The current internet is a vast classical network, designed to facilitate global communication and distributed information processing [11, 2, 3, 4]. The inherent robustness of classical information allows for hybrid, flexible network architectures which operate in both optical-fiber and free-space, befitting an environment and mode of application. A future quantum internet will aim to play an analogous role for quantum information technologies [5, 6, 7, 8], but the inherent fragility of quantum information makes achieving high-rates over long distances much more challenging. In pursuit of this goal, theoretical and experimental progress in the study of hybrid quantum networks is fundamental and necessary.
The ultimate limits of fiber-networks are well understood. By means of the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound, it is known that the capacity of a fiber-link decays exponentially with respect to the link-length with a precise law [9, 10]. The PLOB bound has been used to understand the end-to-end network capacities of fiber-based quantum architectures , to assess the limits of realistic, random network structures [12, 13] and idealized, highly-connected, analytical architectures . These investigations have provided essential insight and motivation for the construction of high performance quantum networks, elucidating key physical properties and network characteriztics. Chiefly, to develop a high-performance, fiber-based quantum network one must carefully consider not only connectivity, but nodal density and maximum link length; leading to high resource demands for large-scale designs.
However, quantum networks will not be limited to just optical-fiber but will collaborate with free-space methods of communication. On the ground, the flexibility of free-space links are obviously more suitable for mobile quantum devices and short-range connections. Meanwhile, the ability to establish ground-to-satellite and intersatellite free-space connections offers remarkable short-cuts for global quantum communications [15, 16, 17, 18, 19, 20, 21]. Such connections bypass many decibels of loss that would be otherwise experienced on the ground, and utilize the dynamic nature of satellites to achieve high-rates over global distances.
Determining the ultimate limits of free-space quantum channels is difficult, requiring tools from quantum information theory [22, 23, 24], optics [25, 26, 27] and turbulence theory [28, 29, 30, 31]. Recent advancements have placed tight upper-bounds on the quantum capacities of point-to-point free-space channels, using a modified PLOB bound that accounts for atmospheric fading processes [132, 133]. With these results in hand, we have the ingredients to go beyond the point-to-point scenario and quantitatively study the ultimate limits of communications in free-space quantum networks.
In this work, we combine results from Refs.  and [132, 133] in order to place bounds on the end-to-end capacities of generally hybrid quantum networks. In particular, we put forward a formalism for studying the capacities of quantum networks whose channels are described by free-space, fiber, or any medium that can be generalized as a fading channel. This treatment is then specified to fading processes that are experienced by optical transmissions through the atmosphere, or in space.
Furthermore, we introduce a framework to investigate the ultimate limits of hybrid, modular quantum networks. We focus on a modular network design which consists of disjoint sub-networks (or communities) connected to a large-scale backbone network used to mediate intercommunity quantum communication. For the first time, this provides the tools to investigate highly relevant quantum network models, such as: Globally distant fiber sub-networks connected to a satellite backbone network, offering insight into the resource requirements of a satellite-based quantum internet; and wireless, free-space sub-networks on the ground interconnected via a fiber backbone, presenting a useful model for studying hybrid metropolitan networks.
Extending the techniques of Ref.  we employ ideally connected structures within different parts of the modular network. In doing so, we are able to derive simple, yet powerful analytical constraints which promise distance-independent rates for modular quantum architectures, and thus optimal performance. These results provide valuable insight into the ultimate limits of hybrid networks, can help to motivate future quantum network designs and provide a valuable platform upon which to further develop realistic free-space quantum networks.
i.1 Paper Structure
This paper is structured as follows: In Section II we provide a comprehensive review of quantum networks, point-to-point capacities of general fading channels and the end-to-end network capacities of quantum networks generally composed of fading channels. We then specify this review to optical free-space quantum communications, summarizing recent progress in the determination of ultimate limits for a number of key settings. In Section III we formalize a network architecture for the study of hybrid, modular quantum networks. We further specify an idealized network architecture which allows us to establish properties that guarantee optimal end-to-end performance and distance-independence. Finally, Section IV applies the machinery from the previous sections to investigate the optimal performance of hybrid quantum architectures. In particular, we establish network constraints for communication between remote fiber-based sub-networks connected to a satellite backbone, and for ground-based free-space sub-networks connected to a fiber backbone. Concluding remarks and future investigative paths are then discussed.
ii.1 Quantum Networks
An arbitrary quantum network can be described as a finite, undirected graph where is the set of all network nodes (points/vertices) on the graph, and is the set of all edges within the network. Each node represents a local register of quantum systems which can be exchanged with connected neighbors. Meanwhile each edge in the network is denoted by the unordered pair , and is used to represent a quantum channel through which users can exchange quantum systems. Two nodes and are connected if the edge exists within .
It is important to note that the physical orientation of each channel in the network can be a forwards or backwards and need not be specified. Under the assistance of two-way classical communications (CCs), the optimal transmission of quantum information is connected with optimal entanglement distribution. It does not depend on the physical direction of system exchange but the Local Operations (LOs) that are applied at each point, and thus the direction of teleportation. This refers to the logical direction of quantum communication. If an undirected edge represents a physically asymmetric channel (i.e. ) then the users can always enable a teleportation protocol that uses the most efficient physically directed channel. Hence, the logical flow of quantum information can always be made independent of the physical flow of quantum systems.
Consider two end-users Alice and Bob, who reside at remote nodes within the quantum network, and respectively. An end-to-end communication protocol between Alice and Bob can be most broadly captured by a general adaptive protocol. Alice and Bob propagate the exchange of quantum systems between nodes throughout the network in accordance with an overarching protocol in order to establish some global, target quantum state. This may involve point-to-point, or point-to-multi-point exchanges dependent on the nature of the protocol. Quantum interactions between any two nodes is alternated with adaptive network-wide LOCCs, allowing for consistent optimization of the network protocol. Communication is complete when the target state is eventually established between the end-users after a number of uses of the network.
