address four-dimensional subsystems
called “ququarts” that express an extension of qubits. Each output arm of the
Sagnac interferometer consisted of two paths, i.e., a1, a2 in the upper arm
and a3, a4 in the lower arm. The quantum state of the upper-arm photon was
specified as follows: (H, a1) $\rightarrow|0\rangle$, (V, a1)
$\rightarrow|1\rangle$, (H, a2) $\rightarrow|2\rangle$, and (V, a2)
$\rightarrow|3\rangle$, while the quantum state of the lower-arm photon was
specified as: (H, a3) $\rightarrow|0\rangle$, (V, a3) $\rightarrow|1\rangle$,
(H, a4) $\rightarrow|2\rangle$, and (V, a4) $\rightarrow|3\rangle$. Alice
exploited four computer-controlled liquid crystal variable retarders to
produce five entangled ququart states that could be discriminated by Bob’s
measurement apparatus:

$\displaystyle|\Psi\rangle_{11}$
$\displaystyle=\frac{1}{2}\left(|00\rangle+|11\rangle+|22\rangle+|33\rangle\right)$
(53) $\displaystyle|\Psi\rangle_{12}$
$\displaystyle=\frac{1}{2}\left(|00\rangle-|11\rangle+|22\rangle-|33\rangle\right)$
(54) $\displaystyle|\Psi\rangle_{13}$
$\displaystyle=\frac{1}{2}\left(|00\rangle+|11\rangle-|22\rangle-|33\rangle\right)$
(55) $\displaystyle|\Psi\rangle_{14}$
$\displaystyle=\frac{1}{2}\left(|00\rangle-|11\rangle-|22\rangle+|33\rangle\right)$
(56) $\displaystyle|\Psi\rangle_{23}$
$\displaystyle=\frac{1}{2}\left(|01\rangle+|10\rangle-|23\rangle-|32\rangle\right).$
(57)

The theoretical channel capacity subject to the encoding scheme was
$\log_{2}5=2.32$ bits per channel use. The measurement apparatus consisted of
a polarizing beam splitter to interfere the two photons followed by additional
polarizing beam splitters and beam displacers to extract the polarization and
path information. The experimental data for quantum-state discrimination are
plotted in Fig. 37b, showing success probabilities in excess of 90% for all
five entangled quqart states. This led to a measured $2.09\pm 0.01$ bits per
channel use, exceeding the 2 bit per channel use limit for qubit-based quantum
superdense coding protocol.

Figure 37: Experiment of superdense coding with quqarts. (a) Experiment setup
comprising a source for entangled quqarts, encoding module based on liquid
crystal variable retarders, and a measurement apparatus for quantum-state
discrimination. (b) Experimental data for discrimination between the five
entangled quqart states, showing high success probabilities. Figures reprinted
from Ref. [451].

###  9.2 Communication Assisted by CV Entanglement

The CV version of quantum superdense coding schematically shown in Fig. 38 was
proposed by Braunstein and Kimble [452] and first demonstrated by Li et al.
[453]. In the protocol, Alice and Bob establish pre-shared entanglement in a
TMSV state via an entanglement-distribution channel. To encode, Alice performs
a displacement operation $\hat{D}(Q+iP)$ to shift her share of the TMSV state,
the signal, in the phase space. The displacement on the $\hat{q}$ ($\hat{p}$)
quadrature is parametrized by $Q$ ($P$). The encoded signal is sent to Bob
through a quantum channel. Bob’s quantum receiver comprises a 50:50 beamp
splitter and two homodyne detectors that measure, respectively, the $\hat{q}$
and $\hat{p}$ quadrature. The beam splitter mixes Bob’s share of the TMSV
state, the idler, with the received signal, generating two single-mode state
squeezed in the $\hat{q}$ and $\hat{p}$ quadrature. The mean of the the two
single-mode squeezed state is $Q^{\prime}+iP^{\prime}$, where
$Q^{\prime}=Q/\sqrt{2}$ and $P^{\prime}=P/\sqrt{2}$ assuming an ideal channel.
As such, multiplying the measurement outcomes from both homodyne detectors by
$\sqrt{2}$ yields an unbiased estimator for Alice’s encoded message.

Figure 38: Schematic for quantum superdense coding with continuous variables.
Alice and Bob pre-share a TMSV state prior to communication. Alice’s encoding
entails a displacement operation that shifts the signal in the phase space.
The encoded signal is transmitted to Bob through a quantum channel. Bob
interfers the received signal and the idler on a 50:50 beam splitter to
produce two single-mode displaced squeezed states. Two homodyne measurements
are followed to detect the quadratures and infer Alice’s encoding.

