Abstract
Recent experimental breakthroughs in satellite quantum communications have opened up the possibility of creating a global quantum internet using satellite links. This approach appears to be particularly viable in the near term, due to the lower attenuation of optical signals from satellite to ground, and due to the currently short coherence times of quantum memories. The latter prevents groundbased entanglement distribution using atmospheric or opticalfiber links at high rates over long distances. In this work, we propose a globalscale quantum internet consisting of a constellation of orbiting satellites that provides a continuous, ondemand entanglement distribution service to ground stations. The satellites can also function as untrusted nodes for the purpose of longdistance quantumkey distribution. We develop a technique for determining optimal satellite configurations with continuous coverage that balances both the total number of satellites and entanglementdistribution rates. Using this technique, we determine various optimal satellite configurations for a polarorbit constellation, and we analyze the resulting satellitetoground loss and achievable entanglementdistribution rates for multiple ground station configurations. We also provide a comparison between these entanglementdistribution rates and the rates of groundbased quantum repeater schemes. Overall, our work provides the theoretical tools and the experimental guidance needed to make a satellitebased global quantum internet a reality.
Introduction
One of the most remarkable applications of quantum mechanics is the ability to perform secure communication via quantumkey distribution (QKD)^{1,2,3,4}. While current global communication systems rely on computational security and are breakable with a quantum computer^{5,6,7}, QKD offers, in principle, unconditional (informationtheoretic) security even against adversaries with a quantum computer. With several metropolitanscale QKD systems already in place^{8,9,10,11,12,13,14,15}, and with the development of quantum computers proceeding at a steady pace^{16,17,18}, the time is right to begin transitioning to a global quantum communications network before fullscale quantum computers render current communication systems defenseless^{19,20,21}. In addition to QKD, a global quantum communications network, or quantum internet^{22,23,24,25,26}, would allow for the execution of other quantuminformationprocessing tasks, such as quantum teleportation^{27,28}, quantum clock synchronization^{29,30,31}, distributed quantum computation^{32}, and distributed quantum metrology and sensing^{33,34,35}.
Building the quantum internet is a major experimental challenge. All of the aforementioned tasks make use of shared entanglement between distant locations on the earth, which is typically distributed using singlephotonic qubits sent through either the atmosphere or optical fibers. These schemes require reliable singlephoton sources, quantum memories with high coherence times, and quantum gate operations with low error. It is well known that optical signals transmitted through either the atmosphere or optical fibers undergo an exponential decrease in the transmission success probability with distance^{36,37}. Quantum repeaters^{38,39,40} have been proposed to overcome this exponential loss by dividing the transmission line into smaller segments along which errors and loss can be corrected using entanglement swapping^{27,41} and entanglement purification^{42,43,44}. Several theoretical proposals for quantum repeater schemes have been made (see refs ^{40,45,46} and references therein); however, many of these proposals have resource requirements that are currently unattainable. Furthermore, experimental demonstrations performed so far have been limited^{47,48,49} and do not scale to the distances needed to realize a globalscale quantum internet.
Satellites have been recognized as one of the best methods for achieving globalscale quantum communication with current or nearterm resources^{23,50,51,52,53,54}. Using satellites is advantageous due to the fact that the majority of the optical path traversed by an entangled photon pair is in free space, resulting in lower loss compared to groundbased entanglement distribution over atmospheric or fiberoptic links. Satellites can also be used to implement longdistance QKD with untrusted nodes, which is missing from most current implementations of longdistance QKD due to the lack of a quantum repeater. A satellitebased approach also allows for the possibility to use quantum strategies for tasks such as establishing a robust and secure international time scale via a quantum network of clocks^{55}, extending the baseline of telescopes for improved astronomical imaging^{56,57,58}, and exploring fundamental physics^{59,60}.