The optimal performance over any such network protocol is captured via the generic two-way assisted network capacity which describes the ultimate rate with which a desired target state can be established. If the target is a private state, this refers to the network secret-key capacity measured in secret-bits per network use. If it is a maximally entangled state then this becomes the network entanglement distribution capacity measured in entanglement bits (ebits) per network use.
Note that we consider a general information-theoretic definition of a quantum repeater as a middle third-party helping the quantum communication between a sender and a receiver (therefore not connected by a direct link). In practice, there are many possible physical realizations, e.g., see Refs. [34, 35, 36] among others.
ii.1.1 Network Routing
Thanks to the interconnectivity of quantum networks, there is no unique path that network interactions must follow in order to establish end-to-end quantum communication. However, there exist two fundamental classes of routing strategy under which all protocols can be described: Single-path or multi-path routing.
Single-path routing is the simplest network communication method which utilizes point-to-point communications in a sequential manner. Quantum systems are exchanged from node-to-node followed by LOCC operations after each transmission until eventually communication has been established between the end-users. One may define a single-path network capacity which describes the optimal performance obtained via a sequential end-to-end strategy. A repeater chain forms a particular instance of quantum network under single-path routing where there only exists one unique end-to-end route.
A more powerful strategy is multi-path routing, which properly exploits the multitude of possible end-to-end routes available in a quantum network. A user may exchange an initially multi-partite quantum state with a number of neighboring receiver nodes, who may each then perform their own point-to-multi-point exchanges along their unused edges. Again, every exchange of quantum systems can be interleaved with adaptive network LOCCs, and this process continues until multi-point interactions are carried out with the end-users. A multi-path routing strategy in which all channels in the network are used precisely once per end-to-end transmission is known as a flooding protocol. This is achieved via non-overlapping point-to-multi-point transmissions at each network node, such that receiving nodes only choose to transmit along unused edges for subsequent connections. Therefore, there exists a multi-path network capacity (or flooding capacity) which describes the optimal network performance out of all possible strategies (single or multi-path).
ii.2 Quantum Networking over Fading Channels
The effect of fading
refers to the temporal variation of transmissivity along a bosonic lossy channel. The transmissivity along a fading channel is not fixed, but instead follows a probability distribution described by the dynamics of the environment. For example, the propagation of bosonic modes through low-altitude free-space instigates a fading channel thanks to chaotic processes in the atmosphere. The impact of fading on a communications channel is described via itsspeed
, i.e. the ability for a receiver to resolve the dynamics of the transmissivity fluctuations. Slow-fading implies that the users can resolve the fading dynamics and accurately perform channel estimation because either the fading process is weak or the users possess sufficiently fast detectors. On the other hand, fast-fading refers to the situation where the users cannot reconcile the dynamics of the channel and can only estimate the statistical distribution of the channel transmissivity[37, 38]. It is clear that fast-fading poses a more formidable task for communicators.
More precisely, a bosonic lossy fading channel is defined as an ensemble of lossy channels in accordance with some probability density functionwhich describes the instantaneous transmissivity along the channel. We denote a lossy fading channel as the ensemble
where is a lossy channel with fixed, instantaneous transmissivity and is the maximum transmissivity that is attainable along the channel.
ii.2.1 Capacities of Fading Channels
The absolute maximum rate that two parties can transmit qubits, establish secret-keys, or distribute entanglement over bosonic lossy channels is known exactly via the PLOB bound. This states that generic two-way assisted capacity of a quantum channel is precisely
measured in bits per channel use, and where we introduce as the capacity function for stable lossy channels. While this assumes a fixed transmissivity , the PLOB bound can be readily employed to study fading channels [10, 139]. Thanks to convexity properties of the relative entropy of entanglement (REE) over ensembles of channels, the capacity of a lossy fading channel can be bounded according to,
where we have defined as the capacity function for lossy fading channels. This can be interpreted as a generalization of Eq. (2), modified to include potential fading processes. Indeed, it is simple to retrieve the standard bound
for fixed lossy channels by considering a trivial probability distribution where only one transmissivity value is possible,. Hence, this format is conveniently general and allows one to describe any lossy bosonic channel (with or without fading).
More generally, lossy channels will also be exposed to thermal-noise, resulting in a thermal-loss channel . This channel equates to mixing an input mode with a thermal mode of mean photon number on a beam splitter of transmissivity , effectively adding photons to the signal mode. The capacity of thermal-loss channels is not known exactly, but upper-bounds have been derived through the techniques developed for the PLOB bound. For a fixed thermal-loss channel the capacity can be upper-bounded via 
where . Otherwise when , meaning that there exists a minimum transmissivity at which communication can be reliably secured.
