Physical Layer Aspects of  Quantum Communications: A Survey
Seid Koudia, Leonardo Oleynik, Mert Bayraktar, Junaid ur Rehman, and Symeon Chatzinotas, Fellow, IEEE

The authors are with the Interdisciplinary Centre for Security, Reliability, and Trust (SnT), Luxembourg, L-1855 Luxembourg.
(emails: {seid.koudia, leonardo.oleynik, mert.bayraktar, junaid.urrehman<EMAIL_ADDRESS>
Quantum communication systems support unique applications in the form of distributed quantum computing, distributed quantum sensing, and several cryptographic protocols. The main enabler in these communication systems is an efficient infrastructure that is capable to transport unknown quantum states with high rate and fidelity. This feat requires a new approach to communication system design which efficiently exploits the available physical layer resources, while respecting the limitations and principles of quantum information. Despite the fundamental differences between the classic and quantum worlds, there exist universal communication concepts that may proven beneficial in quantum communication systems as well. In this survey, the distinctive aspects of physical layer quantum communications are highlighted in a attempt to draw commonalities and divergences between classic and quantum communications. More specifically, we begin by overviewing the quantum channels and use cases over diverse optical propagation media, shedding light on the concepts of crosstalk and interference. Subsequently, we survey quantum sources, detectors, channels and modulation techniques.  More importantly, we discuss and analyze spatial multiplexing techniques, such as coherent control, multiplexing, diversity and MIMO. Finally, we identify synergies between the two communication technologies and grand open challenges that can be pivotal in the development of next-generation quantum communication systems.

§ INTRODUCTION

QC have recently emerged as the cornerstone of Quantum Internet, which envisages a network of interconnected quantum sensors and processors, working together to deliver high-performing applications, which outperform the classical counterparts in terms of sensing accuracy or computational capabilities [1, 2, 3, 4, 5]. Even though quantum phenomena can be observed across a variety of particles, in the QC context photons have been established as the preferred option, either transmitted through wired media such as fiber optics or through wireless media, namely free space optics [6, 7, 8]. Nevertheless, independently of the selected medium, the main challenge in QC nowadays is the low rate of error-free communication, which creates an insurmountable bottleneck against enabling Quantum Internet applications.

It is worth noting that classical communications also suffered from similar limitations in their early days [9]. As technology evolved, the classical communication rates have increased tremendously by efficiently using all available degrees of freedom offered by nature, namely intensity, phase, polarization, frequency, and space [10, 11]. In this direction, one of the most important milestones has been the multiplexing of complex signals across interfering channels and their effective demultiplexing at the receiver side, i.e., MIMO communication. Depending on whether redundant or novel source info was multiplexed, MIMO can either decrease the error rate or improve the data rate of the communication channel, an effect which is widely known as DMT [12].

Following this train of thought, it is natural to pose the research question of whether it would be possible to use a similar rationale to improve the performance of quantum communication channels. As a first step in this direction, this paragraph overviews the classical vs quantum differences, when preparing, transmitting, propagating, receiving, and measuring information. To motivate the discussion, a general structure of possible quantum MIMO communication setup is shown in Fig. <ref>.

§.§ Preparation & Transmission

A common assumption in the classic world is that information can be replicated with minimal effort and as a result introducing redundancy in terms of repeating information is trivial [13]. On the other hand, quantum mechanics forbid the cloning of quantum states [14]. While this might not be an issue when quantum states can be repeatedly prepared at the transmitter based on classical info, it is certainly problematic when quantum sources are employed e.g., quantum computers or sensors. A relevant concept in this direction is quantum cloning, which effectively creates a larger number of low quality output quantum states based on a smaller number of input quantum states [15, 16]. Unfortunately, the output states are not identical to the input and in addition they are entangled, ensuring this way that this method cannot be straightforwardly used to generate redundancy [17, 18]. The second relevant approach is QEC, which finds applications in both computing and communication models [19]. We refer the reader to [20, 21] for an overview.
The second common assumption in the classical information is that the intensity or power of the prepared signal is an analog degree of freedom that can be increased arbitrarily (under public safety constraints) to compensate for the energy that is lost to the environment. However, in the quantum world, the information is encoded in particles (e.g. photons, wavepackets) whose energy levels are dictated by discrete quanta and should not be scaled arbitrarily to avoid the transition into a classical state. For higher energy pulses, distinguishing between different photon number states becomes increasingly difficult due to technological limitations and classical detection remains the only possibility [22].

The general structure of a quantum MIMO channel. Preprocessing and post-processing are required to take full advantage of the higher dimensions offered by the MIMO setup. DV to CV and CV to DV blocks might be required only in some select scenarios, e.g., when connecting DV quantum nodes while transmitting CV modes over the channel.

§.§ Propagation

The most important process towards controlling and mitigating  propagation impairments is channel estimation. In the classic world, multidimensional communication channels can be mathematically represented in the form of complex (intensity and phase) matrices, which capture all possible combinations of input and output dimensions [23, 24, 25]. Assuming a sufficiently long coherence time, during which the matrix remains fixed, each complex coefficient is measured by preparing known sequences of information (pilots), propagating them and measuring the outcome at the receiver [26, 27, 28]. On the other hand, quantum systems are inherently discrete and probabilistic. For example, the propagation environment can impair quantum communications by probabilistically injecting (creation) or absorbing (annihilation) a discrete number of photons [29]. This effect creates a large combinatorial set of probabilities, which is very challenging to estimate (quantum process tomography) without excessive pilot overhead [30, 31]. To be mathematically more exact, quantum channels or processes are modelled through operators over often infinite-dimensional Hilbert spaces. Similarly to their classic counterparts, quantum channels can very well be time-varying, so it remains an open question whether channel resources should be devoted for quantum process tomography or simpler (but less accurate) input-output relationships should be used for quantum channel models [32, 33, 34]. For example, by focusing on specific observables at the receiver and not the full quantum state, classical or semi-classical channel models could be employed [35]. It is worth mentioning that blind feedback-based approaches have also shown promise in quantum channels. Blind refers to the fact that there are no specific pilots or states propagated with the purpose of learning channel coefficients. Feedback-based means that this is usually an iterative process where (classic or quantum) feedback is regularly sent from the receiver to the transmitter, in order to guide the next preparation phase [36, 37]. Such approaches fall under the umbrella of SCAPE [38, 32].

§.§ Reception & Measurement

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child  node I. Introduction
child[xshift=0.1cm]  node I-A. Preparation and Transmission
child[xshift=0.1cm]  node I-B. Propagation
child[xshift=0.1cm]  node I-C. Reception & Measurement
child[xshift=0.1cm]  node I-D. Multidimensional Quantum
Communications & Degrees of Freedom
child[xshift=0.1cm]  node I-E. Reception & Measurement

child  node II. Preliminaries
child[xshift=0.1cm]  node II-A. CV vs DV Systems
child[xshift=0.1cm]  node I-D. Probabilistic Cross-Talk (Classical Interference)
vs Interference (Quantum) and Use Case
child[xshift=0.1cm]  node II-C. Crosstalk in Free Space Optics
child[xshift=0.1cm]  node II-D. Crosstalk in Quantum Circuits

child  node III. Quantum Information in Different Media
child[xshift=0.1cm]  node III-A. Fiber Optics
child[xshift=0.1cm]  node III-B. Free Space Optics
child[xshift=0.1cm]  node III-C. Integrated Photonics

child  node IV. Quantum Communication
child[xshift=0.1cm]  node IV-A. Quantum Sources
child[xshift=0.1cm]  node IV-B. Optical Channels
child[xshift=0.1cm]  node IV-C. Detection
child[xshift=0.1cm]  node IV-D. Quantum Modulation Techniques

child  node V. Interacting Quantum Channels
child[xshift=0.1cm]  node V-A. Quantum Coherent Control
child[xshift=0.1cm]  node V-B. Multiplexing and Diversity
child[xshift=0.1cm]  node V-C. Quantum MIMO

child  node  VI. Connecting DV Quantum Nodes with CV
Quantum Links
child[xshift=0.1cm]  node VI-A. Cat Codes
child[xshift=0.1cm]  node VI-B. GKP Codes
child[xshift=0.1cm]  node VI-C. Truncation of Infinite-Dimensional Hilbert Space

child  node VII. Open Challenges
child[xshift=0.1cm]  node  VII-A. Interference of Photons: The Hong–Ou–Mandel
child[xshift=0.1cm]  node VII-B. Physical Phenomenon Beyond Losses
child[xshift=0.1cm]  node  VII-C. Performance Assessment of Quantum
Communication Systems
child[xshift=0.1cm]  node VII-D. Selective Fading
child[xshift=0.1cm]  node VII-E. Diversity and Multiplexing in Quantum MIMO
child[xshift=0.1cm]  node VII-F. Interacting Quantum Channels
child[xshift=0.1cm]  node VII-G. The Interplay Between DV and CV Systems

child  node VIII. Conclusions

child[missing] ;

Paper Structure

The concept of measurement is fundamental in QC systems, because it signifies the process which extracts classical information from a quantum state [39, 40]. Measurement does not necessarily happen right after the reception, for example when the received quantum state is fed directly to a quantum computer for further processing. However, eventually a post-processed quantum state has to be measured to extract information that can be perceived by humans or used for a classical computer interface. Compared to classic world, the performance metrics are not surprising, namely the quantum communication rate (qudits/sec) and the quantum state fidelity—a similarity metric between the transmitted and received state. Nevertheless, the actual process of measurement differs fundamentally. In the classic world, received information can be stored and measured an arbitrary number of times with the same outcome. In the quantum world, every time a measurement is performed, it unavoidably alters/evolves the quantum state. More importantly, unless the correct measurement operator is applied, there is a chance that no useful classical information will be extracted. In the special case, where the prepared quantum states were encoded based on classic information (e.g. output of digital sensors), the classical performance metrics can be also used, namely classic communication rate (bits/sec) and error probability/rate.

§.§ Multidimensional Quantum Communications & Degrees of Freedom

Before proceeding with this discussion, it is crucial to define the concept of a channel use or slot. In the classic world, a channel slot refers to an event where the communication channel is occupied to propagate information without, in theory, affecting the adjacent slots. The most obvious examples are time or frequency slots, where the orthogonality with adjacent slots is determined by pulse shaping. Another example is spatial dimensions, where channel slots can be seen as spatial directions, achieved by creating orthogonal directive beams across the propagation environment. Independently of the DoF used to create multiple channel slots, the effect of orthogonality is achieved through pre-processing at the transmitter and/or post-processing at the receiver. The aforementioned DoF have been already successfully exploited in real-life and are widely deployed in commercial services that are used on a daily basis (e.g. digital communications over copper, fiber, wireless). However, the extension of these paradigms to the quantum domain are not straightforward. Confining transmissions into a constrained time, frequency or spatial DoF highly depends on our capability to design and implement ideal sources. In addition, the interaction of light quanta over the propagation medium differs considerably depending on the employed type of quantum states (discrete variables vs continuous variable states). Finally, demultiplexing and detecting quantum states at the receiver side requires a fundamentally different approach, due to the no-cloning theorem. In the rest of this manuscript, we elaborate on these intricate points and we point to promising system models, transceiver architectures and open research challenges in this domain.

The main contributions of this manuscript include:

* We review the characteristics of optical beam propagation in free space from classical optics point-of-view and find its applications for quantum communications in FSO. More concretely, we highlight the use of classical techniques, such as adaptive optics, and identify the situations where these can be used for quantum communications as well.

* We draw parallels between the classical and quantum communication systems with the main aim of identifying situations where classically developed methods have a potential to be implemented in quantum communication systems for future developments. The main hypothesis here is that we do not need to develop wireless quantum communication systems from scratch but some of the already developed (classical) techniques can be repurposed with little effort for wireless quantum communication systems of the future. Some of the identified techniques in this article include channel modeling based on physical phenomenon, MIMO for classical wireless communications, and interference/channel cross-talk management.

* We identify fundamental models for the essential components of an end-to-end wireless quantum communication systems that need to be defined for effective utilization of classical communication techniques in quantum domain.