Several proposals for satellitebased quantum networks have been made that use satellitetoground transmission, groundtosatellite transmission, or both^{50,53,54,61,62,63,64,65,66,67,68,69}. Recent experiments^{65,70,71,72,73,74,75,76} (see also ref. ^{77} for a review) between a handful of nodes opens up the possibility of building a globalscale quantum internet using satellites. As shown in Fig. 1, this means having a constellation of orbiting satellites that transmit either bipartite or multipartite entanglement to ground stations. These ground stations can act as hubs that then distribute entanglement to neighboring ground stations via short groundbased links. In order to successfully implement such a globalscale satellitebased quantum internet, many factors must be taken into account, such as economics, current technology, resource availability, and performance requirements. Ideally, the satellite network should have continuous global coverage and provide entanglement on demand at a reasonably high rate between any two distant points on earth. Given this performance requirement, important questions related to economics and resources arise, such as: How many satellites are needed for continuous global coverage? At what altitude should the satellites be placed? What entanglementdistribution rates are possible between any points on earth, and how do these rates compare to those that can be achieved using groundbased quantum repeater setups?
In this work, we address these questions by analyzing a globalscale quantum internet architecture in which satellites arranged in a constellation of polar orbits (see Fig. 2) act as entanglement sources that distribute entangled photon pairs to ground stations. The nearestneighbor entangled links can then be extended via entanglement swapping to obtain shared entanglement over longer distances. We start by determining the required number of satellites for such a network to have continuous global coverage. Since satellites are a costly resource, continuous global coverage should be achieved with as few satellites as possible. To that end, our first contribution is to define a figure of merit that allows us to investigate the tradeoff between the number of satellites, their altitude, the average loss over a 24h period, and the average entanglementdistribution rates. By running simulations in order to optimize our figure of merit, we obtain one of our main results, which is the optimal number of satellites needed for continuous global coverage, as well as the optimal altitude at which the satellites should be placed such that the average loss is below a certain threshold. We then compare the resulting entanglementdistribution rates to those obtained via a groundbased entanglement distribution scheme assisted by quantum repeaters. This leads to another key result of our work, which is that the satellitebased scheme (without quantum repeaters) can outperform groundbased quantum repeater schemes in certain cases. We also consider entanglement distribution to major global cities over intercontinental distances. The key result here is that, with a constellation of 400 satellites, entanglement distribution at a reasonably high rate is not possible beyond approximately 7500 km.
We remark that our approach is similar to the approach taken in ref. ^{64}, in which ground stations are placed only on the equator and there is a single ring of satellites in an equatorial orbit around the earth. Our work goes beyond this by considering a genuine network scenario in which multiple ground stations are placed arbitrarily on the earth and there is a constellation of satellites in polar rather than equatorial orbits, as shown in Fig. 2. Furthermore, while prior work has considered satellite constellations for entanglement distribution^{69,78}, to our knowledge, the type of dynamic quantum network simulation with satellite constellations that we consider, along with optimization over different constellation configurations, has not been previously studied.
We expect the results of this work to serve as a guide for building a globalscale quantum internet, both in terms of the number of satellites needed as well as the expected performance of the network. In particular, our results comparing satellitebased entanglement distribution to groundbased repeaterassisted entanglement distribution suggest that, at least in the near term, satellites are indeed the most viable approach to obtaining a globalscale quantum internet.
Results
Network architecture
Our proposed satellite network architecture is illustrated in Fig. 2. We consider N_{R} equally spaced rings of satellites in polar orbits. We allow for N_{S} equally spaced satellites in each ring, so that there are N_{R}N_{S} satellites in total, all of which are at the same altitude h. This type of satellite constellation falls into the general class of Walker star constellations^{79}, and we consider it mainly for its simplicity, but also because this constellation is similar to the Iridium communicationssatellite constellation^{80,81}. Prior works have examined various other types of satellite constellations for the purpose of continuous global coverage^{79,82,83,84}. The recent Starlink constellation^{85} is also being used to provide a global satellitebased internet service. Investigations of these other satellite constellation types, and comparisons between them in the context of a global quantum internet, is an interesting direction for future work.