Braunstein and Kimble showed that the CV quantum superdense coding protocol
outperformed a classical communication scheme based on transmitting displaced
coherent states with the same power in tandem with a heterodyne detector
[452]. However, one would need 6.78 dB of “break-even squeezing” to beat the
ultimate classical capacity, and in the limit of infinite squeezing CV quantum
superdense coding enables a two-fold enhancement in the communication rate
over the optimal classical scheme. Such a squeezing level would be quite
challenging to achieve in the presence of many imperfect components in a
system-level demonstration. Worse, the amount of break-even squeezing is
derived based on a perfect quantum channel with no loss and additive noise.
Channel loss and noise is known to quickly diminish the squeezing at the
quantum receiver, thereby questioning the practical benefit of the CV quantum
superdense coding protocol.

The challenge faced by CV quantum superdense coding over practical channels
behooves researchers to quantify the advantage of EACOMM over the ultimate
classical capacity. To this end, let us compare the EA classical capacity and
the classical capacity without EA, i.e., the Holevo-Schumacher-Westmoreland
(HSW) capacity. The EA classical capacity over a lossy channel plagued by a
large noise background is found to be

$C_{E}(\mathcal{L}^{\kappa,N_{B}})=\frac{1}{N_{B}}\kappa
N_{S}(1+N_{S})\log_{2}\left(1+\frac{1}{N_{S}}\right),$ (58)

where $N_{B}\gg 1$ is the mean photon number of the background mode, $\kappa$
is the transmissivity of the channel, and $N_{S}$ is the mean photon number
signal or idler mode of the TMSV state. By comparison, the classical capacity
without EA, i.e., the Holevo-Schumacher-Westmoreland (HSW) capacity reads

$C(\mathcal{L}^{\kappa,N_{B}})=g(\kappa N_{S}+N_{B})-g(N_{B}),$ (59)

where $g(N)=(N+1)\log_{2}(N+1)-N\log_{2}(N)$ is the von Neumann entropy of a
thermal state with mean photon number $N$. In the limit of a low-brightness
entanglement transmitter,

$\lim_{N_{S}\rightarrow
0}\frac{C_{E}(\mathcal{L}^{\kappa,N_{B}})}{C(\mathcal{L}^{\kappa,N_{B}})}=\infty,$
(60)

showing that entanglement can, in principle, enable an infinite-fold advantage
in the communication rate over a very noisy channel. To reap the quantum
advantage, a recent theoretical work devised the entanglement sources,
encoding formats, and quantum receivers [8]. Among the investigated quantum
receivers, the phase-conjugate receiver (PCR) presents an experimentally
viable route toward surpassing the HSW capacity to achieve a quantum advantage
in communication. Ref. [10] reported an EACOMM experiment based on low-
brightness non-degenerate TMSV source and a PCR, as sketched in Fig. 39a. The
entanglement transmitter entailed a PPLN crystal that generated the signal and
idler with $N_{S}\ll 1$. The signal and idler were distributed to Alice and
Bob through low-loss optical fibers. Alice’s encoded her classical information
by performing binary phase-shift keying on the signal and sent the encoded
signal to Bob through a very lossy and noisy channel. Akin to the optical QI
experiment, the channel noise was emulated by mixing thermal light with the
signal. Bob’s PCR first exploited a second PPLN crystal to produce the phase
conjugate of the received signal. A 50:50 beam splitter then interfered the
idler and the phase conjugate and transmited the light on its two output ports
to a pair of high-efficiency photodiodes in a balanced setting. The difference
photocurrent of the two photodiodes was used to infer Alice’s encoded bit. The
experimental data plotted in Fig. 39b show that the EACOMM rate surpassed the
classical limit set by the HSW bound. Moreover, the achieved EACOMM rate
substantially outperformed the rate attained by a practical classical
communication setup also built in the experiment.

Figure 39: Entanglement-assisted communication experiment. (a) The EACOMM
experimental setup comprising a TMSV source, a phase-modulation encoder
operated by Alice, and a PCR at Bob’s terminal. (b) Normalized information
rate for EACOMM (blue) and classical communication (red) at different channel
background noise levels. Black solid line is the classical limit specified by
the HSW capacity. Solid line: theory; dot: experimental data; dashed line:
theory with ideal experimental parameters. (b) reprinted from Ref. [10].