* We review promising techniques in quantum communication systems that may have a large role/impact in the quantum communication systems of future. These include the quantum equivalents of classically developed techniques, e.g., MIMO channels and WDM, as well as some uniquely quantum methods, e.g., quantum coherent control.

The paper structure is summarized in Fig. <ref> and is organized as follows. In Sec. <ref> we define some preliminaries and make a distinction between some terms that are used both in classical and quantum communications with different meaning. In Sec. <ref> we different mediums that act as a communication channel for quantum information and the basic modeling of flow of information in these mediums. In Sec. <ref>, we review the basic components of quantum communications, i.e., sources, channel models, and detectors. The techniques to combine quantum channels are discussed in Sec. <ref>. Open challenges are discussed in Sec. <ref> followed by conclusions in Sec. <ref>.

Table of Acronyms

Acronym  Definition

MIMOmultiple-input and multiple-output

QCquantum communications
DMTdiversity-multiplexing trade-off
DOFdegree of freedom
SCAPEsimultaneous communication and parameter estimation
DoFdegree of freedom
CVcontinuous variable
DVdiscrete variable
FSOfree space optics
KPIkey performance indicator
MCFmulticore fiber
QKDquantum key distribution
RFradio frequency
SKRsecret key rate
QWDMquantum wavelength division multiplexing
SMFsingle-mode fiber
WCPweak coherent pulse
SPPsingle-photon pulse
BSbeam spliter
MDImeasure-device independent
PACprobably approximately correctly
RRDPSround-robin differential-phase shift
SLMspatial light modulator
OAMorbital angular momentum
QECquantum error correction
SPAsignal-processing approach
QMLquantum machine learning

§ PRELIMINARIES

In this section, we set some notation and define some key terminology that we use throughout the manuscript. A list of acronyms used throughout the manuscript are provided in the Table <ref>.

§.§ CV vs DV Systems

Quantum information processing systems can be characterized into one of the two broad categories, i.e., discrete- and continuous-variable quantum system. The term discrete or continuous refers to the spectra of measurable/observable quantities. Encoding and processing based on the number of photons in a pulse is discrete variable processing, whereas encoding and processing in field variables such as amplitude or phase results into a continuous-variable system [22].

A quantum system is said to be a DV system when its associated observables have a discrete spectrum. That is, when the measurement outcomes result into one of the finitely many outcomes. In DV quantum information systems, the basic unit of information is a quantum bit (qubit), which is a system with a two-level DOF. A qubit can be expressed as a superposition of two basis states[A qudit is the qubit generalization to $d$ orthogonal states quantum superposition rather than two states], e.g.,

\begin{align}
\ket{\psi} = \alpha \ket{0} + \beta\ket{1},
\label{eq:qubit_ket}
\end{align}

where $\left\{ \ket{0}, \ket{1}\right\}$ is an orthonormal basis; and $\alpha, \beta \in \mathds{C}$ normalize to $\left| \alpha \right|^2 + \left| \beta \right|^2 = 1$. In (<ref>), the implicit assumption is that the qubit is in a pure state, i.e., in a well-defined quantum state with no statistical mixing. A more general representation of a qubit is the density operator form that takes into account the possible classical mixing of multiple pure states. A density matrix $\rho$ is a convex combination of one or more pure states, i.e.,

\begin{align}
\rho = \sum_{i = 1}^{n} p_i \ket{\psi_i}\bra{\psi_i},
\end{align}

where the pure state $\ket{\psi_i}$ appears in the ensemble with probability $p_i$. A quantum state that is not pure is called a mixed state. Examples of DV systems include spin 1/2 particles such as electrons, the two lowest energy states of semiconductor quantum dots, or two polarization states of a single photon [41].

A CV quantum system is defined over an infinite-dimensional Hilbert space and the associated observables have continuous spectra [41]. Examples of CV quantum systems include quantized modes of bosonic systems such as the different DOF of electromagnetic field, vibrational modes of solids, and nuclear spins in a quantum dots [41]. Similar to the DV case, a CV quantum system in pure state can be described as the superposition of eigenstates, i.e.,

\begin{align}
\ket{\psi} = \int \braket{x}{\phi}\ket{x}\,\mathrm{d}x,
\end{align}

where $\braket{x}{\phi}$ is the wave function of the state in the continuous basis $x$ [22].

The best known example of continuous variable is the quantized harmonic oscillator that is described by CV quadratures such as position and momentum [41]. The Hamiltonian $\hat{H}_k$ of a quantum harmonic oscillator can be expressed in terms of annihilation and creation operators[We are considering normalized creation and annihilation operators leading to quadratures of the same dimension]

\begin{align}
\hat{H}_k = \left( \hat{a}_k^{\dagger}\hat{a}_k + \frac{1}{2}\right),
\end{align}

This Hamiltonian can be represented in terms of position $\hat{x}_k$ and momentum $\hat{p}_k$ operators

\begin{align}
\hat{H}_k = \frac{1}{2} \left( \hat{p}_k^2 + \hat{x}_k^2\right),
\end{align}

\begin{align}
\hat{a}_k &= \frac{1}{\sqrt{2}}\left( \hat{x}_k + \dot{\iota} \hat{p}_k\right),\\
\hat{a}_k^{\dagger} &= \frac{1}{\sqrt{2}}\left(  \hat{x} - \dot{\iota} \hat{p}_k\right)
\end{align}

and the commutation relation $\left[\hat{x}_k, \hat{p}_{k'}\right] = \dot{\iota}\delta_{k, k'}$ holds.

While working with DV quantum systems,  the density operator formalism is preferred, in CV systems other representations are often more convenient. The Wigner function, for example, is suitable to describe the effects on quantum observable which may arise from quantum theory and statistical effect. The Wigner function behaves like classical probability and enabling calculations of mean and variances of the quadratures [42]. In contrast to classical probability, Wigner function can become negative.[The negativity of the Wigner function is considered as a measure of a genuinely quantum non-Gaussianity [43]. ] For a given CV quantum state with density operator $\rho$, the Wigner function can be obtained [42, 44]

\begin{align}
W\left(x, p\right) = \frac{2}{\pi}\int e^{\dot{\iota}4yp}\bra{x - y}\rho \ket{x + y}\, \mathrm{d}y,
\end{align}

where the integration is from $-\infty$ to $+\infty$. Gaussian states are the ones whose Wigner representation is Gaussian and can be completely specified with the first two moments [41].

Indeed, any CV state can be expanded in the Fock space basis representation, comprising photon number states \(\{\ket{n}\}_{n=0}^{\infty}\). Different representations of CV systems can be used depending on the context. For instance, when studying the entanglement of Gaussian states or their behavior under different Gaussian transformations, the Wigner or the Glauber-Sudarshan representations are appropriate. Conversely, when orthogonality is an important consideration, the Fock representation is more suitable as it provides orthogonality through its inherent orthonormal basis.

§.§ Probabilistic cross-talk (classical interference) vs interference (quantum) and Use Cases

The term “interference” is used in both wireless communication systems and quantum information systems, though with different meanings. In quantum information systems, interference refers to the interaction of matter waves. This interaction can occur between matter waves associated with different (sets of) physical particles or, more commonly, through the self-interference of a matter wave associated with a single (set of) physical particle. Designing quantum information processing protocols involves careful tracking of constructive and destructive interference as a result of this interaction, both of which can be desirable. The most well-known experiment for demonstrating quantum interference is Young's double-slit experiment where the wavefunctions of single photons interact with themselves to give rise to an interference pattern on the screen. The most well-known application/use case is Grover's quantum algorithm for database search. In Grover's algorithm, a marked element is to be searched in a database. Through a carefully designed quantum routine, the probability of measurement outcome corresponding to the marked element (other elements) is increased (decreased) by utilizing quantum interference.

In classical communication systems, interference generally refers to the disruptive modification of a signal caused by the presence of another signal in its vicinity. Interference in classical communication systems can be of different types, e.g., co-channel interference, inter-symbol interference, adjacent-channel interference, multi-user interference and spatial interference. Similar to quantum interference, interference in classical communication systems can be among different electromagnetic waves (e.g., in co-channel and adjacent channel interference) or self-interaction of an electromagnetic wave (e.g., inter-symbol interference, temporal and spectral dispersion through the propagation channel).

In contrast to quantum interference, classical interference is almost always undesirable and degrades the performance of the communication system, unless intentionally manipulated at the trasceivers. In the same context, untreated interference affecting optical systems (free-space or fiber) has been so far modelled as an additional source of noise under the term crosstalk.

To avoid ambiguity in this paper, we reserve the term interference for the quantum interference discussed in the previous paragraph and reserve the term cross-talk for the interaction of different signals regardless of their classical or quantum nature.

§.§ Crosstalk in Free Space Optics

Classical optical communication techniques play a crucial role in characterizing and mitigating the effects of turbulence on quantum communication systems. Turbulence in the atmosphere can cause random phase and intensity variations in propagating light beams, leading to unwanted distortions in the received field. These distortions affect most of the degrees of freedom of the light, including intensity, phase, and polarization, which are critical in quantum communication. Classical beam propagation methods can be adapted and enhanced to improve the performance of quantum communication systems in turbulent environments. These methods allow for calculating the received field and intensity expressions not only in free space but also in turbulent environments. By modeling the effects of turbulence on the transmitted quantum states, it becomes possible to predict and quantify the level of distortion experienced by the quantum signals. Moreover, classical techniques enable the calculation of the transmissivity of the beam, taking into account the impact of turbulence. By accurately assessing the transmissivity, researchers can determine the key rate of the quantum communication system, which is essential for evaluating its overall performance. In essence, by leveraging classical optical communication methodologies, researchers can develop strategies to mitigate the effects of turbulence on quantum communication systems. This includes understanding how turbulence affects the propagation of quantum states, modeling its impact on key parameters such as transmissivity, and ultimately optimizing the system to achieve reliable and efficient communication even in turbulent environments.

§.§ Crosstalk in Quantum Circuits

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Markovian cross-talk acting on different gates that are spacelike separated.

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Non-Markovian cross-talk acting on different gates that are timelike separated.

An illustration of different types of cross-talk that can affect quantum circuits

Cross talk in quantum circuits refers to the undesired interaction among qubits, leading to errors and reduced fidelity in quantum computations. In an ideal quantum system, qubits should interact only with intended control signals or with other qubits in a controlled manner during the execution of quantum gates. However, due to physical proximity and the complex nature of quantum hardware, qubits can influence each other. This unwanted interaction can manifest as cross talk, which can introduce noise, causes decoherence, and results in computational errors. As a matter of fact, we distinguish two types of cross talk, Markovian and non-Markovian cross talks. Markovian cross talk is a memoryless process that afffects the computational qubits in a spacelike way, creating undesirable correlations which affects the readout after measurements as illustrated in Fig. <ref>. Differentely, non-Markovian cross talk is a process that affects the computational qubits in a timelike manner, creating a memory of the individual systems of their past evolution as illustrated in Fig. <ref>. quantum
Cross talk is particularly problematic in scalable quantum computing, where maintaining high coherence and precise control over numerous qubits is crucial. Mitigating cross talk involves careful design and isolation of qubits, improving error correction techniques, and developing more sophisticated control mechanisms to ensure that qubits only interact as intended. Addressing cross talk is essential for advancing the reliability and efficiency of quantum circuits, paving the way for more robust quantum computing applications. In [45] the authors designed a cross talk model and benchmarked it with the IBM QPU , in order to investigate the presence of cross talk noise and characterize it. The influence  of cross talk on different gates and circuits and the efficiency of the designed model were addressed  by a tomography on the IBM hardware, showing that the latter is significantly influenced by cross talk effects. A different approach was taken by the authors of [46] who considered cross talk effects arising from noise in a measurement readout stage where errors appear not only on individual qubits but also on multiple ones collectively. The authors have presented an error mitigation protocol consisting of a preproccessing stage, where local quantum gates are applied before the measurement setup, followed by a post-processing stage treating measurement outcomes to recover noiseless data. A similar approach using post-processing techniques for readout error mitigation in NISQ quantum devices was established in [47]. The authors considered a correlated noise model
based on Continuous Time Markov Processes to simulate the cross talk effect. The proposed error mitigation methods
were demonstrated experimentally for measurements of up to 20 qubits. Indeed, in a real world, cross talk effects in NISQ devices might follow non-markovian models. The authors of [48] provided a way for characterizing non-markovian errors (arising from crosstalk or from other sources) and experimentally testing it on the IBM hardware, using the process tensor formalism. In the same spirit, using tensor-network quantum processes [49], non-markovian effects in quantum processors can be characterized and mitigated. This has been reported in [50, 51].