The satellites act as source stations that transmit pairs of entangled photons to lineofsight ground stations for the purpose of establishing elementary entanglement links. The ground stations can act as quantum repeaters in this scheme—performing entanglement purification and entanglement swapping once the elementary links have been established. In this way, we execute longdistance entanglement distribution between ground stations. Note that we could alternatively use the satellites as quantum repeaters^{86,87}, which would require uplinks. It has been shown in, e.g., ref. ^{63}, that uplinks are more lossy and lead to lower key rates for QKD. For this reason, we consider downlinks only. The photon sources on the satellites produce polarizationentangled photon pairs. Stateoftheart sources of entangled photons are capable of producing polarizationentangled photons on a chip with a fidelity up to 0.97^{88,89,90,91}.
Overview of simulations
We obtain our results by running several entanglement distribution simulations using the satellite network architecture illustrated in Fig. 2. We consider as our baseline requirement that a satellite network should provide continuous coverage to two ground stations located on the equator. We thus start by running a 24h simulation with two ground stations at the equator separated by distances d between 100 and 5000 km, and satellite configurations ranging from 20 to 400 satellites at altitudes between 500 and 10000 km. We choose ground distances starting from 100 km because 100 km is roughly the longest distance at which groundbased entanglement distribution can be successfully performed at a reasonable rate without quantum repeaters; see, for e.g., refs ^{92,93,94,95}. Our choice of satellite altitudes encompasses both low earth orbits and medium earth orbits, which are the orbits currently being used for most satellite communications systems^{81,85}.
A satellite configuration is given by the number N_{R} of satellite rings, the number N_{S} of satellites per ring, and the altitude h of the satellites. Our requirement of continuous coverage means that both ground stations must be simultaneously in view of a satellite at all times. We also impose an additional requirement that, even when in view of both ground stations, the total transmission loss between a satellite and the ground station pair should not exceed 90 dB, in order to keep ebit rates above 1 Hz. (See the Methods section for further simulation details.) Note that, based on the satellite constellations that we consider here, two ground stations at the equator is the worstcase scenario, in the sense that two ground stations at higher or lower latitudes will always have less satellitetoground loss on average (we show this in Fig. 5 below).
For all of our simulations, we take into account attenuation due to the atmosphere; see the “Methods” section for a description of our loss model. However, we assume clear skies, hence no rain, haze, or cloud coverage in any area. Including these extra elements would introduce extra attenuation factors (see, for e.g., section 2.1.1.4 of ref. ^{37}and refs ^{96,97}), which would increase the overall satellitetoground transmission loss (see refs ^{78,98} for an analysis of satellitetoground quantum key distribution in a localized area that incorporates local weather conditions). We also point out that, especially in the daytime, background photons (e.g., from the sun) can reduce the fidelity of the distributed entangled pairs, because the receiver will collect those background photons in addition to the signal photons from the entanglement source. This source of background photons is perhaps the most difficult obstacle to continuous global coverage. Timing information, as well as information about the spectral and spatial profile of the signal, can help reduce the noise via filtering, but only to a certain extent (see, for e.g., refs ^{70,99}). Furthermore, because the probability to transmit single photons from satellite to ground is quite low, the communicating parties must ensure that the probability to collect background photons is even lower in order to ensure a high signaltonoise ratio (SNR), and thus a high fidelity for the received quantum state. In the Methods section we show how the fidelity of the transmitted states is affected by spurious background photons.