Analogous to the pure-loss setting, a thermal-lossy fading channel can be described by the ensemble
where it is possible that both transmissivity and thermal noise are probabilistic and described within a probability density function . Typically, thermal noise can always be considered constant by either assuming stable operational conditions, or by minimising (maximising) its potential value for best-case (worst-case) rates. This allows us to consider the simpler ensemble on which we place the following upper-bound of its capacity [132, 133],
Here we have defined as a tight capacity bounding-function for thermal-lossy fading channels 111The word tight in this context refers to how close the upper-bound is from its best known lower-bound. Indeed, there exists a lower-bound on the capacity of a point-to-point thermal-loss channel based on its reverse coherent information (RCI) . Hence, throughout our work we implicitly refer to tight upper-bounds on thermal-loss channel capacities (and subsequently, network capacities) as those which in conjunction with the RCI can tightly sandwich the exact capacity.. Intuitively, one can never outperform the pure-loss PLOB bound in the presence of thermal-noise, hence we can always write
ii.2.2 Capacities of Fading Networks
We can combine the theory from these previous sections in order to provide a general model for quantum networks with fading channels. Indeed, we may construct a quantum network such that all edges are generally associated with a unique thermal-lossy fading channel,
In this way, each network edge not only possesses a unique maximum transmissivity and thermal-noise properties , but also a unique instantaneous transmissivity probability density function through which each edge can adopt its own fading dynamics (or lack thereof). This allows for a description of network channels within different environmental media such as fiber channels, ground-based free-space channels, or free-space channels beyond the atmosphere. Furthermore, we can retrieve pure-loss fading channels via .
It has been shown that the capacities of quantum networks can be derived through a combination of quantum information theoretic tools and ideas from classical network theory . By transforming the notion of classical network cuts into entanglement cuts of a quantum network, one can determine compact, analytical expressions for the network capacities of arbitrary architectures. Consider a pair of end-users within the fading network . We define an entanglement cut as a means of disconnecting/partitioning the network into two disjoint super-users A and B such that and and . A network cut generates an associated cut-set,
defining a collection of network edges which when removed successfully partitions the network.
As discussed in previous sections, single-path routing can be thought of as a generalization of repeater-chains, where end-to-end communication is established via the sequential exchange of quantum systems along a designated path. The single-path network capacity is bounded by determining the network cut that generates the smallest, maximum single-edge capacity in the cut set. For lossy and thermal-lossy fading networks, we may define the single-path capacity quantities ,
which upper-bound the single-path network capacity,
In the absence of thermal noise, the single path capacity is achievable and equates to performing sequential communication along the optimal route in the network. For an arbitrary network, finding the optimal route is equivalent to solving the well known widest-path problem and can be solved efficiently . It is unknown whether the thermal upper-bound in Eq. (6) is achievable, hence in the presence of thermal noise presents a tight upper-bound.
More powerful network protocols employ multi-path routing . The multi-path network capacity is associated with an optimal flooding protocol, and is found by locating the entanglement cut which minimizes the multi-edge capacity over all cut-sets. For lossy and thermal-lossy fading networks we can compute the multi-path quantities ,
which upper-bound the multi-path network capacity,
Once more, for pure-loss based networks the flooding capacity is achievable, and equates to solving the classical maximum-flow minimum-cut problem according to a network of capacity achieving links. For general quantum networks with an arbitrary architectures, this problem requires a numerical treatment and can be solved efficiently [43, 44, 45]. Once more, since the thermal-loss upper bound is not guaranteed to be achievable is instead a tight upper-bound.
ii.3 Free-Space Quantum Communication
Consider two remote parties Alice and Bob who are separated by a distance , and employ quantum communications based upon a quasi-monochromatic optical mode (-nm large and -sec long). This may be characterized by a Gaussian beam with wavelength , initial field spot-size and curvature . Communication consists of transmitting a directed beam towards a receiver with circular aperture of radius . Here we assume that the initial spot-size is sufficiently small with respect to the transmitter aperture of radius so that there is no relevant diffraction caused by the transmitter.
The atmospheric effects which characterize free-space channels are variable with respect to altitude, due to changes in atmospheric density. Therefore specifying the trajectory of a Gaussian beam through free-space is pivotal in capturing channel quality. To this end, for any point-to-point communications task we may assume a general beam trajectory and introduce the following altitude/propagation functions respectively, and . Using these functions we can retain a geometry independent framework for our study until we wish to specify to a particular setting.
ii.3.1 Free-Space Transmissivity
Free-space diffraction is a universal contributor to loss. As a beam propagates in free-space its waist will widen as a function of the distance it travels,
where is the Rayleigh range. A target receiver will then only detect a portion of the spread beam since its aperture is finite in size, inducing a diffraction-limited transmissivity,
It is also useful to define a diffraction induced transmissivity in the far-field regime, , making the approximation . This loss quantity exists regardless of the specific environmental setting considered, from ground-based links to intersatellite connections.
Propagation through the atmosphere incurs further loss due to aerosol absorption and Rayleigh/Mie scattering; an effect known as atmospheric extinction. At a fixed altitude , this loss can be accurately described via the Beer-Lambert equation . Since beam trajectories may be variable in altitude, we can generally define the extinction-induced transmissivity as
where is the extinction factor, , and is the extinction factor at sea-level. For it follows that
Finally, there exist inevitable internal losses associated with the detector setup, due to imperfect fiber-couplings, sub-optimal quantum detector efficiency, and more. This inefficiency-induced transmissivity can be as low as and must be considered to capture realistic performance. All of these effects can be used to describe a fixed, maximum transmissivity of a free-space connection,
Importantly, can be readily modified to consider variable altitude beam trajectories and written as a function of a chosen spatial geometry to account for different extinction properties throughout the atmosphere.
ii.3.2 Atmospheric Fading
It is remarkably optimistic to assume that a free-space transmission deterministically undergoes a pure-loss channel characterized by Eq. (19) only. The chaotic behavior of air-flow, temperature and pressure throughout the atmosphere invites further complications for free-space transmissions, causing inaccuracies in the point-to-point trajectory known as beam wandering. As a result, we must incorporate fading for a more accurate characterization.