§ QUANTUM INFORMATION IN DIFFERENT MEDIA

In this section, we review different media that acts as a communication channel for quantum communication systems, i.e., fibre optics and free-space.

§.§ Fibre Optics

§.§.§ Fiber Optics MIMO

MCF whose cross-section is shown in Fig. <ref> have garnered significant research attention recently due to their promising advantages in telecommunications and data transmission. These fibers offer the potential for achieving higher data rates while also providing space efficiency, fault tolerance, and reduced power consumption compared to traditional single-mode fibers. However, challenges such as crosstalk between the cores have been identified alongside these advantages. Crosstalk can occur due to interference between the cores, potentially impacting signal quality and transmission efficiency. Despite this challenge, researchers have been actively working on improving MCF performance. Early studies measured inter-core cross-talk (XT) at around -40dB/km[52], but advancements have improved this to approximately -60dB/km[53]. These improvements are influenced by the coupling methods used. Both non-coupled and coupled MCFs have been thoroughly analyzed. The impulse response width is also presented as a function of the operating wavelength and core diameter[54]. The influence of crosstalk on the signal-to-noise ratio(SNR) is critical. SNR penalty in MCF is derived and proved experimentally considering the crosstalk effect [55].

§.§.§ Quantum Fiber Optics

Additionally, researchers have investigated crosstalk power caused by forward Raman noise and forward Raman scattering noise resulting from inter-core crosstalk. Proposals have been made to model and evaluate QKD performance in multi-core and few-mode fibers, considering factors like dark current and inter-core crosstalk as background noise sources [56]. In these studies, authors have analyzed the feasibility of QKD transmission in multi-core fibers with quantum channels alone or both quantum and classical channels. Such research contributes to understanding the potential and limitations of MCFs in practical applications, paving the way for their future deployment in advanced telecommunications systems[57].One notable area of focus is QKD, where MCFs have become a trending topic. For instance, a 52km QKD link was successfully established using a 4-core MCF, achieving a transmission rate of 51.5kbps[58]. Furthermore, CV-QKD is less affected by crosstalk overall. However, when the CV-QKD wavelength overlaps with a classical channel, the excess noise from crosstalk rises sharply. Nevertheless, a secret key rate of 94.2 Gbps is still anticipated[59].

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Crosstalk in six-core fiber. Signals transmitted through neighboring cores at the same wavelength can crosstalk with each other, as illustrated with cores $C_1$ and $C_2$.

§.§.§ Crosstalk Mitigation

The mitigation of crosstalk in MCF has recently become a trending topic.  MCF design, core effective area, and propagation configurations are some of the crucial aspects that should be considered to mitigate the crosstalk. Different MCF designs with trench or moat-assisted cores are evaluated, highlighting that larger trench/moat volumes significantly reduce inter-core crosstalk and radiation leakage loss. However, these designs pose manufacturing challenges and hydrogen sensitivity issues. The study shows that bidirectional transmission can further reduce crosstalk compared to co-propagating transmission. Optimal inter-core distances for minimizing crosstalk and maximizing SNR are identified, with specific values for different core effective areas. The results indicate that, despite potential higher SNR for larger effective area designs, smaller effective area designs may offer higher overall capacity due to accommodating more uncoupled cores [60]. In another study, key strategies include bi-directional transmission, which can suppress crosstalk by about 20 dB per core pair, and specialized core designs like trench-assisted (TA) and step-index (SI) MCFs that lower inter-core interference through careful geometric and refractive index adjustments. The proposed approaches also involve advanced routing and spectrum allocation algorithms, such as core prioritization and spectrum-splitting, to optimize spectral resource use and mitigate crosstalk.  New wavelength-dependent crosstalk models are presented, and their effectiveness is evaluated through simulations, highlighting that TA-MCFs with bi-directional transmission significantly enhance network capacity and link efficiency compared to traditional methods[61]. Orthogonal filtering techniques suppress inter-core crosstalk (IC-XT) in weakly-coupled multicore fiber (WC-MCF) transmission systems. This method simplifies the reception of signals in WC-MCF systems by eliminating the need for MIMO digital signal processing, enabling individual lane extraction. The study demonstrates effective IC-XT mitigation using a pair of orthogonal filters in a 2-core MCF, achieving a maximum Q factor improvement of 6 dB for 600-Gbps 64-QAM signal transmission over 100 km. The research employs a transmission model based on coupled nonlinear Schrödinger equations to simulate and verify the proposed method's efficacy. Additionally, the impact of spatial mode dispersion on filter orthogonality performance is analyzed, highlighting the proposed system's considerable tolerance to IC-XT and its potential to enhance design flexibility for ultra-high-density WC-MCFs [62]. Another method is designing and fabricating ultra-low crosstalk and low-loss MCF, focusing on reducing crosstalk in multicore fibers through various methods. It is achieved significantly low crosstalk and attenuation levels by employing trench-assisted pure-silica cores arranged hexagonally and utilizing fiber bends. Using an approximation model based on a coupled-mode theory with the equivalent index model, a relationship is derived among crosstalk, fiber parameters, and fiber bend. The mean crosstalk is estimated to be less than -30 dB even after 10,000 km of propagation. This study highlights the importance of managing fiber bends and using trench-assisted cores to achieve these results[63]. The other methods are the development and advantages of trench-assisted multi-core fiber (TA-MCF) for reducing crosstalk in high-density fiber optic systems. TA-MCF is proposed as an alternative to traditional step-index multi-core fibers (S-MCF), highlighting its ability to significantly reduce crosstalk through an index trench structure surrounding each core. This trench structure minimizes the overlap of electric fields between adjacent cores, thereby reducing crosstalk. The simulation results demonstrate that TA-MCFs can achieve crosstalk values up to 20 dB lower than S-MCFs over a 100 km distance. Additionally, the study includes the fabrication and testing of two types of TA-MCFs, which confirmed the simulation results, showing excellent crosstalk reduction, bending loss, and cutoff wavelength characteristics. The findings suggest that TA-MCFs are promising candidates for future high-capacity, long-haul optical communication systems[64]. A hybrid analytic-numerical method reduces crosstalk in multicore fibers by optimizing the amplitude and frequency of random core radius fluctuations along the propagation direction. This method employs coupled mode theory and first-order perturbation theory to analytically determine noise amplitude parameters that minimize power transfer between inner and outer cores, validated by numerical experiments using finite difference beam propagation methods. The study reveals that moderate-noise regions effectively suppress crosstalk, with noise amplitude and fluctuation frequency parameters tailored to achieve optimal suppression. The proposed approach is generalizable to various optical systems described by coupled mode theory, facilitating efficient design and prediction of crosstalk suppression in multicore fibers and similar structures [65].

§.§ Free space optics

§.§.§ MIMO FSO

The efficacy of FSO and free space QKD is intricately tied to atmospheric turbulence. The unpredictable refractive index variations, random intensity fluctuations (scintillation), and erratic beam movements across the transverse receiver plane (beam wander) significantly impact communication performance. These atmospheric effects can be mitigated utilizing MIMO configuration as given in <ref>. It is presented that the scintillation index of MIMO FSO systems is mitigated by up to 1/3 as compared to the single-input single-output(SISO) configuration; therefore, it brings \(10^{-2}\) times less BER [66]. Aperture average scintillation of the FSO system is also inversely proportional to MIMO complexity as it is in pointlike scintillation [67].   Similarly, the BER of multiple-input single-output (MISO) decreases as the number of transmitters increases [68]. Furthermore, array-type beams are more resistive against turbulence [69].

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Crosstalk in a MIMO FSO system operating in atmospheric turbulence. Two optical beams (depicted in green and red), transmitted by $T_1$ and $T_2$, crosstalk each other before reaching the receiver's detectors, $R_1$ and $R_2$.

In addition to the mentioned DoF, the performance of FSO systems is significantly influenced by spatial, spectral, and polarization DoF.
Spatial DoF is particularly affected by the divergence of the optical beam caused by the turbulent environment. Atmospheric turbulence induces fluctuations in the refractive index, leading to random variations in the beam's propagation direction and spreading, thereby impacting the spatial focusing of the transmitted signal.
Polarization DoF is another critical aspect influenced by the atmosphere, primarily through phenomena like polarization mode dispersion (PMD). PMD occurs due to variations in the refractive index along different polarization axes, causing differential delays and phase shifts in the optical signal. Consequently, this leads to polarization fading and degradation in signal quality, especially in high-data-rate FSO systems.
Moreover, the spectral DoF of FSO systems is affected by specific wavelengths within the optical spectrum being attenuated by atmospheric constituents such as aerosols and water vapor. These attenuation windows can lead to wavelength-dependent losses, spectral broadening, and selective fading, impacting the transmission of information over specific frequency bands.
Understanding and mitigating the effects on these DoFs are essential for designing robust FSO communication systems capable of maintaining reliable and high-quality links in diverse environmental conditions.

§.§.§ Free Space QKD

Scientists leverage the degrees of freedom inherent in classical light to amplify data transmission capacity within the quantum realm. Consequently, they employ classical light for characterizing quantum channels. Moreover, it is stated that broken quantum states cannot be recovered without quantum tomography. As a result, the concurrence variation in quantum channels aligns with both classical light behavior and theoretical expectations [70]. In pioneering work, the first atmospheric turbulence model for quantum light was introduced in [71], defining the transmission coefficient's interplay with input mode through the Glauber-Sudarshan P function. Thermal fields exert a significant influence on the behavior of light. In the context of phenomena like the Hong-Ou-Mandel Effect, where precise control over photon interactions is essential, the impact of thermal turbulence becomes particularly pronounced. The theoretical visibility limit, typically estimated at around 33%, reflects the maximum visibility achievable under ideal conditions. However, experimental constraints often impose a more stringent limit, typically around 28%, owing to the practical challenges posed by the thermal field[72].
Subsequently, a model utilizing Huygens-Kirchhoff integration [73]. emerged, offering a versatile validation spanning from weak to strong turbulence. This model commences its analysis by characterizing the elliptic beam within the transmitter aperture while incorporating the Kolmogorov power spectrum to simulate turbulent refractive index fluctuations.
Adjusting the atmospheric coherence length is achieved by modulating the refractive index structure constant, reflecting turbulence strength.
Furthermore, beam wander, denoting undesired beam deviations from the line of sight [74], is scrutinized, with emphasis placed on its impact on nonclassicality [75].

Leveraging the Kolmogorov power spectrum, this research computes the scintillation index of the Laguerre-Gaussian beam along a slant path, while considering refractive index variations via the Hufnagel-Valley model, elucidating the relationship between attitude and turbulence strength [76].
With a similar approach, log-normal power spectrum and gamma-gamma turbulence power spectrums are utilized to calculate the key rate of continuous wave QKD in weak and strong turbulences, respectively.
The inverse relation between the propagation distance and the key rate is emphasized due to the accumulated turbulence along the propagation distance [77]. Accordingly, it is shown in [78] that there is a 20% difference in link distance between the fiber-based and free space QKD systems in the targeted secret key rate of $10^{-4}$.

The numerical approach random phase screen(RPS) is used to obtain the transmissivity of the satellite to ground optical link. Then, the inverse relation between the zenith angle and the key rate is presented, benefiting from the transmissivity[79].

From an experimental standpoint, a QKD setup is established, where turbulence is induced through a SLM.
Random phase fluctuations are modeled by leveraging the inverse Fourier Transform of a frequency domain delta-correlated zero-mean Gaussian random complex function, combined with Kolmogorov power spectral density multiplication  [80].
In one early free-space QKD experiment, a 210m link was established, employing twisted photons and yielding a mean QBER of 3.81% [81].
Subsequently, a significant milestone was achieved with the establishment of a 144km free-space entanglement-based QKD link between La Palma and Tenerife, wherein the European Space Agency's ground station played a pivotal role as the receiver. This endeavor shed light on the phenomenon wherein noise escalates as the detection of photons from uncorrelated pairs increases[82].