Optimal network configurations for global coverage
Given two ground stations separated by a distance d and situated at the equator, along with a particular satellite constellation defined by (N_{R}, N_{S}, h), as described above, how do we evaluate the performance of the given satellite constellation? Since satellites are currently an expensive resource, we would like to have as few satellites as possible in the network while still maintaining complete and continuous coverage. We could therefore take as our figure of merit the total number of satellites in the network. Specifically, given an altitude h of the satellites and distance d between the two ground stations, we define N_{opt}(h, d) to be the minimum total number of satellites needed to have continuous 24h coverage for the two ground stations (see the Methods section for details). We could then minimize N_{opt}(h, d) with respect to altitudes. On the other hand, we also want high entanglement distribution rates. We let \(\overline{R}({N}_{{\rm{R}}},{N}_{{\rm{S}}},h,d)\) denote the average entanglementdistribution rate over 24 h for the satellite configuration given by (N_{R}, N_{S}, h) and two ground stations at the equator separated by a distance d (see the Methods section for the formal definition). The rate is calculated in a simple scenario without multimode transmission from the satellites and without multimode quantum memories at the ground stations. We could then take the quantity
as our figure or merit, which is the average rate (in ebits/s) over a 24h period for a given altitude h and a given distance d, where the optimization is over satellite configurations with a fixed h such that there is continuous coverage for 24 h and the loss at any time is less than 90 dB (see the Methods section for details). Now, as one might expect, with fewer satellites the average loss would increase, thus decreasing entanglementdistribution rates, while increasing the number of satellites would decrease the loss, hence increasing the average entanglementdistribution rate. In order to balance our two competing goals—minimizing the total number of satellites and also maximizing the average rate—we take as our figure of merit the ratio of the average entanglementdistribution rate to the total number of satellites:
which has units of ebits/s per satellite. Then, the goal is to take the satellite configuration that maximizes this figure of merit. In other words, our goal is to find
for any given distance d between the two ground stations, where the optimization is constrained such that there is continuous coverage to the two ground stations for 24 h and the transmission loss at any given time is less than 90 dB (see the “Methods” section for details). We suppress the dependence of the functions \({N}_{{\rm{R}}}^{\star }\), \({N}_{{\rm{S}}}^{\star }\), and h^{⋆} on the distance d when it is understood from the context. We let
be the figure of merit c optimized over N_{R} and N_{S}, with the constraint that both ground stations have continuous coverage over 24 h and that the transmission loss at any time is less than 90 dB.
The results of our simulations are shown in Fig. 3. The complete set of results for all ground distances and satellite configurations considered is contained in the data files accompanying the paper. We first consider the quantity N_{opt}(h, d) as a function of altitude h for fixed groundstation separations d (left panel of Fig. 3). In terms of the satellite configurations, we find that at higher altitudes more satellites per ring are required in general, while at lower altitudes generally more rings are required. In terms of the total number of satellites, we find that as the altitude increases the total number of satellites decreases. Interestingly, however, as we continue to increase the altitude we find that there are altitudes (between 5000 and 6000 km) at which the total number of satellites reaches a minimum. Beyond this range of altitudes, the required number of satellites increases. The presence of this minimum point gives us an indication of the altitudes at which satellites should be placed in order to minimize the total number of satellites. However, for these altitudes, the average entanglementdistribution rates are generally quite low, on the order of 10 ebits/s.
Next, we consider the figure of merit C(h, d) defined in eq. (4). We plot this quantity for various values of the altitude h and distance d in the central panel of Fig. 3. In the right panel of Fig. 3, we plot the corresponding average entanglementdistribution rate over 24 h. For all distances d, except for d = 500 km, we find that there is an altitude h at which C(h, d) is maximal. These optimal altitudes, along with the values of N_{R} and N_{S} achieving the value of C(h, d) and the corresponding average loss and average entanglementdistribution rate over 24 h, are shown in Table 1. Given a desired distance between the ground stations, these optimal parameters can be used to decide on the number of satellites to put in the network and the altitude at which to put them so that there is continuous coverage, which then leads to particular values for the average loss and the average entanglementdistribution rate. Conversely, given a particular performance requirement (in terms of the entanglementdistribution rate), we can use our results to determine both the required satellite configuration and the required distance between the ground stations in order to achieve the desired rate. For example, using the plot on the right panel of Fig. 3, in order to achieve a rate greater than 10^{3} ebits/s on average in 24 h, the satellite constellation altitude should be less than 2000 km (among the constellations considered), and the distance d between the ground stations has to be roughly less than 1500 km.