Turbulence is used to describe how a free-space propagating beam is perturbed by fluctuations in the atmospheric refractive index, caused by spatial variations in pressure and temperature. Propagating beams interact with small turbulent air-flows on a fast time-scale, too fast for communicators to monitor or resolve. This causes the beam waist to broaden and forces us to define a short-term spot-size which is larger than the diffraction-induced spot size,
. On a slower time-scale, the beam will undergo deflections by significantly larger eddies in the atmosphere. This slower time-scale may be reconcilable by the communicators, and manifests as a wandering of the beam centroid. This wandering can be described by a Gaussian random walk of the centroid with variancewhich is a functional of the beam trajectory, operational setup, conditions, and more.
Wandering is not exclusively caused by turbulence, and one must also consider pointing errors caused by jitter and imperfect targeting. These effects also occur on a reasonably slow time-scale of order , and may be resolved by the receiver. This introduces an additional wandering variance , e.g. a pointing error at the transmitter causes a variance (where is in meters). Overall, these effects combine to induce Gaussian centroid wandering with variance
The ability for communicators to resolve these wandering dynamics is dependent on their time-scale. The behavior of turbulence is variable, with regimes ranging from weak to strong turbulence. Increasing turbulent strength can be modeled as an increasingly faster fading process, such that a receiver loses the ability to reconcile the wandering dynamics. For stronger levels of turbulence, it is possible to define a long-term spot-size which averages over the wandering caused by both small turbulent eddies and larger eddy deflections, . Indeed, the turbulence-induced variance is defined with respect to the long-term and short-term quantities . However, rigorous studies of strong turbulence will require further considerations, for which work is currently underway .
In this work, we focus on the regime of weak turbulence and the concept of short-term beam spot sizes. These can be used to provide precise descriptions of free-space quantum channels on the ground at short-range, and for ground-to-satellite communication along trajectories with small zenith angles [132, 47, 48].
ii.3.3 Weak Turbulence
For communications undergoing weak turbulence, the beam wandering acts on a time scale of and can be fully resolved with a sufficiently fast detector. In this case, analytical expressions can be found for the short-term spot size and the centroid wandering variance . Consider a beam with wavenumber following a free-space trajectory (and its associated altitude function ). Then the spherical-wave coherence length is given by,
where denotes the refractive index structure constant, used to measure the strength of fluctuations in the atmospheric refractive index. This quantity has an explicit dependence on the beam’s trajectory, since this may be variable in altitude, and is typically described via the Hufnagel-Valley model (See Appendix C of Ref. ). Provided that Yura’s condition is satisfied  then we can write ,
The short-term spot size can be used to update the diffraction induced transmissivity to account for fast beam interaction with small turbulent eddies in the atmosphere. That is,
where we have simultaneously introduced a far-field approximation, when the propagation distance is very large .
Updating the diffraction-induced transmissivity in Eq. (19), we may write a new maximum transmissivity incorporating weakly turbulent effects, This represents the optimal transmissivity parameter that can be achieved when the beam centroid is perfectly aligned with the receiver centroid , i.e. the centroid deflection is . However, due to turbulence and pointing errors, the beam centroid now undergoes a Gaussian random walk with variance , invoking a fading channel. We can then connect the non-zero centroid deflection to an instantaneous transmissivity to precisely capture the fading process. Gaussian wandering induces a Weibull distribution for the centroid deflection, which results in an instantaneous transmissivity probability density function . Defining the functions,
where is the modified Bessel function of the first kind for , we can introduce the following shape and scale parameters,
With these, we can now write the instantaneous transmissivity probability density function,
Consequently, we are left with a free-space, lossy fading channel . Using the tools from Section II, we can study the capacities of free-space connections.
Hence, the capacities for free-space quantum communications (entanglement distribution or secret-key distribution) are upper bounded according to 
where represents a correction factor to the PLOB bound due to imperfect alignment,
Through specification to a free-space trajectory, one can easily determine geometry dependent expressions for this ultimate limit. Importantly, for channels which are accurately described as ensembles of pure-loss channels (thermal noise is negligible), then Eq. (28) is in fact an achievable and optimal rate, . For all other scenarios where thermal noise is non-negligible, it remains an effective upper-bound.
ii.3.4 Thermal Noise
As discussed previously, pure-loss based bounds remain ultimate bounds in the presence of thermal noise. Yet, it is still possible to construct tighter performance bounds by considering fading channels which are ensembles of thermal-loss channels. Let be the mean number of input photons transmitted towards a receiver via a single free-space mode. For an instantaneous transmissivity the mean photon number collected at the receiver will be , where describes the total environmental thermal noise added to the signal. It is useful to define contributions to this environmental noise via
where the receiver collected mean background photons with detector efficiency , and accounts for excess setup noise. In the study of ultimate limits, can be considered to be approximately zero, or can be attributed to trusted noise.
For free-space links, the primary source of thermal noise is attributed to natural brightness within the field of view of the transmission, i.e. the sky, Sun, Moon, etc. Using a receiver of aperture , angular field of view , a detector with time window and frequency filter around , then the number of background thermal photons per mode is
Here, describes the spectral irradiance of the environment in units of photons , and is unique to the operational setting and trajectory. Using the general bound from Eq. (6) and specifying to free-space beam wandering dynamics with variance , we can write the free-space thermal upper-bound,
where the thermal correction is given explicitly by,
This result applies to settings of weak and intermediate turbulence, such that one can substitute the appropriate reconcilable wandering variance and maximum transmissivity into this result.
ii.3.5 Noise Suppression and Frequency Filters
As seen in Eq. (31), the number of background thermal photons per mode has a strong dependence on the frequency filter, . The frequency filter assists in blocking out noise, and thus the use of ultra-narrow filters is highly desirable. In discrete-variable quantum communications, physical frequency filters are typically limited to around . However, using CV quantum systems and appropriate interferometric measurements it is possible to achieve much narrower effective filters.