§.§.§ Mitigation Techniques of Atmospheric Turbulence

The research is carried out to mitigate distortions due to atmospheric turbulence. Adaptive optics is a common method to correct the receiver plane wavefront.
Adaptive optics plays a crucial role in QKD systems, especially in scenarios affected by atmospheric turbulence.
This technology typically comprises a wavefront sensor, deformable mirror, and control unit.
The correction order is determined based on the aberrations detected and the actuators in the deformable mirror.
Research, such as [83], has demonstrated that in slant-path QKD systems, open-loop adaptive optics can notably improve signal-to-noise ratio and reduce quantum bit error rates compared to closed-loop configurations.
Building on this, recent studies, like [84], have extended the application of adaptive optics to satellite-to-ground QKD links, both discrete and continuous wave systems.
These studies indicate a significant enhancement in key rates, especially noticeable up to the 10th order of correction, with continued benefits observed up to the 20th order.
Furthermore, investigations such as [85] emphasize the importance of closed-loop control frequency in improving secret key rates in CV-QKD setups.
In challenging environments like turbulent mediums, as explored in [86], adaptive optics effectively mitigates wavefront distortions, thus improving the reliability of QKD operations.
Additionally, integrating orbital angular momentum (OAM) with QKD, as seen in [87], showcases a direct relationship between the correction order provided by adaptive optics and the secret key rate.
Higher correction orders enhance key rates and contribute to reducing quantum bit error rates, thus bolstering the overall security and performance of QKD systems.
In summary, adaptive optics emerges as a vital tool in advancing the capabilities of QKD systems, enabling improved performance and resilience against environmental challenges like turbulence.

Another method to mitigate atmospheric effects in free-space optical communication is by altering the structure of the beams. Some non-traditional beams exhibit reduced scintillation, smaller beam size, and less beam wander, making them advantageous for free-space optical communication. Consequently, shaped beams are utilized in free-space QKD applications. For instance, the propagation properties of a vector Bessel-Gauss beam with a specific polarization state are analyzed in [88]. Changes in polarization and beam evolution are plotted as functions of propagation distance. The experimental setup for generating higher-order beams, in addition to zero-order beams, includes lenses, mirrors, a polarizing beam splitter, half-wave plates, quarter-wave plates, and a phase spatial light modulator.
Similarly, vector Bessel beams are employed in quantum systems due to their self-healing feature. This property enhances the signal-to-noise ratio as the beam can recover itself after encountering an obstacle [89]. When considering the self-healing property of multiple Bessel-Gaussian beams with OAM, multiplexing is performed on the testbed. This study finds that the simulation results for beams with lower OAM modes closely match the experimental results. Additionally, Laguerre-Gaussian beams are generated using a spatial light modulator and are also used for multiplexing [90]. However, the quantum bit error rate (QBER) of Laguerre-Gaussian beams is three times higher than that of Bessel-Gaussian modes, as noted in [91]. Furthermore, the QKD rate of flat-topped beams surpasses that of Laguerre-Gaussian beams under weak and moderate turbulence conditions [92].

§.§ Integrated photonics

Integrated photonics has become a cornerstone technology for advancing quantum communication and computing, leveraging the miniaturization, scalability, and stability of photonic circuits [7]. This technology enables the integration of various quantum components such as single-photon sources, single-photon detectors, and quantum gates on a single chip, facilitating complex quantum information processing and secure communication protocols. For quantum communications, integrated photonics has been effectively utilized in QKD systems, capitalizing on established telecommunication infrastructure. Pioneering work includes SiO2-based optical interferometers for time-bin QKD systems [93]. The first fully integrated chip-to-chip QKD system combined an InP transmitter and a silicon oxynitride (SiOxNy) receiver, achieving state rates up to 1.76 GHz [94]. Moreover, integrated photonic devices have demonstrated the feasibility of Measurement-Device-Independent QKD (MDI-QKD) through Hong-Ou-Mandel (HOM) interference, with high visibility between weak coherent states from different integrated sources [95, 96].

Quantum photonic chips represent a cutting-edge technology in quantum computing and quantum information processing, utilizing the unique properties of photons such as coherence and entanglement. These chips integrate various optical components on a single substrate, providing advantages in scalability, stability, and processing speed. Key platforms in this field include Xanadu's X-Series [97], PsiQuantum's chip [98], and Quandela's Ascella [99], each offering distinct technologies and features tailored for different applications.
Xanadu's X-Series chips leverage silicon photonics and continuous-variable (CV) quantum computing, supporting a wide range of quantum applications with high-fidelity operations and reconfigurable circuits. PsiQuantum's chip and quandela's Ascella  focuse on  quantum computing with single-photon sources and detectors integrated on silicon photonic chips, emphasizing fault tolerance and scalability.
Advances in single-photon sources and detectors, such as spontaneous parametric down-conversion (SPDC) and superconducting nanowire single-photon detectors (SNSPDs), which will be discussed in Sec. <ref> have significantly improved the efficiency and reliability of quantum operations. Waveguide integration in materials like silicon, silicon nitride, and lithium niobate allows for precise photon routing with minimal loss, enabling the creation of complex photonic circuits. Quantum interference and entanglement generation on-chip have achieved high levels of fidelity, essential for scalable quantum networks. QEC on photonic chips involves encoding logical qubits into multiple physical qubits using error-correcting codes, with techniques such as the surface code and bosonic codes under active investigation. Implementing fault-tolerant operations requires low error rates and efficient error detection and correction, which remain critical research areas.
Applications of photonic quantum chips span quantum computing, communication, and sensing. In quantum computing, these chips promise scalable systems with inherent parallelism and low decoherence rates, targeting optimization, simulation, and cryptography. Quantum communication benefits from integrated photonics for secure communication channels and robust quantum repeaters. Quantum sensing applications include precision measurements in various fields, with integrated photonic chips offering compact and robust sensor solutions. Challenges in scaling up the number of qubits, maintaining high fidelity, and ensuring efficient integration with classical systems persist. Advances in manufacturing precision and hybrid quantum-classical architectures are essential. Establishing standards and benchmarks for performance evaluation will facilitate technology comparison and guide future development.

§ QUANTUM COMMUNICATION

Due to ease of implementation and low losses at small distances, optical fibers are the preferred medium for implementing short quantum channels [100, 101, 102, 103, 104, 105, 106, 107]. However, the photon losses in fiber optic grow exponentially with the length of the channel making this approach non scalable [108]. The vision of quantum internet incorporates a global-scale network that necessarily includes quantum links going beyond a few hundred kilometres. In parallel, a large percentage of the current digital networks operate over wireless links, either due to the infeasibility of underground or underwater cabling or due to the inherent mobility of the end nodes.  Therefore, the free-space links appear as a necessary component of this future quantum internet. In this section, we survey key components to achieve optical quantum communications, i.e., quantum sources, optical quantum channels, and detectors.

§.§ Quantum sources

§.§.§ DV Sources

Key Performance Indicators (KPIs) for Different Single-Photon Sources

KPI     Value and Source
Production rate     Millions of single photons per second [109]

Generation probability     $10^{-6}$ (SPDC and FWM)

Visibility in two-photon interference     Exceeds 90% (Diamond color centers) [110]

Zero phonon line incidence (ZPL)     4% (Diamond color centers)

Branching ratio     80% (Diamond color centers)

Collection efficiency     Up to 20% (PAHs) [111]

Photon generation rate     Just under 50 million photons/s (PAHs)

Intensity squeezing     Up to 2.2 dB (PAHs) [112]

Brightness (at detector)     53% (Quantum dots) [113]

Indistinguishability (raw)     About 91% (Quantum dots) [113]

Photon coincidences     One per year (six single-photon Fock states)

Single-photon purity ($g^{(2)}(0)$)     $0.00(3)$ (PAHs) [114]

A single-photon source ideally produces a single photon with high probability when requested. The generation of single-photon states is essentially based on two possible approaches, respectively relying on nonlinear effects in optically pumped crystals or on emission from individual quantum emitters. The main difference between the two  lies in the probabilistic nature of generation processes for nonlinear effects, compared to the potential of deterministic processes. Hereafter, we will provide a literature review on single photon sources with more focus on deterministic ones relying on quantum emitters.

Some systems utilize nonlinear interactions between the electromagnetic field and engineered materials to generate correlated photon pairs, such as through spontaneous parametric down-conversion (SPDC) or four-wave mixing (FWM). These interactions involve one or two pump photons decaying into pairs of photons. In these setups, one photon of the pair often serves as a herald, meaning its detection confirms the presence of a single photon in the other branch. Modern sources can produce millions of single photons per second in controlled states [109] with specific spectral properties and well-regulated spatial modes. However, due to the probabilistic nature of these nonlinear optical processes, achieving both a high probability of photon production and high purity (the ratio between single-photon generation probability and the probability of multiple-photon detection) simultaneously is challenging.

Probabilistic single-photon sources are still widely used due to fundamental characteristics such as simple integration into photonic circuits [115, 116, 117], tailored spectral properties, and high indistinguishability. By employing many low-probability but high-purity heralded single-photon sources, systems can increase the probability of successful single-photon generation. Multiplexing in these sources utilizes time, space, and/or frequency degrees of freedom to parallelize photon creation in different modes, with active switching based on heralding event detection [118]. This approach can achieve very high brightness (generation rate over pump power) [119], though still lower than deterministic emitters.

Single-atom emissions are utilized to generate single-excitation states in the electromagnetic field [120]. These emissions stem from isolated quantum emitters, which are two-level systems excited by short, intense laser pulses to ensure a single excitation per cycle. The resulting spontaneous emission produces a single-photon state, crucial for applications requiring well-defined modes of the field. Quantum emitters are commonly integrated with optical cavities, single-mode fibers, or waveguides for enhanced processing capabilities [121]. This approach is promising due to its capability to generate photons on demand, driving exploration across various platforms to minimize experimental losses. Key technologies include color-center-based sources, quantum dots, and molecular sources.

Point defects in solids, or color centers, in a transparent host matrix have gained interest for applications like single-photon sources and spin qubits due to their unique combination of single-photon emission and long spin state coherence. Advanced sources include color centers in diamonds, such as nitrogen vacancies and silicon centers. Nitrogen vacancy centers have reported visibility in two-photon interference processes exceeding 90% [110], even with emission from distinct centers. The main challenge is the low emission probability in the zero phonon line (ZPL) spectral region, at only 4%. Silicon centers show an 80% branching ratio but suffer from low radiative decay efficiency. Additionally, extracting light from diamond, a high refractive index material, is difficult, resulting in low brightness.

Photonics structures crafted from diamond aim to improve collection efficiency but often at the expense of coherence properties. An alternative method involves implanting impurities into structured materials using ion or electron beams, creating cavities, photonic crystals, or waveguides. Diamond-based sources are complex to fabricate, with only a select few groups worldwide consistently achieving favorable characteristics [122]. These sources primarily serve as spin-photon interfaces for applications in quantum communication and distributed quantum computing protocols, as discussed further in [123].

Sources including single molecules of polycyclic aromatic hydrocarbons (PAHs) in transparent matrices,  exhibit sub-Poissonian statistics of anti-bunching [124]. PAH molecules show near-unit quantum yield and high purity values (\( g^{(2)}(0) = 0.00(3) \)) [114], making them competitive in the solid state. Fluorescence intermittency is due to a non-zero probability of decaying to a triplet state, weakly allowed by spin-orbit coupling. For example, dibenzoterrylene (DBT) molecules in anthracene (Ac) matrices have a triplet state decay probability of \( 10^{-7} \) at low temperatures. This source emits in the near-infrared at around 785 nm and is stable at room and cryogenic temperatures, with an off-time of less than 1%. Combined with intelligent photonic engineering, it produces a regular flow of single photons [112], achieving intensity squeezing up to 2.2 dB at room temperature. Collection efficiency is typically the limiting factor, with simple photonic structures improving efficiency from a small percentage to 20% in a single spatial mode in cryogenic experiments [111]. Near-unit collection efficiency has been demonstrated at room temperature with a planar antenna and high-numerical-aperture optics, producing nearly 50 million photons per second. Molecular sources do not require optical cavities for high photon flux and indistinguishability, though strong coupling with a single emitter has been shown with open Fabry-Perot cavities [125].
Molecular systems excel in coherence time, intensity, frequency stability, and the ability to interfere with photons from distinct molecules [126].