In Fig. 4, we plot the entanglementdistribution rate to two ground stations at the equator separated by a distance d = 1000 km with a satellite constellation given by N_{R} = 9 satellite rings, N_{S} = 10 satellites per ring, and altitude h = 1500 km. We also plot the distances of the ground stations to a satellite. We find that the rate exhibits a distinct oscillatory behavior with periodic bumps. In each bump, the rate increases as a satellite gets closer to the ground stations and decreases as the satellite passes by. All of the bumps in the rate have slightly different duration and slightly different peaks due to the fact that, at each time, the ground station pair is generally in view of multiple satellite, and we pick the satellite with the lowest transmission loss to the ground station pair (see the Methods section for further details). In general, therefore, each bump corresponds to a different satellite distributing entanglement to the two ground stations.
Let us now consider optimal entanglementdistribution rates to the two ground stations, i.e., let us consider the quantity \({\overline{R}}^{\text{opt}}(h,d)\) defined in eq. (1). The results are shown in the top panel of Fig. 5. We assume that the satellites transmit entangled photon pairs at a rate of R_{source} = 10^{9} ebits/s^{100}. Unsurprisingly, for every pair (h, d) of altitudes h and distances d, the quantity \({\overline{R}}^{\text{opt}}(h,d)\) is attained by the satellite configuration that we considered that has the highest number of satellites, namely N_{R} = 20 rings and N_{S} = 20 satellites per ring. However, despite the sharp increase in the number of satellites, the rates are not much higher than those in the right panel of Fig. 3, which are obtained by optimizing our main figure of merit C(h, d). The highest rate among all distances is around 4.6 × 10^{4} ebits/s, which is attained for a distance of 500 km and altitude of 500 km.
In the bottom panel of Fig. 5, we display the results of our entanglement distribution simulations when both ground stations are at a different latitude, with N_{R} = N_{S} = 15. Due to the fact that the satellites follow polar orbits in our network architecture, meaning that they congregate at the poles, the entanglementdistribution rates are higher for latitudes closer to the north and south poles than for the equator. This result also confirms that placing two ground stations at the equator is the worstcase scenario in terms of average loss (and thus average rate).
Before continuing, let us remark that our technique for obtaining optimal satellite configurations for continuous global coverage, via optimization of the quantities defined in eqs. (2) and (4), can be straightforwardly extended to an optimization procedure that consists of more than two ground stations; see the Methods section for details.
Multiple ground stations
We now present the results of an entanglement distribution simulation consisting of multiple ground stations. We place 42 ground stations in a gridlike arrangement, with horizontal separation (i.e., separation in longitude) of approximately 18° and vertical separation (i.e., separation in latitude) of approximately 18°. We use a satellite constellation of N_{R} = 15 rings and N_{S} = 15 satellites per ring, for a total of 225 satellites. In Fig. 6, we display the average loss for nearest neighbor pairs over a simulation time of 24 h.
In the top plots of Fig. 6, we consider all possible nearestneighbor pairs in the simulation. As expected, the loss is lowest away from the equator (latitude 0°), because neighboring ground stations are closer to each other away from the equator, due to the curvature of the earth, and because of the nature of our satellite constellation (satellites congregate at the poles). We also find that diagonal nearestneighbor pairs have higher losses compared to pairs that are horizontally or vertically separated. This can be explained by the fact that diagonallyseparated ground stations are farther away from each other than horizontally or verticallyseparated groundstation pairs. Our strategy for assigning a satellite to a groundstation pair (see the Methods section) thus favors pairs that are horizontally or vertically separated. We also find that the maximum loss for a satellite altitude of h = 1000 km is around 90 dB and the minimum loss is around 50 dB. For h = 5000 km, the maximum loss is around 105 dB and the minimum loss is around 75 dB.