Many CV-based protocols rely on the use of a local-oscillator (or phase reference) in order to perform homodyne or heterodyne measurements at the output. This phase reference may be co-propagated with signal pulses, or alternatively reconstructed at the receiver. This reconstruction method involves interleaving the signal pulses with strong reference pulses that carry information about the local-oscillator . Since the output of a homodyne measurements is proportional to the mean photon number in the local-oscillator modes, the ability to utilize bright references pulses over free-space channels introduces an effective homodyne filter. Thermal noise mode-matching with the local-oscillator and the signal will be detected, but all other noise will be filtered out. This allows for the implementation of ultra-narrow effective filters on the order of with practical CV protocols, and can dramatically reduce the magnitude of the thermal background noise (see Ref.  for more details).
ii.4 Important Free-Space Channels
ii.4.1 Ground-Based Channels
Wireless classical communication networks are ubiquitous and fundamental to everyday modern life. Thus the desire for a free-space quantum analogue is obvious, enabling access to future wireless quantum technologies. Nonetheless, it is intuitive that such communication will be limited to short-range thanks to prominent decoherence obtained at ground-level. At a fixed altitude, beam trajectories are horizontal paths with the simple altitude/propagation functions . The absence of a variable altitude in the beam path simplifies a number of key quantities such as the extinction-induced transmissivity
and the spherical-wave coherence length
which can be used to accurately describe decoherence and fading dynamics on the ground. Here, turbulence is a major factor and must be stringently considered. A useful parameter for assessing the validity of turbulent regimes on the ground is the Rytov variance,
In particular, weak turbulence requires that . Using a Gaussian beam with and altitudes close to sea-level during typical day-time conditions, weak turbulence is only guaranteed for distances of . Beyond this, as in the intermediate () and strong () turbulent regimes, the long-term spot size must be adopted, leading to poorer ultimate channel capacities .
Fig. 1(a) illustrates the behavior of transmissivity in ground-based free-space channels with respect to propagation length. Within the weak-turbulence regime the loss properties of free-space channels limited to dB for communications over km, encouraging the utility of short-range, optical free-space quantum communications.
For the assessment of thermal bounds, the primary source of thermal-noise at ground-level is attributed to the brightness of the sky. This provides a spectral irradiance ranging from
in units of . Using this information, the expressions in Eqs. (34) and (35), and the general capacity bounds developed in the previous sections, we can accurately assess the ultimate limits of free-space quantum communications on the ground (see Ref.  for further details and derivations).
ii.4.2 Ground-Satellite Channels
For communication between ground/satellite stations, there are two unique configurations that must be considered: Transmissions directed from the ground towards a satellite (uplink) or from a satellite towards the ground (downlink). The quantum channel descriptions of these configurations are very different.
Consider a Gaussian beam propagated in uplink. The beam immediately undergoes turbulence upon generation at low altitude, and thus has a large decohering impact which must be carefully considered. However, pointing errors are less critical thanks to the availability of adaptive optics to optimize the beam trajectory from the ground station. Therefore we must model uplink as a fading channel predominantly due to turbulent effects. Meanwhile, a Gaussian beam in downlink experiences the opposite; the beam does not undergo serious levels of turbulence until it reaches lower altitudes. But by this point, its spot-size has already been spread by diffraction, hence turbulence does not present a serious factor and . Yet, in this setting pointing errors become much more relevant due to the lack of onboard access and optimization ability. Hence, atmospheric decoherence associated with uplink and downlink is physically asymmetric, invoking two unique fading channels.
We specify the trajectory of ground-satellite communication according to a target satellite altitude and zenith angle , which describes the angle formed between the zenith point at the ground station and the direction of observation towards the satellite. The zenith angle takes values , such that when the satellite is at the zenith. The distance that the beam physically travels from its point of generation (known as its slant distance) can then be expressed with respect to this geometry. Defining the functions,
we may then introduce the altitude/propagation functions with respect to uplink and downlink communications ,
Fig. 1(b) illustrates the behavior of transmissivity in ground-satellite channels with respect to uplink, downlink and satellite altitude. Here we plot both the the expected transmissivity when averaged over the respective fading processes and maximum transmissivity (a best-case loss in the absence of fading). Crucially, it can be shown that for beam trajectories with relatively small zenith angles , we can assume the regime of weak turbulence for the ground-satellite fading channel (see Appendix C of ). Within this angular window we can accurately resolve the fading dynamics, and by inserting the beam trajectory expressions into the machinery of Section II.3.2, it is possible to derive loss-based ultimate limits for both uplink and downlink quantum communications using the Eq. (28), and thermal-loss-based limits using Eq. (32).
The sources of environmental thermal-noise are also unique to both uplink and downlink configurations, and operational settings such as the time of day and weather. In uplink during the day, the primary source of thermal-noise is sunlight being reflected from the Earth to the satellite detector. Meanwhile, at night, this noise is diminished but there still exists sunlight being reflected from the Moon to the Earth, and back towards the satellite. For uplink, we may write
Here is a parameter that accounts for the Earth/Moon albedos and ranges from for a clear night with a full Moon, to during clear day-time. Meanwhile, for the optical wavelength , we can approximate that in uplink the solar spectral irradiance is .