Quantum-dot sources, are notable for generating single photons on demand. These exciton systems, with highly confined electron-hole pairs, behave like artificial atoms. The best single-photon sources based on quantum dots use III-V semiconductors grown epitaxially. Growth techniques leverage strain at interfaces between layers with different bandgaps, like gallium arsenide and indium arsenide. III-V QDs have large optical dipoles, facilitating broad coupling with confined or guided optical modes, essential for bright sources. While tunability is possible during growth, quantum dots are typically selected after optical characterization due to control challenges. Significant advancements in fabrication and excitation techniques (cross-polarization, phonon-assisted excitation, SUPER scheme, rapid adiabatic passage), coupling to photonic structures (bullseye cavity, open cavity, waveguide, micropillar), and local charge control in p-n junctions [127] position quantum dots as promising single-photon sources. They achieve an overall brightness (at the detector) of 53% at a 76.3 MHz pump rate and raw indistinguishability of about 91% [113]. Other studies have demonstrated the use of demultiplexed quantum dot sources for photonic-quantum computing [128][129]. For instance, Boson Sampling with up to 14 detected photons in an interferometer with 60 modes (see Table. <ref> for a summary of single-photon sources KPI and their values).

§.§.§ CV Sources

Continuous variable quantum states are typically generated using various optical components and nonlinear interactions. Some common sources include:

* Laser Sources: Coherent states are produced by highly stable laser sources, where the amplitude and phase can be precisely controlled.

* Squeezed states:
Squeezed states of light are crucial for continuous variable quantum information processing, offering reduced quantum noise in one quadrature while increasing noise in the conjugate quadrature. This property is essential for quantum-enhanced metrology, quantum communication, and quantum computation. Below is an overview of the state-of-the-art techniques for generating squeezed states, focusing on advancements in optical parametric oscillators (OPOs), waveguide sources, and microresonators.

* Optical Parametric Oscillators:
Optical parametric oscillators remain the most widely used and versatile source for generating squeezed states. In an OPO, a nonlinear crystal (such as periodically poled lithium niobate, PPLN) is placed inside an optical cavity. When pumped with a laser at frequency \(\omega_p\), the crystal mediates the down-conversion process, generating signal and idler photons at frequencies \(\omega_s\) and \(\omega_i\), respectively, satisfying \(\omega_p = \omega_s + \omega_i\). The Hamiltonian governing this interaction is:

\begin{equation}
\hat{H}_{\text{OPO}} = i\hbar \kappa (\hat{a}_s^{\dagger} \hat{a}_i^{\dagger} e^{-i\Delta t} - \hat{a}_s \hat{a}_i e^{i\Delta t}),
\end{equation}

where \(\kappa\) is the nonlinear coupling coefficient, and \(\Delta = \omega_p - \omega_s - \omega_i\) is the detuning.
High level of squeezing exceeding $15db$  have been achieved using PPLN crystals and loss-cavities [130].  Moreover, the Development of chip-scale OPOs integrating PPLN waveguides with microresonators, has enabled compact and stable squeezed light sources [131]. Other experiments have been carried in this direction, staying within the state-of-the-art levels of squeezing [132, 133].

* Cubic non-linearities: Cubic nonlinearity offers an alternative to second-order nonlinear processes for generating squeezed light. Typically, third-order nonlinear processes are much weaker than second-order ones; however, they are present in amorphous materials, broadening the selection of nonlinear media. In this context, optical fibers made from silica or specially designed highly-nonlinear glasses are particularly promising. The low intensity of interaction can be offset by longer interaction lengths, as advanced fiber production technology ensures low loss over large distances. The first observation of squeezing in an optical fiber occurred in 1986 using continuous radiation and a specialized detection scheme. This scheme involved reflecting the output beam from an optical resonator to achieve significant dispersion for necessary phase corrections of the sideband frequencies, enabling the measurement of squeezing. Despite considerable efforts to minimize noise, the observed squeezing was minimal, measuring only $0.5$ dB [134]. Many experiments, afterwards, have been carried in this regard using different setups like optical solitons [135, 136] or sagnac fiber interferometers [137].

§.§ Optical Channels

§.§.§ DV Channel Models

Quantum channel models in discrete variable systems are fundamental for understanding the dynamics and performance of quantum information processes. These models mathematically describe the various mechanisms through which quantum states undergo perturbations due to interactions with their surroundings. The theoretical basis of quantum channels originates from the unitary interactions between a quantum system and its environment.

Consider a quantum system \( S \) and its environment \( E \), initially separate, with the composite system described by the Hilbert space \( \mathcal{H}_S \otimes \mathcal{H}_E \). The evolution of the combined system is governed by a unitary operator \( U \) acting on this composite Hilbert space. The density operator \( \rho \) of the system evolves as:

\begin{equation}
\rho \rightarrow \rho' = \text{Tr}_E \left( U (\rho \otimes \rho_E) U^\dagger \right),
\end{equation}

where \( \text{Tr}_E \) denotes the partial trace over the environment's degrees of freedom and \( \rho_E \) is the initial state of the environment.

This partial tracing process represents an averaging over the environment's behavior, leading to reduced dynamics described by the quantum channel \( \mathcal{E} \). The action of the quantum channel on the system's density matrix \( \rho \) is typically expressed in the Kraus representation:

\begin{equation}
\mathcal{E}(\rho) = \sum_i K_i \rho K_i^\dagger,
\end{equation}

where \( K_i \) are the Kraus operators satisfying the completeness relation \( \sum_i K_i^\dagger K_i = I \).

Different types of unitary interactions between the system and the environment yield different types of quantum channels:

* Phase Damping Channel: This channel arises from the interaction of a two-level system (qubit) with a reservoir of quantum harmonic oscillators through a linear coupling where the spin of the two-level system is a constant of motion. The Kraus operators for this channel are:

\begin{align}
&K_0 = \begin{pmatrix} \sqrt{1 - \lambda}  & 0 \\ 0 & \sqrt{1 - \lambda} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\lambda} \end{pmatrix}, \nonumber\\
& K_2 = \begin{pmatrix} \sqrt{\lambda} & 0 \\ 0 & 0 \end{pmatrix},
\end{align}

where \( \lambda \) is the phase damping parameter, which has a time dependence as $1-\lambda=e^{-\Gamma(t)}$, where $\Gamma(t)$ can be different when different approximations of the interaction of the system with the environment are considered. For instance, in the markov approximation, $\Gamma(t)=C\cdot t $ with $C$ a constant depending on the spectral properties of the environment.

* Amplitude Damping Channel: This channel results from a linear interaction between a two-level system and a thermal bath, where the spin of the system is not a constant of motion. The Kraus operators for this channel are:

\begin{equation}
K_0 = \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1 - \gamma} \end{pmatrix}, \quad K_1 = \begin{pmatrix} 0 & \sqrt{\gamma} \\ 0 & 0 \end{pmatrix},
\end{equation}

where $\gamma$ is the amplitude damping parameter. This channel can be encountered in superconducting platform for quantum computing. This channel has a similar time dependence to the dephasing channel in the Markovian approximation given explicitly as $1-p=e^{-C\cdot t}$ with $C$ a constant depending on the spectral properties of the environment.

Quantum channels such as the depolarizing channel, bit-flip channel, and phase-flip channel are analogs of classical noise models like the white noise and binary symmetric channel.

* Depolarizing Channel: This channel represents a situation where the qubit is replaced by a maximally mixed state with probability \( p \). The action of the depolarizing channel is:

\begin{equation}
\mathcal{E}(\rho) = (1 - p)\rho + \frac{p}{3} (X \rho X + Y \rho Y + Z \rho Z),
\end{equation}

where \( X, Y, Z \) are the Pauli matrices. The depolarization channel can perfectly model free space communications where the information is encoded in the polarization degree of freedom of the beam.

* Bit-Flip Channel: This channel flips the state of a qubit with probability \( p \). Its action is:

\begin{equation}
\mathcal{E}(\rho) = (1 - p)\rho + p X \rho X.
\end{equation}

* Phase-Flip Channel: This channel flips the phase of a qubit with probability \( p \). Its action is:

\begin{equation}
\mathcal{E}(\rho) = (1 - p)\rho + p Z \rho Z.
\end{equation}

These channels play a critical role in the study of QEC and the robustness of quantum information protocols under various noise conditions.

§.§.§ CV Channel Models

Bosonic channel models cater for optical quantum communication both in an optical fibre or in a free-space link [138, 139]. The standard model for bosonic quantum communication caters for two noisy processes: photon loss and dephasing. A single-mode pure-loss bosonic channel is modeled by a beamsplitter that has the transmittivity $\eta$. The signal from transmitter is injected into one of the two input ports of the beamsplitter where the other input is injected with a fixed environment state, typically a vacuum state for pure-loss channels or thermal state to model noise [140, 141]. Similarly, one of the two output ports is fed to the receiver whereas the other output is lost to the environment that models the lossy process. The transmittivity $\eta$ of the beamsplitter under different transmission conditions can be obtained by taking the physical channel conditions into account, e.g., haze, rain, and turbulence [142, 141]. After characterizing $\eta$ by taking all relevant factors into account, the input-output relation of quantum states traversing through this channel can be written as [142]

\begin{align}
P_{\mathrm{out}}\left( \alpha\right) = \int_{0}^{1} \frac{1}{\eta}\mathcal{P}\left( \eta\right)P_{\mathrm{in}}\left( \frac{\alpha}{\sqrt{\eta}} \right)\, \mathrm{d}\eta
\end{align}

where $P_{\mathrm{out}}$ and $P_{\mathrm{in}}$ are quasiprobability distribution that characterize the output and input states, respectively, and $\mathcal{P}\left( \eta\right)$ is the probability distribution of transmittance.

The decoherence of quantum state during transmission is modeled by dephasing that provides the mapping (written in Fock basis)

\begin{align}
\rho = \sum_{n, m = 0}^{\infty} \rho_{n, m}\ket{n}\bra{m} \rightarrow \sum_{n, m = 0}^{\infty} \rho_{n, m} e^{-\frac{\gamma}{2}\left( n - m\right)^2}\ket{n}\bra{m},
\end{align}

where $\gamma \in \left[ 0, 1\right]$ models the dephasing strength [138]. The dephasing process can take into account several forms of decoherence occurred in the medium, e.g., temperature fluctuation, Kerr nonlinearities in the fibre, imprecision in the path length, or the lack of shared phase reference between the communicating parties [143]. A general bosonic channel model that incorporates both loss and dephasing can be obtained by the order-independent composition of the two processes [138].

These types of optical quantum channel models are studied in detail in the literature. For example, exact quantum and private capacities of bosonic dephasing channels have been obtained recently [143]. Anther recent work took diffraction, atmosheric extinction, turbulence, pointing errors, and background noise into accout to find the analyze the composable finite-size security of coherent state protocols in fibre optics and free-space fading quantum channels [141]. A similar model for bosonic Gaussian channels was considered in [144] to obtain classical capacity of optical quantum channels.

A relevant type of bosonic channel is the thermal-loss channel.
This class of channels is particularly important in optical and microwave communication scenarios, where laser noise and background radiation [145, 146] have a significant impact, respectively. However, thermal-loss channels are more challenging to model than dephasing channels. They are neither degradable nor anti-degradable [147, 148], and only lower and upper bounds for the quantum and private capacities are known [149].