In the bottom plots of Fig. 6, we simulate a network such that the satellites can only distribute entanglement to diagonallyseparated nearestneighbor pairs. Now, since we do not allow entanglement distribution between horizontally and vertically separated pairs, we find that the maximum average loss decreases and the minimum average loss increases. We still find that groundstation pairs at latitudes farther away from the equator have lower loss.
In the central panels of Fig. 6, we plot average entanglementdistribution rates in a simple scenario without multimode transmission from the satellites and without multimode quantum memories at the ground stations. We assume that the satellites transmit entangled photon pairs at a rate of R_{source} = 10^{9} ebits/s^{100}. In the case of entanglement distribution to all nearestneighbor pairs (top part of the central panel of Fig. 6), the maximum average rate is around 4000 ebits/s, and this occurs for horizontally separated ground stations at latitudes of 54°N and −54°N. For entanglement distribution only to diagonallyseparated nearestneighbor pairs (bottom part of the central panel of Fig. 6), the maximum average rate is around 450 ebits/s. It is possible to compensate for the loss by having multimode signal transmission from the satellites and by including multimode quantum memories at the ground stations, which would increase the average rates.
Entanglement distribution between major global cities
Although the ultimate goal of a satellitebased quantum internet is to have satellites distribute entanglement between any collection of nodes on the ground, an example of which we considered above, satellitebased quantum communication networks will likely have a hybrid form in the near term. In a hybrid network, the satellites distribute entanglement to major global cities, which act as hubs that then distribute entanglement to smaller nearby cities using groundbased links (see Fig. 1). With this in mind, we now consider entanglement distribution between pairs of major global cities. We run a 24h simulation with a satellite constellation of 400 satellites, with N_{R} = N_{S} = 20, at altitudes of h = 500, 1000, 2000, 3000, 4000, and 5000 km. We include the following cities in the simulation: Toronto, New York City, London, Singapore, Sydney, Auckland, Rio de Janeiro, Baton Rouge, Mumbai, Johannesburg, Washington DC, Lijiang, Ngari, Delingha, Nanshan, Xinglong, and Houston. The LijiangDelingha pair is chosen for comparison to a recent experiment^{71}. The simulation results are shown in Table 2.
From Table 2, we see that at around a distance of 6300 km, which is the distance between Singapore and Sydney, we can only obtain an average loss less than 90 dB for altitudes greater than 2000 km. Similarly, entanglement distribution between London and Mumbai (which are 7200 km apart) at an average loss less than 90 dB is possible only for an altitude greater than 3000 km. These results suggest that, using our constellation of 400 satellites, a distance of around 7500 km is the highest for which entanglement distribution at a loss less than 90 dB can be achieved. Indeed, for Houston and London (which are 7800 km apart), we find that the average loss is greater than 90 dB for all of the satellite altitudes that we consider.
Comparison to groundbased entanglement distribution
Let us now compare the entanglementdistribution rates obtained with satellites to the rates that can be obtained via groundbased photon transmission through optical fiber with the assistance of quantum repeaters. In particular, we compare the rates in the top panel of Fig. 5 for two ground stations at the equator separated by a distance d between 100 and 2000 km to groundbased repeater chains with endpoints the same distance d apart. For the latter, we suppose that the distance d between the endpoints is split into M elementary links by (M − 1) equallyspaced quantum repeaters. We place a source at the center of each elementary link that transmits entangled photon pairs to the nodes at the ends of the elementary link. We assume that the probability of establishing an elementary link is \(p={{\rm{e}}}^{\alpha \frac{d}{M}}\), where \(\alpha =\frac{1}{22\,\text{km}\,}\)^{101}, and we also assume that all repeater nodes are equipped with N_{mem} quantum memories facing each of its nearest neighbors. Under these conditions, the rate \({R}_{M,{N}_{\text{mem}}}\) (in ebits/s) of entanglement distribution between the endpoints is
where c is the speed of light and
(See the Methods section for details.) Note that our assumption that \(p={{\rm{e}}}^{\alpha \frac{d}{M}}\) is the bestcase scenario in which the sources fire perfect Bell pairs (so that no entanglement purification is required) and the Bell measurements for entanglement swapping are deterministic. Furthermore, the formula in eq. (5) holds in the case that the quantum repeaters have perfect readwrite efficiency and have infinite coherence time.