##  10 Outlook

The next-generation entanglement-based QIT beyond the three classes of
protocols and their associated applications as discussed in this article is on
the verge, thanks to the remarkable strides across the spectrum of functional
quantum modules including large-scale entanglement sources [120, 129, 216, 64,
63, 62], efficient quantum transducers [454, 455, 290, 456, 457], long-lived
quantum memories [458, 459, 460, 461, 462, 463], high-fidelity quantum gates
[464, 465, 466, 467], near-unity-efficiency [468] and photon-number-resolving
[469, 470] detectors, and novel quantum receivers [471, 10, 472, 473, 474,
475, 476, 477, 478, 479, 480, 481], in conjunction with auxiliary classical
real-time processing capabilities. An envisaged architecture for QIT protocols
based on the emergent enabling quantum modules is illustrated in Fig. 40. The
quantum transmitter entails sources of encoded entangled states to address
distributed sensing, data processing, and decision problems. The encoded
entangled states, in tandem with the quantum receiver and processor, are
tailored to combat imperfections such as environmental loss and noise [33,
482, 417, 483, 484, 485, 486]. Over recent years, large-scale deterministic
quantum sources have been extensively investigated to serve future scalable
QIT [216, 63, 64, 61, 129, 127, 120]. The signals from the quantum transmitter
interface with physical processes, which are operations realized by quantum
transduction to interconnect various quantum information processing platforms
[288]. The idlers emitted from the quantum transmitter are loaded into quantum
memories, awaiting on-demand retrieval by a next-generation quantum receiver
that features the following characteristics. First, it allows for more general
joint measurements on the signals and idlers, which are proven an essential
ingredient to approach the ultimate quantum-limited performance [487]. Second,
the measurements are non-destructive, producing signals and idlers for the
next stage of processing [488, 489, 490, 491, 489]. Notably, new materials on
the horizon exhibit extraordinary nonlinearities [492, 493], rendering the
quest for quantum non-demolition measurements more practical. Third, with
state-of-the-art electronics, the quantum receiver is capable of generating
real-time feedforward to configure a quantum processor comprising high-
fidelity quantum gates to conduct a joint operation on the signals and idlers.
The quantum processor enhances, as needed, the entanglement shared by the
signals and idlers for the next cycle of information processing. Such an
adaptive architecture based on real-time measurements, feedforward, and
reconfiguration of quantum operations has been demonstrated in quantum
receivers for semi-classical state discrimination. As a recent advance in this
subject, machine-learning techniques were utilized to assist in the design and
implementation of quantum receivers to tackle environmental noise [471]. One
can envision that the adaptive architecture, in conjunction with classical
artificial intelligence, will empower a wealth of entanglement-based QIT
protocols.

Figure 40: Outlook for entanglement-based QIT architecture. Future QIT
protocols would build on multiple stages of entanglement preparation,
processing, and detection, in tandem with an adaptive feedforward mechanism to
unlock the full power of quantum information processing.

The prospective QIT architecture based on the emergent functional quantum
modules will give rise to unprecedented communication, sensing, and data
processing capabilities. For instance, with the quantum transducers quantum
information can be transferred between stationary qubits and flying photonic
qubits to facilitate long-distance quantum communications [455, 494, 495]. The
quantum error correction components can ensure the fidelity of such conversion
[33, 486]. Moreover, the joint encoding at the quantum transmitter and joint
measurement at the quantum receiver will unlock quantum-communication regimes
with no classical analogue, such as superadditive quantum capacities [27, 29,
30] and superactivation over quantum channels [496, 497, 498]. On the sensing
front, recent experiments have demonstrated that entanglement in the optical
domain together with transducers can be leveraged to enhance the performance
of arrayed RF [18] or optomechanical sensors [90]. One may imagine that large-
scale optical entanglement in tandem with efficient quantum transducers,
quantum error correction, and quantum-limited detector arrays would constitute
new powerful metrological tools for a variety of realms such as the search for
new physics [499, 282, 500, 501, 422, 502], inertial navigation [503, 504,
505], and remote sensing based on microwave photonics [506, 507, 508, 509]
enhanced by QIT. With respect to data processing, recent proof-of-concept
experiments have leveraged quantum resources to tackle physical simulation and
machine-learning problems [217, 431, 20, 433, 432]. The potential of quantum-
enhanced data processing will be further unleashed with the new QIT
architecture by virtue of the capability of generating a wide variety of
multipartite entangled state, general joint measurements, real-time
feedforward, and quantum error correction.

Over the past three decades, the humankind has witnessed entanglement’s
transition from a verifiable physical phenomenon to a profound resource
underpinning a plethora of new technologies. With the highly
interdisciplinary, synergistic, and holistic efforts devoted by the scientific
and technological communities, entanglement-enhanced QIT is set to boom in the
years to come, bringing excitements that have yet to imagine.

Funding National Science Foundation Award No. 1920742, 2134830, 2144057, and
2304118, Defense Advanced Research Projects Agency (DARPA) under Young Faculty
Award (YFA) Grant No. N660012014029, and University of Michigan. C. Y. and O.
S. M. L. acknowledge support from the Army Research Office (ARO), through the
Early Career Program (ECP) under the grant no. W911NF-22-1-0088 and the U. S.
Department of Energy for supporting this research through the Office of Basic
Energy, Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering under the Award DE-SC0021069.

Acknowledgments We thank Sophia Economou for the feedback on the manuscript.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement No data were generated or analyzed in the
presented research.

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