§.§ Detection

§.§.§ Detection of DV Quantum Signals

Key Performance Indicators (KPIs) for Different Single-Photon Detectors

KPI     Value and Source
System efficiency     50-60% at 800 nm (Si-based SPAD) [150]

25-35% (InGaAs/InP SPAD) [151]

>90% (SNSPD) [152] [153] [154]

Operating bandwidth     Visible to near-infrared (SPAD) [155]

Telecom wavelengths (SNSPD) [152] [153] [154]

Timing jitter     50-180 ps (Si-based SPAD) [150]

70-150 ps (SNSPD) [152] [153] [154]

As low as 40 ps (Commercial SNSPD)

Maximum count rate     Up to 37 MHz (Si-based SPAD) [150]

Exceeding 10 MHz (SNSPD) [152] [153] [154]

Dark count rate     Up to 25 Hz (Si-based SPAD) [150]

Approximately 1-2 kHz (InGaAs/InP SPAD) [151]

Up to 10 Hz (SNSPD) [152] [153] [154]

Single photon detectors are classified into two primary categories based on their response types: Threshold Detectors and Photon-Number Resolving Detectors. Threshold detectors function by establishing a threshold level for photon detection. When the incoming signal surpasses this threshold, the detector records an event, signifying the presence of at least one photon. However, these detectors cannot distinguish between one photon and multiple photons; they only provide a binary output indicating detection above the threshold. They are typically used in applications where the precise number of photons is not critical, such as in quantum key distribution and certain forms of quantum computation like boson sampling. In contrast, Photon-number resolving detectors measure the energy or charge associated with each detected photon. By analyzing this information, they can determine the number of photons in the incoming signal.

The key performance indicators (KPIs) for assessing a detector's performance include system efficiency (the overall probability that an incident photon is detected by the detector, accounting for quantum efficiency and losses such as single-mode fiber coupling), operating bandwidth (the range of incident light wavelengths within which the detector maintains high system efficiency), timing jitter (the temporal uncertainty in the detector's response, where lower time jitter corresponds to higher temporal resolution, allowing more accurate identification of photon arrival times), maximum count rate (the highest number of counts per second the detector can register, depending on the reset time after each detection event), and dark count rate (the number of counts per second due to thermal effects, not from actual photon incidence). A comparison of different detectors against these KPI is provided in Table <ref>

Single-Photon Avalanche Diodes (SPAD)
Single-photon avalanche diodes (SPADs) are a type of photon detector based on semiconductor technology [155]. They consist of a p-n junction to which an appropriate voltage (reverse-bias) is applied to remove charge carriers from the depletion region . In this way, the incidence and absorption of a single photon are sufficient to initiate an avalanche ionization process, resulting in a macroscopic current. A greater number of photons does not lead to a significant difference in the generated current, and therefore, the detector's response does not provide information about the number of incident photons. The detector remains "blind" until it is reset to return to its initial condition. Among the main advantages of detectors based on this technology is their operation at room temperature. This characteristic makes them more suitable for portable applications where cryogenic equipment, such as those based on superconducting materials, is not feasible, in addition to lower manufacturing costs.

The choice of material for constructing the detectors depends on the required wavelength range. According to current technology, detectors with better specifications (quantum efficiency, dark count rate) can be obtained in the visible wavelength range, with performance tending to be lower as they move into the near-infrared region, especially in the O-band (1260-1360 nm) and C-band (1530-1565 nm) of the telecom wavelength. Among the best results reported in the literature, Si-based detectors [150] can achieve quantum efficiencies of approximately 50-60% at a wavelength of 800 nm, with dark count rates up to 25 Hz, jitter of 50-180 ps, and maximum counting rates up to 37 MHz. In the telecom band, the choice of material mainly falls on InGaAs and InGaAs/InP detectors [151], with quantum efficiencies of about 25-35%, but associated with higher dark count rates (approximately 1-2 kHz). SPAD detectors are also commercially available from various companies.

Superconducting Detectors
The second class of single-photon detectors is based on superconduncting technology. Within this category, several types based on different architectures have been developed, including transition edge sensors (TES) [156], superconducting tunnel junctions (STJ) [157] and microwave kinetic inductance detectors (MKID) [158]. Of particular interest both for quantum computing and quantum communications due to recent technological advancements and their characteristics are superconducting nanowire single-photon detectors (SNSPD) also known as nanowire detectors [159].

These detectors utilize nanowires, approximately 100 nm in diameter, fabricated from ultra-thin film with thickness about 5-10nm, made of superconducting material. The detection mechanism is based on the creation of cooper pairs, followed by a transition of the material from a superconducting state to a non-superconducting state, generating a detectable macroscopic current [160]. Subsequently a reset time is required to return the detector to its initial superconducting state and become active again. SNSPDs have been reported in the literature with high detection efficiency, low dark count levels, and high count rates. For instance, in the telecom wavelength range in the C band, several works have been reported in the literature [152, 153, 154] on various materials such as NbN, WSi, and MoSI, with performance parameters including efficiency >90%, dark count rates up to 10 Hz, timing jitter of 70-150 ps, and maximum count rates exceeding 10 MHz. Similarly, many companies have introduced detectors of this type to the market, with typical parameters ranging from system efficiency of 80-90%, timing jitter as low as 40 ps, and operating wavelengths ranging from telecom wavelengths to visible/near-infrared. Typically, a compromise needs to be made between the wavelength band of operation and detector efficiency. Unlike SPAD detectors, superconducting detectors require the use of a cryogenic apparatus at low temperatures. Therefore, this type of technology currently has a higher cost impact and lower portability.

§.§.§ Detection of CV Quantum Signals

Continuous Variable (CV) quantum detection is a crucial aspect of quantum information processing, particularly in quantum communication and computation. It involves measuring quantum states that are described by continuous variables, such as the quadratures of the electromagnetic field. The primary detection techniques in CV quantum systems are homodyne and heterodyne detection for Gaussian states, and PNRD for non-Gaussian states.

Homodyne Detection:
Homodyne detection is used to measure the quadrature components of a light field, typically for Gaussian states. It involves mixing the signal field with a strong local oscillator (LO) at a beam splitter and measuring the resulting interference.
The output photocurrent $I(t)$ of a homodyne detection is proportional to the quadrature component:

\begin{equation}
\end{equation}

where $\eta$ is the detection efficiency, $\theta$ is the phase of the local oscilator  and $\xi(t)$ and $\langle\hat{x}_\theta\rangle$ are the noise term and the expected value of the quadrature respectively.

Heterodyne Detection:
Heterodyne detection measures both quadratures simultaneously by mixing the signal with a local oscillator at a frequency offset. The signal is down-converted to an intermediate frequency, which can be processed to extract both quadratures. In heterodyne detection, the measured field $\hat{\beta}$ is a complex combination of the quadratures:

\begin{equation}
\hat{\beta}=\frac{1}{\sqrt{2}}\big(\hat{X}+i\hat{P}\big)
\end{equation}

The heterodyne photocurrent yields information about both quadratures:

\begin{equation}
\end{equation}

with $\langle\hat{x}_\theta\rangle$ and $\langle\hat{p}_\theta\rangle$ are the expectation values of the quadratures where $\theta$ is the phase of the local oscillator and $\eta$ and $\xi(t)$ are respectively the detection efficiency and the noise term.

§.§ Quantum Modulation Techniques

In qauntum communications, modulation techniques are essential for encoding information into quantum states. Here we discuss several quantum modulation techniques, that are summarized in Table. <ref>, and are relevant for different discussions throught the manuscript.

§.§.§ Frequency encoding

Frequency encoding uses different frequencies of photons to represent qubits. This method is beneficial for multiplexing and integrating with classical communication channels. Different frequency modes $\ket{f_1}$ and $\ket{f_2}$ represent logical states $\ket{0}$ and $\ket{1}$.

§.§.§ Time bin

Time-bin encoding uses the arrival time of photons to encode quantum information. This technique is robust against decoherence over long distances. A Mach-Zehnder interferometer with a delay line can be used to create time-bin encoded states where the early arrival time $\ket{e}$ and the late arrival time $\ket{l}$ represent logical states $\ket{0}$ and $\ket{1}$ as illustrated in Fig. <ref>

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[arrow] (A) – (B) node[midway,above] $\ket{e}$ node[midway,below=.5cm,text width=2cm] Mach-Zehnder interferometer;
[arrow] (C) – (D) node[midway,above] $\ket{l}$;
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[shift=(A1), scale=.4,samples=100, smooth, domain=μ-2:μ+2, variable=, mixedcolor!60, fill=mixedcolor!20] plot (, 2 * exp(-(- μ)^2 / (2 * σ* σ)) );

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[right=2*.4 of A1, above] (sup) ;
[above=.7cm of sup] $\alpha\ket{l} + \beta\ket{e}$;

[right=3*.4 of B, above] (l) ;
[above=.7cm of l] $\ket{l}\equiv\ket{0}$;

[right=7*.4 of B, above] (e) ;
[above=.7cm of e] $\ket{e}\equiv\ket{1}$;

Time bin encoding

§.§.§ Polarization encoding

Polarization encoding uses the polarization states of photons to represent qubits. This is one of the most common methods due to the ease of manipulating polarization states. Fig. <ref> illustrates this type of encoding where for instance, the vertical $\ket{V}$ and the horizontal $\ket{H}$ polarizations can represent respectively the states $\ket{1}$ and $\ket{0}$. Other polarization states like the left and right circular polarizations might be used as well instead oif the horizontal and vertical polarizations.

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[stan] at (+.8,1) Unpolarized Photon;
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An illustration of polarization encoding

§.§.§ Orbital angular momentum

OAM encoding uses the helical phase front of photons to encode information. Photons with OAM carry angular momentum along their propagation direction. Photons can have OAM values $\ket{l}$ where $l$ is an integer respresenting the number of twists in the phase front per wavelength. Spatial light modulators (SLMs) or q-plates can generate OAM
states. These states might be detected using mode sorters or forked gratings can detect OAM states.

§.§.§ Spatial encoding of structured light

Spatial encoding uses the spatial mode structure of light beams to encode information. Structured light beams such as Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes are used. These modes can carry information based on their spatial structure.  Spatial light modulators can be used to generate and manipulate these modes. This encoding is useful in high-dimensional quantum information processing.

§.§.§ Path encoding

Path encoding uses different spatial paths to represent quantum states. In this technique, the state of a qubit is determined by the path it takes. A simple Mach-Zehnder interferometer can be used to illustrate path encoding as in Fig. <ref>. We highlight that path encoding is different from spatial modulation where the information unit is encoded on the selected antenna/path once at a time. Instead, in path encoding, many antennas transmit the information unit simultaneously through different paths in a quantum way.

A Mach-Zehnder interferometer illustrating path encoding where the photon can be in a superposition of taking two different paths, $\ket{0}$ and $\ket{1}$ can represent the photon taking the left or right path, respectively.

§.§ Modulation techniques and crosstalk

Crosstalk is a significant challenge across various media, from free space and fiver, to waveguides in integrated photonics as was mentioned Sec. <ref>. Indeed, cross-talk highly depends on modulation and encoding techniques used in optical communication. Different from the MIMO configuration, inter-channel, intra-channel, and inter symbol interference(ISI) are taken into account. Its impact varies depending on the specific method used and mitigating crosstalk often requires careful design considerations such as mode purity, synchronization, and the choice of modulation format. Each technique has specific strategies to minimize crosstalk, ensuring reliable and high-fidelity transmission of information. When it comes to modulation techniques, each has its unique vulnerabilities to cross-talk, which can degrade the performance of an optical communication system. For carrier-free signals, such as quadrature amplitude modulation (QAM), the crosstalk power is nearly constant under expected conditions in multi-core transmission systems [161][The expected comditions refers to  the modulation formats, symbol rates, and the nature of the transmitted signals. For instance,  Different modulation schemes, such as quadrature amplitude modulation (QAM) and on-off keying (OOK), have distinct characteristics that affect how they interact with crosstalk. QAM, for example, is noted for its constant power over time, which can make its crosstalk behavior more predictable compared to OOK, which has inherent fluctuations due to its binary nature. Moreover,   the symbol rate, or the speed at which symbols are transmitted, also influences crosstalk dynamics. Higher symbol rates can lead to different temporal and spectral characteristics of the crosstalk compared to lower symbol rates. Additionally, the relative delay between signals in different cores, known as inter-core skew, impacts the crosstalk. Variations in this skew can cause fluctuations in the crosstalk power, affecting the overall signal quality.]. In contrast, carrier-supported signals, such as on-off keying (OOK), always induce time-varying crosstalk powers due to the strong carrier component, necessitating a performance margin for these systems [161]. Polarization states can experience crosstalk when different polarization modes couple. This can occur in systems using polarization multiplexing where maintaining the orthogonality of the polarization states is critical to minimize crosstalk. Crosstalk between polarization states can degrade the performance of the system and reduce the fidelity of the transmitted information. Crosstalk in dual polarization scenarios is influenced by random polarization rotations between phase-matching points. The crosstalk variance is affected by the inter-core skew and the frequency characteristics of the signal [161]. OAM modes are susceptible to crosstalk, especially when transmitted through fibers. In [162] it has been reported that the separation of modes (5, 6, 7) in specially designed air-core fibers helps reduce crosstalk. However, lower order modes (<4) do not have sufficient separation, leading to higher crosstalk. The mode purity and extinction ratio are key indicators of crosstalk levels, with higher extinction ratios indicating lower crosstalk. In time-bin encoding, crosstalk can occur due to overlapping time bins, especially in systems with high symbol rates or poor synchronization. Proper time-bin separation and precise timing are essential to minimize crosstalk and ensure clear distinction between time bins [163]. Path encoding, used in quantum key distribution (QKD) and other optical communication systems, can experience crosstalk when there is unwanted coupling between different paths. This is particularly challenging in multi-core fibers where inter-core coupling can induce crosstalk. The level of crosstalk is influenced by the inter-core skew and the modulation format, with higher order modulation formats showing lower crosstalk variations [164]. Path encoding, particularly in quantum photonic interconnects, can suffer from cross talk. In these systems, path-entangled states are generated on one chip and distributed to another chip by interconverting between path and polarization degrees of freedom using a two-dimensional grating coupler. Furthermore, cross talk can introduce noise and affect the fidelity of entanglement distribution [165].