In Fig. 7, we compare the rate in eq. (5) with N_{mem} = 50 to the rates shown in the top panel of Fig. 5. For an altitude of 500 km, we find that the quantum repeater scheme with M = 50 elementary links outperforms the satellitebased scheme for all distances up to 2000 km. However, for M = 10 and 20 elementary links, we find that there are critical distances beyond which satellites can outperform the groundbased repeater schemes. For example, for an altitude of 500 km, the satellitebased scheme outperforms the M = 20 quantum repeater scheme beyond approximately 600 km and the M = 10 scheme beyond approximately 300 km. For an altitude of 1000 km, the satellitebased scheme outperforms the M = 20 repeater scheme beyond approximately 1200 km. Similarly, for an altitude of 2000 km, the satellitebased scheme outperforms the quantum repeater scheme beyond approximately 900 km. For an altitude of 4000 km, the satellitebased rates are lower than the quantum repeater rates for all values of M considered.
Currently, satellitebased schemes are arguably are more viable, because highcoherencetime quantum memories (which are not widely available) are not required. However, the monetary cost of the satellites, along with other overhead monetary costs (e.g., launch costs) can make implementing a satellitebased entanglementdistribution network challenging. Furthermore, local weather conditions and background photons during the daytime make it difficult to achieve the continuous coverage assumed here, which ultimately results in lower entanglementdistribution rates. On the other hand, groundbased quantum repeater schemes can achieve higher rates than satellitebased schemes, but this occurs only when the number of repeater nodes is quite high, the number of quantum memories per repeater node is high, and the coherence times of the memories is high. In addition, quantum memories currently exist mostly in a laboratory environment and are not at the stage of development that they can be widely deployed in the field, and they certainly do not have high enough coherence times to achieve the rates presented here.
Discussion
In this paper, we explored the possibility of using satellites for a globalscale quantum communications network. Our network architecture consists of a constellation of satellites in polar orbits around the earth that transmit entangled photon pairs to ground stations (see Fig. 2). By defining a figure of merit that takes into account both the number of satellites as well as satellitetoground entanglementdistribution rates, we provided estimates on the number of satellites needed to maintain full 24h coverage at a high rate based on the maximum value of the figure of merit. Using our figure of merit to decide the number of satellites in the network, we estimated the transmission loss and entanglementdistribution rates that can be achieved for two ground stations placed at various latitudes, for multiple ground stations at various locations in a gridlike arrangement, and for multiple major global cities in a hybrid satellite and groundbased network in which the cities can act as hubs that receive entanglement from satellites and disperse it to surrounding locations via groundbased links. Finally, we compared the achievable entanglementdistribution rates for two ground stations using satellites to achievable entanglementdistribution rates using groundbased links with quantum repeaters. With a large enough number of repeater nodes, along with a high enough number of highcoherencetime quantum memories at each node, it is possible to obtain entanglementdistribution rates that surpass those obtained with satellites. However, satellitebased schemes operating without quantum repeaters can, in certain cases, outperform quantum repeater schemes, with drawbacks being that a relatively high number of satellites is required and that adverse weather conditions can prevent continuous operations and thus reduce the rate. These drawbacks appear to be less prohibitive in the near term than the major drawback of groundbased, repeaterassisted entanglement distribution, which is that quantum memories with very high coherence times are simply not widely available. Therefore, it appears that a satellitebased scheme will remain the preferred option over groundbased repeater schemes into the near term, especially with the improving miniaturization and increasing fidelity of entanglement sources^{65,90} and the decreasing cost and miniaturization of satellites^{51,53,54}.