For downlink, the receiver is now a detector on the ground and the main source of noise is more simply attributed to the sky (as it was in the ground-based scenario). In this setting, and for , the spectral irradiance of the sky follows Eq. (37). For a much more detailed analysis, see Appendix D, Ref. .
ii.4.3 interSatellite Channels
Finally, we can consider free-space quantum communication between satellites in orbit. This represents a high quality free-space quantum channel which is free from atmospheric decoherence, and thus will not experience losses due to extinction nor undergo turbulence. Indeed, these intersatellite link losses are characterized by free-space diffraction only. Assuming negligible pointing errors, then the intersatellite channel is simply a lossy channel with transmissivity given by as a function of the propagation distance between satellites, . This lets us write an ideal upper-bound on the intersatellite channel capacity,
Due to the lack of onboard access and adaptive optics, it is possible that pointing errors become important and must be considered. If pointing errors are non-negligible, , then we instead must consider a lossy fading channel with maximum transmissivity . As discussed in earlier sections, pointing errors occur on a sufficiently slow time-scale such that they are reconcilable by the receiver. Hence, the capacity for this channel can be accessed via Eq. (28), such that
where acts as a correction factor to the PLOB bound. It is clear that when we retrieve Eq. (42). In Fig. 1(c) the loss properties of an optical intersatellite channel are illustrated with respect to distance between communicating satellites. This depicts similar transmissivity behavior to ground-satellite downlink channels with zenith angle without the additional degradation associated with atmospheric interactions.
We have some important considerations to note. First of all, intersatellite channels can only be formed between satellites that fall within each other’s line-of-sight. This naturally implies a limit to the maximum distance over which an intersatellite channel can be physically established. For any two satellites in circular orbits, at some point the Earth blocks the free-space between them, prohibiting transmittance. If two satellites orbit at altitudes and , then we find this limit to be
This is derived through basic geometric considerations (see the Supplementary Material for a derivation).
Secondly, let us justify the modeling of intersatellite channels as pure-loss channels. The number of thermal photons impinging upon a satellite detector is determined by the orientation and field of view of the detector. For communication between satellites, the transmitters and detectors do not occupy fixed orientations with respect to the main sources of brightness. Indeed, there will exist best case and worst case orientations: In the best-case scenario, the satellite detector will face completely away from the Earth or Moon, so that their albedos are not within the detector’s field of view whatsoever. In a worst case scenario, the detector will be oriented directly facing the Earth (as in uplink).
However, point-to-point quantum communication can always be optimized by choosing the physically directed channel which results in less thermal background photons at the detector; irrespective of the logical direction of communication. That is, each intersatellite channel can exchange quantum systems in the direction which achieves the best detector orientation with respect to background noise. By optimizing the physical orientation of an intersatellite quantum network, each receiver will only ever experience a fraction of the worst background noise experienced by satellite uplink channels for which thermal corrections are minimal for link lengths of . We leave more formal treatments of these channel properties to future works, with the confidence that pure-loss channels accurately model such free-space links.
Hence, we can reliably model intersatellite free-space links as pure-loss channels. As such, we treat the upper-bound in Eq. (43) as an achievable rate so that can be accomplished by an optimal point-to-point protocol.
Iii Modular Quantum Networks
iii.1 Network Model
In this work, we construct a simple model for the study of modular quantum networks. Namely, we consider a global network which consists of a collection of sub-networks called communities, where the community is denoted by the undirected sub-graph
Here, defines a subset of all network nodes that compose the community, while denotes the subset of all network edges that connect them. For now, we consider each community network to be completely general, and can adopt an arbitrary topology. Here, we focus on quantum networks which observe spatial-modularity , such that communities are spatially separated. This means that each community is completely disconnected from every other community, i.e. the community node sets are all pairwise disjoint , for all .
In order to mediate communication between different communities, we introduce a backbone network . This is a large-scale network for which none of its nodes are user nodes, used purely to facilitate end-to-end communications between users contained in different communities. Crucially, we assume that each community possesses a set of undirected edges which connect a set of community nodes to backbone network nodes. We refer to these as intercommunity edges, such that the set of intercommunity edges
gives each community access to the backbone.
More precisely, we can define an intercommunity sub-network , which describe the undirected graph that emerges between the community and the backbone. The set defines the complete collection of nodes that are interconnected between the community and the backbone. However, the nodes are already contained within or ; hence, it is important to distinguish between the community nodes and the backbone nodes which comprise this sub-network. For this, we introduce the notation
Intuitively, can be thought of as the subset of nodes from the community conditioned on being connected to (and vice versa for ).
This modular structure takes a very intuitive form and is remarkably useful for modeling realistic, hybrid quantum networks. When an equivalence relation is enforced between nodes in similar communities, the network quotient graph can be viewed as a star-network 222In the Supplementary Material we present more general aspects of networks with community structures from which this modular network emerges as a useful and highly desirable class. It allows us to completely separate communities and the backbone from one another. This makes it easier to compartmentalize different sub-network structures which may operate in completely different physical domains. Furthermore, it helps to derive independent network conditions on each of the sub-networks in accordance with some global objective. We summarize this architecture in the following definition which has also been illustrated in Fig. 2(a).
(Modular Network): A modular network is a network architecture constituent of community sub-networks , and a backbone sub-network . Each community sub-network is connected to the backbone via a set of edges , described by the intercommunity sub-networks , and there are no direct links between communities.
iii.2 Modular Network Capacities
As discussed in Section II, the optimal end-to-end performance within a quantum network is quantified by its multi-path or flooding capacity , which describes the optimal number of target bits that can be transmitted between end-users per use of a flooding protocol. Any quantum network , including the modular designs introduced, can be represented by a global distribution of channels and a corresponding distribution of single-edge channel capacities , where . For general fading networks, it is always possible to use these distributions and the general expressions from Eqs. (13) and (14) in order to determine the flooding capacity.