§ INTERACTING QUANTUM CHANNELS

In classical communication theory, noise models play a pivotal role in understanding the degradation of transmitted signals over communication links. Common noise models include additive white Gaussian noise (AWGN) which represents random fluctuations added to the received signal during detection. Other models encompass impairments on the propagation channel, such as thermal noise through the receiving aperture, multipath fading, and co-channel interference. To mitigate the effects of noise in modern classical communications, various techniques are employed. These include error-correcting codes like convolutional codes and Reed-Solomon codes, which introduce redundancy to the transmitted signal data, enabling receivers to detect and correct errors [166]. These are usually blind to the exact physical impairment that generated the errors and are optimized for specific impairments such as the AWGN or erasure channels. Additionally, diversity techniques like spatial diversity, frequency diversity and time diversity are implemented to combat fading effects and enhance signal reliability [167]. Diversity techniques are normally aware of the underlying statistics of the heterogeneous channel streams/slots in order to apply intelligent forms of combining. Advanced modulation schemes such as quadrature amplitude modulations (QAM) and phase-shift keying (PSK) are also utilized to improve spectral efficiency and robustness against specific kinds of impairments e.g. phase noise, and non-linear channels. Furthermore, adaptive modulation and coding (AMC) techniques dynamically adjust modulation and coding parameters based on channel conditions, ensuring optimal performance under varying SNR conditions. Through a combination of these techniques, modern classical communication systems can achieve reliable and efficient data transmission over real-world communication links [168].

Despite the huge efforts that the community is putting into the migration towards quantum communications, quantum communication systems are still susceptible to various sources of impairments, which can compromise the fidelity and security of transmitted quantum states. As such, noise mitigation techniques are indispensable in quantum communications to ensure reliable and secure transmission. Similar to noise mitigation techniques in classical communications, quantum communications systems employ strategies such as QEC and quantum repeaters to combat noise and enhance the resilience of transmitted quantum states.  QEC techniques enable the detection and correction of errors that arise due to noise and decoherence, thereby preserving the integrity of quantum information. Quantum repeaters extend the range of quantum communication networks by mitigating the effects of signal attenuation and noise accumulation over long distances, thus enabling efficient long-distance quantum communication. These noise mitigation techniques are crucial for realizing the full potential of quantum communications and advancing towards practical applications such as secure quantum networks and quantum internet. We refer the reader to [169, 170] for more details on QEC and error mitigation (see Table. <ref> for a comparison between classical vs quantum techniques for noise mitigation and correction).

Other more forward-looking techniques consider the employment of the genuinely quantum resource of quantum coherence to manipulate the propagation of the information between the transmitter and the receiver. In particular, the quantum coherent control of the causal orders between given channels and the coherent control of the path in which the information propagates have been investigated. Despite the differences and the similarities between the two schemes, they have proven to be very important for mitigating noise, when some or full knowledge about the communication channel is known to the transmitter [The indefinite causal paradigm of quantum channel requires in some cases only partial knowledge of the channel to provide an advantage, like its nature, without knowing the full behaviour of the channel. This is the example of the entanglement breaking channel. In some other cases full knowledge of the channel is needed, like the noise parameters of the channel.]. Using these quantum coherent control techniques, better transmission rates beyond the ultimate bounds posed by quantum Shannon theory can be achieved.

Despite the different trials to come up with an efficient scheme to mitigate the impairments on information transfer in the quantum realm, communication strategies akeen to diversity, are still to be deeply investigated  due to the different nature of the interaction of information carriers with their environment causing a different effective behavior of the transmission lines on a quantum level [171]. Another problem is related to the nature of information to be transmitted. If the information is classical and is encoded in quantum states, it can be easily prepared in many copies and sent through different transmission lines. Instead, when the information is purely quantum and is being an outcome of a previous quantum process that might be not known, the information cannot be copied due to the no-cloning  theorem.

Classical and Quantum Correspondence in Communication Systems

Concept     Classical Communications     Quantum Communications
Forward Error Correction (FEC)     Classical Error Correction Codes (e.g., Hamming, Reed-Solomon, LDPC)     QEC Codes (e.g., Stabilizer codes Codes, Surface Codes, Holographic codes, QLDPC)

Modulation     Amplitude, Frequency, Phase Modulation, spatial, time     path, frequency, polariziation, OAM, time, structured light

Adaptive Modulation and Coding (AMC)     Adaptive schemes based on channel conditions     Quantum Communications with Feedback

Quantum Coherent Control     None     Quantum Coherent Control Techniques, indefinite causal ordering

§.§ Quantum Coherent Control

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Quantum coherent control of quantum operations capitalizes on the distinctive attribute of quantum coherence to manipulate quantum operations or channels. Leveraging quantum coherence enables the simultaneous transmission of a single information carrier through multiple potential trajectories coherently, a capability absent in classical communications. This is inherently different from multipath propagation scenarios, where undesired side effects of information transfer resulting from reflection and diffraction, result in a constructive or destructive interference of the signal at a waveform level. The difference is mainly in the quantum nature of superposition between individual paths.
Employing such quantum coherent strategies for information transmission offers significant advantages in error and noise mitigation during transmission. For instance, pioneering research has showcased that coherent manipulation of the access point of the information carrier and its propagation path facilitates a scheme that exploits channel multiplexing in the path degree of freedom without signal multiplexing or redundancies, ultimately resulting in enhanced transmission rates [172]. This innovation, termed "error filtration," has recently been extended to the domain of quantum coherent control of quantum channels [173]. In this seminal study, a completely depolarizing channel, which inherently lacks the capacity to transmit any information due to its diminishing classical and quantum capacities, becomes viable for information transfer when a single qubit traverses two instances of the channel coherently. To illustrate, consider the scenario where the information carrier is a photon with its information encoded in its polarization degree of freedom. Utilizing a polarizing beam splitter (PBS), one can concurrently transmit the polarization qubit through both channels, with upward polarization traversing one completely depolarizing channel and downward polarization traversing the other, while maintaining coherence between the two polarization states.
A more comprehensive depiction of coherent control of quantum channels using polarizing beam splitters (PBS) is elaborated in [174], introducing a generalized version of PBS within a novel graphical framework inspired by optical configurations involving PBS. This approach aims to formally characterize quantum coherent control through a new descriptive formalism that extends beyond classical completely positive trace-preserving (CPTP) maps, which inadequately capture quantum coherent control. The motivation behind this endeavor is to provide a more nuanced and accurate representation of quantum coherent control, particularly in scenarios involving complex quantum channels and operations.

In recent years, another framework gaining increasing attention in the realm of quantum coherent control is the concept of Quantum Switching, also known as quantum coherent control of the causal order of quantum channels. This area of study emerged as an extension of quantum information theory, aiming to transcend the limitations of conventional completely positive and trace-preserving maps [175]. The latter are applicable only under the assumption of a dynamical reduced dynamics on the information carrier, either initially correlated with the environment or during its transmission. The novel formalism expands upon this premise to encompass more exotic and theoretically plausible scenarios. Beginning with the notion of quantum combs [176], which may describe memory effects during transmission leading to potential information backflow from the environment to the system [The notion of a quantum comb is a generalization of the notion of a quantum channel. The latter does not exist only under the assumption that the system is initially not quantumlly correlated with the environment – not in an entangled or discordant state– as the reduced dynamics on the system cannot be recovered or the complete positivity of the channel is not guaranteed [177, 178]. As such, the quantum comb comes as an extension of the notion of a quantum channel or a quantum map in these extreme scenarios.], the concept of quantum supermaps emerges as higher-level quantum maps [179].Unlike conventional maps that act solely on the states of the information carrier, quantum supermaps operate on the configuration of quantum channels' placement, allowing for the exploration of genuinely quantum configurations that were previously inaccessible.

A comparison between the performance of the two schemes has been subject of investigation. One may be over-driven by the idea that the advantage of both schemes for mitigating noise comes from the constructive interference of the two signals sent through different legs or different orders in the case of quantum coherent control of paths and indefinite causal ordering respectively. This is only partially true! In indefinite causal ordering there is another additional effect coming from the commutation and anticommutation relations between the Kraus operators describing the individual channels. This has been reported in  [180], where a comparison between the performance of indefinite causal ordering namely the quantum switch and quantum coherent control in randomly sampled channels. Another question of debate is the regime in which quantum coherent control of path of indefinite causal ordering can indeed provide an advantage over the ultimate limits of quantum Shannon theory of point-to-point communications. It has been observed that under some noise parameters of individual quantum channels these novel schemes can be advantageous, whereas in other regimes of the noise parameter, these schemes fail to provide any advantage. For instance, the quantum switch perfectly activates the capacity of a qubit entanglement breaking channel while if fails to do so for less drastic types of noise like the bit-flip and phase-flip noises. In [181], the authors proposed to think of the quantum switch as a particular instance of an environment assisted communication strategy which is optimal in the case of entanglement breaking channels, leading to perfect capacity activation, and fails to be so in the majority of cases where it might be even detrimental for information transmission.

§.§ Multiplexing and Diversity

Spatial multiplexing  involves the simultaneous transmission of multiple data streams through separate spatial paths within the propagation medium. Unlike traditional multiplexing techniques that allocate different time slots, frequencies, or wavelengths, spatial multiplexing exploits the spatial directions in the atmosphere or interfering cores in fiber cables to transmit multiple independent information streams concurrently. This is achieved by employing multiple transmitters and receivers strategically positioned to establish parallel communication links. By distributing information across distinct spatial paths, space multiplexing enhances spectral efficiency, increases energy efficiency, and mitigates the effects of atmospheric turbulence, crosstalk and other impairments. The key concept behind spatial multiplexing is that advanced joint pre/post-processing of information can transform energy leakage among spatial channels: from unwanted crosstalk treated as additional noise to useful signals which can enhance the communication performance.

On the other hand, there is spatial diversity. Spatial diversity refers to the utilization of multiple spatially separated antennas or optical apertures to improve the reliability and robustness of communication links in the presence of propagation impairments. In FSO systems, spatial diversity can be achieved by deploying multiple transmitters and receivers at different locations or orientations. By leveraging spatially separated paths, FSO systems can exploit variations in atmospheric conditions across different paths to enhance signal reception and mitigate the effects of fading and distortion caused by atmospheric turbulence and obstructions. Both methods, assume mutually quantum mechanically incoherent channels [By quantum mechanically incoherent channels we mean that the channels cannot propagate information simultaneously through both of them while having quantum coherence of the information carrier, making the channels placed in a parallel independent configuration used in multiplexing].   Similarly, in fiber optic communication systems, spatial diversity can be employed by utilizing multiple optical fibers or deploying multiple spatially separated cores within a single fiber. This approach allows the system to distribute the optical signal across different paths, thereby reducing the impact of localized fiber impairments such as microbends, macrobends, and scattering losses. By taking advantage of spatially diverse paths, fiber optic systems can achieve improved signal integrity and robustness against physical layer impairments, ensuring more reliable and higher-quality transmission. This method also assumes that the channels are mutually quantum mechanically incoherent, allowing independent transmission paths to be used for mitigating the overall effect of signal degradation.