However, the translation into a modular architecture means that there exist particular classes of network cuts which are performed on different sub-networks. It becomes very useful to formally define a number of the important multi-edge capacities associated with these classes of cuts. In each of the following settings, we consider a pair of end-users contained within remote communities of a generic, global modular network i.e. and such that . It is now useful to denote community sub-networks with respect to the end-user that they contain, i.e. we may write and respectively. We assume each sub-network to adopt arbitrary topologies and capacity distributions.
iii.2.1 Local-Community Capacities
We define a local-community cut as that which partitions two end-users within the network by exclusively collecting edges within one of the user communities , for either . That is, a local-community cut-set takes the form This restricted form of network cut will generate an associated multi-edge capacity, which we label a local-community capacity,
For end-user nodes which do not share a direct connection with the backbone (i.e. ) then this form of restricted cut always exist.
However, if an end-user node does share a direction connection with the backbone, then a valid local-community cut will not exist. In this case it is never sufficient to remove edges solely from the community networks, and one must cut at least one edge from the set of intercommunity edges . To this end, we must slightly modify the local-community cut so that it removes any direct connections from the user node to the backbone, and then to identify the optimal set of edges to be removed from the community. Hence, a valid cut-set becomes We can then define an analogous local-community capacity according to this class of network cut.
iii.2.2 Backbone Capacities
A backbone cut is a network cut that exclusively collect edges on the backbone network in order to partition the two end-users. This kind of cut-set takes the form which generates an associated multi-edge backbone capacity,
In the modular network architecture we are investigating, when considering end-users contained in unique communities, these kinds of cuts always exist. It is always sufficient to perform a cut on the backbone since there does not exist any other collection of edges that can be used to form a valid path between communities.
iii.2.3 Global-Community Capacities
Finally, we can formalize a multi-edge capacity associated with exclusively collecting intercommunity edges. The end-user communities and are connected to the backbone via the sets of intercommunity edges and respectively. If we removed either of these sets of edges, then the two remote users would be automatically partitioned. Hence, the edge sets and both correspond to valid cuts on the network and each generate a multi-edge capacity
for . We can then minimize over the end-users to define a multi-edge capacity,
Clearly, this form of network cut always exists. We refer to this kind of partitioning as community isolation, since it isolates a community sub-network entirely from the rest of the network. Furthermore, we name the global-community capacity, as it refers to globally isolating the entire community sub-network 333While it might be more convenient to call this the intercommunity capacity, such a name might be confused as a more general term for the capacity when the end-users are located in different communities (which is implied). The global-community capacity is intended to be more distinct than this, and specify a particular network cut..
iii.3 Idealized Modular Networks
Arbitrary architectures can always be treated using the capacity expressions from Section II for general fading networks. However, the generality of these arguments make it difficult to present rigorous analytical statements about specific features or tangible network properties. In order to understand the ultimate potential of quantum networks, we need to simultaneously optimize the point-to-point channels and the network architecture in which they are arranged. Hence, it is desirable to strike a balance between realism and ideality in such a way that allows us to derive informative results about quantum networks and end-to-end performance. In the following we propose sub-network connectivity constraints that strike this balance.
iii.3.1 Backbone Regularity
Firstly, we can impose regularity on the network backbone, demanding that the degree of each node is constant. This leads to a highly-connected network structure which is ideal for multi-path routing strategies. Let the function compute the degree of the node . Then we impose
which defines the regularity parameter of the backbone. It is important to make clear that these constraints only apply to intra-network connections. Indeed, a node on the backbone can have connections to neighbors on the backbone network, but also possess additional intercommunity connections via the sub-network , without any further constraint. It is useful to quantify the number of intercommunity connections permitted between the backbone and communities using the notation,
While regularity is an idealized property of realistic networks, in the context of a non-user repeater network such as the backbone it is very much feasible and extremely useful in order to understand the limits of quantum networks.
iii.3.2 Community Connectivity
Community sub-networks are likely to be smaller scale and less predictable structures than the backbone, partly due to the presence of user-nodes. Thus flexibility in their design is important. Here, we do not impose regularity but instead define classes of communities in accordance with the smallest local-community cut that they contain.
(-connectivity): Consider a community sub-network . We say the community is if is the smallest number of edges that must be removed in order to disconnect a pair of community nodes, minimized over all possible node pairs . More precisely,
where denotes a community cut-set between the nodes and .
Hence, defines the minimum local community cut-set cardinality, given some network topology and choice of end-users. This is a completely general property which is unique for all community networks, using the most easily disconnected pair of nodes in the network as a metric for how well it is connected. Regular networks are an example of an architecture for which their -connectivity is simply equal to the network regularity. Hence, we can consider community sub-networks to be -connected while encompassing a very large set of architectures.
iii.3.3 Idealized Modular Network
Combining the constraints of regularity on the backbone and -connectivity on the community sub-networks, it is possible to define an ideal modular quantum network architecture in terms of these parameters. This generates a structure that can be investigated analytically in the following sections.
(Ideal Modular Network): An ideal modular network is a network architecture constituent of community sub-networks each of which are -connected, and a backbone sub-network which is -regular. Each community sub-network is connected to the backbone via edges, described by the intercommunity sub-networks , and there are no direct links between communities.
An illustration of this architecture can be found in Fig. 2(b). When focusing on a particular pair of end-user nodes from two remote communities in the global network, we can then specify their -connectivity properties.
iii.4 Minimum-Cut as Community Isolation
Care must be taken when constructing this form of modular network so to ensure not only high-rate communication within each community, but also high-rate communication between different communities mediated by the backbone. If the backbone network is poorly connected, or possesses weak links, it will not effectively assist long-distance communication. Meanwhile, even if communities are connected to a high quality backbone, insufficiently strong capacities in a local-community can compromise its use. Hence, there exists a careful balance between all of the sub-networks in the modular model, and their connectivity/capacity properties throughout. It is therefore highly desirable to identify a relationship between the quality of channels within the backbone and the quality of channels within the communities.
In order to better grasp these relationships, we can investigate the ideal modular networks defined in Definition 3. Regular networks (such as that on the backbone) possess very convenient qualities which allow for useful insight into minimum network cuts. As such, they can be analytically studied as highly connected, ideal network structures and used to reveal fundamental limitations for end-to-end communication.
Our mission becomes the following: To derive conditions on each of the sub-networks such that the flooding capacity between the remote users is always their global-community capacity. In this way, the minimum cut is always achieved by community isolation on either of the end-user communities. Equivalently, it means that the minimum cut can always be found on a simplified quotient graph of the modular network, vastly simplifying its analysis . When this is the case, the end-to-end capacities between any two unique communities are always distance-independent, i.e. the ultimate rate between two end-user communities does not change with respect to the physical separation of those communities. This is an extremely desirable property of a quantum network, particularly on large-scales.
If a modular network satisfies this property, it means that (i) the backbone network is of sufficiently high quality that it never impedes the network performance over (potentially very) long distances, and (ii) that the local-communities are of sufficiently high quality that neither compromises local or network-wide communication. Furthermore, by imposing that the minimum cut be the intercommunity edges, it allows us to reveal unique constraints on each sub-network, which are summarized in the following theorem.
Consider an ideal modular network of the form introduced in Definition 3. Select any pair of end-users contained in remote communities and . For all , there exist single-edge threshold capacities on the communities and backbone sub-networks for which the network flooding capacity is given by the global-community capacity,
The threshold capacities are given by,
where is the minimum cut-set cardinality on the backbone network. If these threshold capacities are violated, then the global-community capacity becomes an upper-bound on the end-to-end capacity, .
A detailed proof can be found in Section I of the Supplementary Material. Thanks to backbone regularity and community connectivity, the minimum cut-set cardinalities that occur within each sub-network can be easily identified. Then, it is straightforward to enforce single-edge capacity constraints which ensure that the local-community/backbone capacities are always larger than the global-community capacity.
In this theorem, we have used the fact that the cardinality of the smallest backbone cut-set between two end-users in remote communities can be analytically derived, thanks to network regularity. This minimum cardinality takes the form
where is a function that computes the minimum number of edges that must be cut to isolate all the nodes on the backbone which are also connected to the community . The explicit form of this expression in found in the Supplementary Material, and depends on the precise spatial arrangement of connections from the community to the backbone. However, we can generally bound this quantity using
The lower-bound corresponds to a worst-case spatial distribution of community-to-backbone connections, when all the community nodes are connected to the same node on the backbone, i.e. . Then it is sufficient to isolate just one backbone node to perform a valid end-user cut, collecting only edges (since the backbone is -regular). The upper-bound corresponds to a best-case scenario; when all the community nodes are connected to backbone nodes which don’t share any neighbors or edges. In this case, the smallest cut-set restricted to the backbone is found by isolating all nodes individually. As a result, this cut collects exactly edges.
Fig. 3 depicts a number of examples of minimum backbone cut-sets for remote communities connected to a Manhattan backbone (). In these figures we display only one end-user community and assume that the other end-user community is sufficiently distant that it does not share intercommunity connected nodes on the backbone.
As a result, we can always present a best and worst-case single-edge threshold capacity for the backbone network, . That is, we can sandwich the backbone threshold capacity according to
The more effectively that the intercommunity connections are dispersed across the backbone, the weaker the single-edge constraint that must be forced upon it.
Iv Hybrid Free-Space Networks
In this final section we combine results and theory from Section II and III in order to investigate hybrid fiber/free-space modular quantum networks. Here we study two pertinent cases in an effort to reveal tangible resource requirements for future quantum networks: A fiber/satellite modular configuration and a ground-based free-space/fiber model.
iv.1 fiber/Satellite Configuration
An interesting modular configuration consists of fiber-based community networks which are interconnected via a backbone satellite network. This model captures a realistic satellite-based model of the quantum internet, in which dynamic intersatellite links are used to facilitate long distance quantum communication at high-rates. In this scenario, the weakest links are typically the ground-to-satellite free-space connections, due to the impact of atmospheric decoherence and turbulence on a transmitted beam. Therefore, the constraints revealed in Theorem 2 are very realistic, as community isolation is likely to be the minimum cut in many settings.
In Theorem 2 we devise single-edge capacity lower bounds on the community networks which guarantee the network flooding capacity is equal to the global-community capacity. For fiber-based networks, these single-edge lower bounds can be used to identify a maximum tolerable fiber-length, that is permitted within the fiber-network. In the context of a satellite-based backbone network, the single-edge capacity lower bound can be translated into a maximum intersatellite separation, which describes the maximum propagation distance that is permitted for free-space channels between satellites in the backbone. These are critical quantities which directly motivate the construction of ground-based and satellite-based networks for global quantum communication.
iv.1.2 Optimal Performance
We wish to enforce that the minimum cut is always achieved by community isolation, generating the global-community capacity . In this physical setting, each intercommunity edge is described by ground-to-satellite channel which may be an uplink or downlink channel. Thanks to teleportation, a network protocol can always choose the physical channel direction that maximizes its point-to-point capacity independently from the desired logical direction of community. Downlink channels are always superior to uplink, and therefore we can simply model the global community capacity as the sum of a downlink capacities. This multi-edge capacity will be bounded by
where and capture the fading dynamics and maximum transmissivity of each downlink channel that connect to the backbone, and depend on beam trajectory. Meanwhile, infers community-wide thermal-noise conditions. Since all of the intercommunity edges in are connected to a relatively small area, we can assume identical operational conditions for all downlink edges. However, these operational conditions will not be consistent for both end-users; when communicating on a global scale, one user may be in night-time while the other is in day-time with independent weather conditions.
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