We should highlight that quantum coherence between quantum channels should not be confused with phase coherence in optical systems. The latter is a property of a wave-like system where the phase relationship between different points or components of the system remains constant over time. This is crucial in classical waves such as optical systems. For instance, in laser beams, phase coherence ensures that the light waves are in step, producing a beam with a narrow frequency range and minimal dispersion. Maintaining phase coherence is essential for applications requiring high precision and stability, such as interferometry and coherent communication systems. Quantum coherence, on the other hand, is a fundamental concept in quantum mechanics. It describes the ability of a quantum system to exhibit superposition, where a particle can exist in multiple states simultaneously. Quantum coherence is characterized by the preservation of the relative phase between components of a quantum state. This property is crucial for phenomena such as quantum entanglement and quantum interference, which are the basis for quantum computing, quantum cryptography, and other quantum technologies. Quantum coherence is more delicate than classical phase coherence as it can be easily disrupted by interactions with the environment, leading to decoherence and loss of quantum information.

Quantum Coherent control in the counterpart, can harness the advantage of both schemes, namely, spatial multiplexing and spatial diversity by harnessing the genuinely  quantum superposition of different states during the transmission simultaneously [Quantum coherent control relies on the fact that the genuinely quantum coherence, which has no classical counterpart, stored in the information carrier, is mapped to the transmission channels putting them in a new quantum configuration that does not have a classical counterpart in its turn - it is not equivalent to sequential or parallel placement of channels.  ]. One can easily notice that by increasing the number of coherent modes used for transmission, for instance with d-modes, a qudit codeword can be coherently transmitted, increasing the throughput. In the same time, without copying the information and transmitting it on different independent paths to the receiver, as in spatial diversity, quantum coherent control can indeed reduce the effect of noise arising from atmospheric effects in free space communications, as was proven by theory and experiment [182].
Indeed, the signal from different spatial paths should be interfered at the receiver and coherent detection plays an important role to harness the full advantage of quantum coherent control. One can see this as an interferometric setup, where the interference of the signals at the end allows to project the global system into a constructive interference and destructive interference, leading to noise mitigation and filtration as was detailed in [172]. One can of course tune the weights of each propagation path manually to achieve the desired constructive interference effect if a priori knowledge on the channels is given to optimize the transmission.
Indeed, this scheme holds similarities with spatial multiplexing, in which the quantum state is encoded onto different spatial modes or paths of propagation, such as different optical fibers or waveguides. Each spatial mode carries a distinct component of the quantum state, and at the receiver, the components are combined or measured separately to retrieve the original quantum state.
This scenario has been investigated in different studies demonstrating that high-dimensional states exhibit greater resilience to noise, thereby suggesting a higher error tolerance in quantum communications.  Similarly, Quantum Coherent Control holds also similarities with path-based multiplexing, which involves encoding quantum information directly onto different paths of propagation, rather than using spatial modes. Each path corresponds to a distinct quantum state and the information is encoded in the choice of path taken by the quantum system which is decoded at the receiver by an interferometric detection cmprising single photon detectors. Path encoding shows promise due to its compatibility with photonic integrated circuits, facilitating the generation, manipulation and detection of path-encoded quantum states [165, 183]. However, reliablely transmitting such states remains a major challenge. One potential solution is coupling each path to a SMF each of which would be affected by varying random phase drifts due to temperature changes, bends, and mechanical stress could disrupt transmission, especially over long communication channels. Alternatively, coupling each path to different core of a multicore fiber (MCF) suppresses phase drifts among cores, enabling better transmission, particularly of superposition states. Previous experiments explored these fibers for high-dimensional quantum communication, but limitations in stability impacted achievable distances. Recently, this issue was addressed by actively stabilizing the system, achieving high-fidelity transmission of qudits of dimension $d=4$ and $d=7$ over a $2 KM$ long MCF  [164]. For the knowledge of the authors no path encoding experiment has been carried in free space.

Although quantum coherent control shares similarities with spatial multiplexing and path encoding, there are still some differences between the two approaches. Notably, spatial encoding schemes may require additional resources, such as spatial light modulators, optical components for mode manipulation, or multiplexing/demultiplexing devices. Path encoding schemes may have simpler requirements in terms of routing and switching elements. Moreover,  in spatial encoding schemes, the receiver may need to perform spatial mode demultiplexing and spatial mode detection to retrieve the original quantum state. In path encoding schemes, the receiver typically measures the quantum state directly on each path, such as through photon counting or interferometric techniques. Additionally,  in spatial encoding, the quantum state is encoded onto different spatial modes, which can have higher dimensionality compared to path encoding. Each spatial mode can carry multiple degrees of freedom, such as different polarizations, spatial positions, or orbital angular momentum states. In path encoding, the quantum state is typically encoded onto discrete paths, which may have lower dimensionality compared to spatial modes.

§.§ Quantum MIMO

MIMO systems proved to bring a paradigm shift in wireless communication systems. In particular, the presence of multiple antennas at the transmitter and the receiver can either be used to increase the diversity or to increased the number of degrees of freedom in wireless communication systems. A seminal work  [184] introduced a fundamental tradeoff between the diversity and multiplexing. Discussion on MIMO channels with main focus on QKD has also begun recently [185, 186]. The main component of the channel model, again was a quantum optical circuit composed of beamsplitters where different inputs and outputs were controlled by different parties in a typical QKD protocol, i.e., Alice, Bob, and Eve (cf. Fig. <ref>). These channel models were utilized to analyze the performance of QKD in Terahertz regime under different assumptions.

One challenge in designing a MIMO communication scheme with quantum information is the no-cloning theorem [14]. One fundamental operation to harvest spatial diversity gain from classical MIMO communication is to transmit multiple copies of the signal. On the receiver end, the multiple copies of received signal are combined together to extract a higher quality of the received signal. Due to the no-cloning theorem, preparing multiple high-fidelity copies of a quantum signal is not possible, thus making it challenging to harness an advantage from MIMO quantum channels. There are still a number of cases where transmitting multiple copies of a quantum state remain possible, e.g., when the underlying information is classical, e.g., in QKD or when the transmitter has access/knowledge of the state preparation process. However, in the presence of multiple parallel channels (spatial or otherwise) it is possible to multiplex several streams of quantum signals together to harvest the multiplexing gain.

A $2\times 2$ MIMO channel model for THz QKD introduced in [185]. The interception of Eve is modelled by the two beamsplitters in the channel. Eve can arbitrarily change the input/output as well as the transmittivity of these beamsplitters.

WDM is a popular and effective method in optical communication for sending multiple data streams of different wavelengths on a single physical medium.[The frequency-division multiplexing is the same approach in the RF communication.] A typical optical WDM system consists of multiple optical sources of different wavelengths fed into an optical multiplexer that combines the input signals. The resulting output is fed into a SMF for transmission. At the output, a demultiplexer splits the individual signals, which are then transmitted to the intended receiving ports.

WDM remains to be an important approach for the co-existence of classical and quantum networks on the same infrastructure [187, 188, 189, 190]. Recent works have demonstrated the coexistence of classical and quantum signals on a single fiber using WDM to achieve QKD SKR of more than 4 kbps at 65.5 km [187]. Similarly, another recent experiment demonstrated the coexistence of a quantum channel and $8\times 200$ Gbps 16-QAM optical channels with launching power as high as -9dBm/channel [189]. Due to a strong disparity between the typical strengths of classical and quantum signals special care must be given to the channel spacing to ensure minimal cross-talk between the two [191, 192].

Another important application of WDM is to multiplex multiple quantum signals in a SMF. A recent work has proposed an all-QWDM for the multiplexing of quantum signals [193]. The work develops general models for the involved components in a complete QWDM communication system. Specifically, the operations of quantum transmitters, a passive quantum wavelength distributor, and quantum receivers are analyzed in fully quantum terms. The model is adaptable to various scenarios; possible sources of quantum signal include WCP, SPP, or Poissonian mixed states; the wavelength distributor component is modeled as a unitary operator that admits various configurations including mux-fiber-demux configuration or a star coupler; finally the receiver also admits multiple configurations as a combination of demux or Filter followed by single-photon detectors or conjugate homodyne detectors. The analysis shows that QWDM of coherent states gives a tensor product state at the output, i.e., there is no correlation between the states at the output of all quantum receivers. On the other hand, SPP of different quantum receivers will turn out to be entangled in Lambdanet broadcasting communication system with a star coupler as the wavelength distributor. This unintended entanglement is a typical case of crosstalk between different signals and can be avoided by using an ideal router as a wavelength distributor of a router-based communication system.

§ CONNECTING DV QUANTUM NODES WITH CV QUANTUM LINKS

In the rapidly advancing fields of quantum communications and computing, one of the key challenges is to find efficient ways to encode and manipulate quantum information. Quantum harmonic oscillators provide a promising avenue for such tasks. Unlike classical oscillators, which can only occupy discrete energy levels, quantum harmonic oscillators can access a continuous spectrum of energy states, making them ideal candidates for encoding quantum information.

Encoding qudits in quantum harmonic oscillators offers several advantages. By utilizing the infinite dimensional Hilbert space of harmonic oscillators, we can achieve a higher information density compared to traditional qubit-based systems. This increased dimensionality allows for more robust error correction and potentially enables powerful quantum algorithms. The encoding process typically involves mapping the quantum states of the qudit onto energy eigenstates of the harmonic oscillator. This mapping can be achieved through various techniques such as superposition of Fock states or employing coherent states. Once encoded, the quantum information can be manipulated and processed using continuous-variable (CV) quantum gates and measurements acting as logical qudit gates and measurements.

The interconnection between discrete-variable (DV) quantum computing and continuous-variable (CV) quantum communications is particularly noteworthy. DV quantum computing typically involves qubits, which represent quantum information in discrete states. In contrast, CV quantum communications leverage quantum harmonic oscillators to encode information in continuous spectra. By integrating these approaches as shown in Figure. <ref>, we can harness the strengths of both systems: the precise control and established error correction methods of DV systems and the high-dimensional encoding and flexibility of CV systems.

Furthermore, encoding qudits in quantum harmonic oscillators opens up new possibilities for quantum communications, quantum cryptography, and quantum computing. In quantum communications, the ability to encode information in a continuous spectrum allows for more efficient and secure transmission of data. Quantum cryptography benefits from the increased security offered by high-dimensional encoding, making eavesdropping significantly more challenging. In quantum computing, the hybrid approach of combining DV and CV systems can lead to the development of more versatile and powerful quantum algorithms, paving the way for advancements in various applications ranging from cryptography to complex simulations. This synergy represents a significant step forward in the quest for practical and scalable quantum technologies.

§.§ Cat Codes

Cat states or Shroedinger states have been shown to be of paramount importance for quantum sensing [194], quantum communications [195] and quantum computing [196].  Cat states are a coherent superposition of two coherent states with opposite displacement parameters. Although their seemingly simple structure, their generation still only probabilisitic in photonic platforms due to their non-Gaussian nature when only passive optics is used [197]. By including non-Gaussianities through measurement setups [198], like PNRD measurmenets, or evolutions like cubic-gates, photon addition and subtraction, these state can be obtained in the optical domain [199, 200]. Many experimental studies have been devoted to the deterministic generation of these states based on Jaynes-Cummin interaction between light and single atoms in a cavity [201, 202]. Notoriously, macro cat states of mechanical oscillators has been achieved recently in optomechanical systems [203, 204].
Multi-component cat states (cat states with a coherent superposition of more than two coherent states) have shown promess towards loss tolerance, when encoding discrete variable degrees of freedom, making them suitable candidtaes for QEC in a DV/CV interface [195, 205].

§.§ GKP Codes