Our analysis of a global, satellitebased quantum internet opens the door to plenty of further study. For example, our simulations can be refined by taking into account local weather conditions. Our optimization procedure can also be extended to include more than two ground stations (see the Methods section). It would also be interesting to compare other types of satellite constellations, much like those studied in refs ^{83,84}. Finally, to have a genuine quantum network requires efficient routing algorithms. It would be interesting to explore entanglement routing in a satellite network along the lines of, e.g., refs ^{85,102,103} in the classical setting.
In summary, the broadscope vision is to have a quantumconnected world, similar to today’s internet, where users across the globe can share quantum information for any desirable task. In our view, the backbone of such a network is built on local and global quantum entanglement, in which intercontinentallyseparated ground stations located in major cities act as entanglement hubs connecting the local network users of one city to those of another (Fig. 1). Hybrid networks interfacing spacebased quantum communication platforms with groundbased quantum repeaters will make this vision a real possibility.
Methods
Loss model
In the absence of spurious background photons, the transmission of photons from satellites to ground stations is modeled well by a bosonic pureloss channel with transmittance η_{sg}^{104}. For singlephoton polarization qubits (with a dualrail encoding), transmission through the pureloss channel corresponds to an erasure channel^{105}. That is, given a singlephoton polarization density matrix ρ, the evolution of ρ is given as
where \(\,{\text{vac}}\,\rangle\! \langle \,{\text{vac}}\,\) is the vacuum state. Hence, with probability η_{sg}, the dualrail qubit is successfully transmitted and with probability 1 − η_{sg} the qubit is lost. For the transmission of a pair of singlephoton dualrail qubits, let \({\eta }_{\,\text{sg}\,}^{(1)}\) and \({\eta }_{\,\text{sg}\,}^{(2)}\) be the transmittances of the two pureloss channels. Then, with probability \({\eta }_{\,\text{sg}\,}^{(1)}{\eta }_{\,\text{sg}\,}^{(2)}\), both qubits are successfully transmitted and with probability \(1{\eta }_{\,\text{sg}\,}^{(1)}{\eta }_{\,\text{sg}\,}^{(2)}\) at least one of the qubits is lost^{101}. In the following subsection, we consider photon transmission in the presence of background photons.
The transmittance η_{sg} generally depends on atmospheric conditions (such as turbulence and weather conditions) and on orbital parameters (such as altitude and zenith angle)^{96,97,106}. In general, we can decompose η_{sg} as
where η_{fs} is the freespace transmittance and η_{atm} is the atmospheric transmittance. Freespace loss occurs due to diffraction (i.e., beam broadening) over the channel and due to the use of finitesized apertures at the receiving end. These effects cause η_{fs} to scale as the inversedistance squared in the farfield regime. Atmospheric loss occurs due to absorption and scattering in the atmosphere and scales exponentially with distance as a result of the BeerLambert law^{37,107,108}. However, since atmospheric absorption is relevant only in a layer of thickness 10–20 km above the earth’s surface^{37}, freespace diffraction is the main source of loss in spacebased quantum communication. In order to characterize the freespace and atmospheric transmittances with simple analytic expressions, we ignore turbulenceinduced effects in the lower atmosphere, such as beam profile distortion, beam broadening (prominent for uplink communication^{37,63}), and beam wandering (see, for e.g., ref. ^{106}). Note that turbulence effects can be corrected using classical adaptive optics^{37}. We also ignore the inhomogeneous density profile of the atmosphere, which can lead to path elongation effects at large zenith angles. A comprehensive analysis of loss can be found in refs ^{106,108}.
Consider the lowestorder Gaussian spatial mode for an optical beam traveling a distance L between the sender and receiver, with a circular receiving aperture of radius r. Then, the freespace transmittance η_{fs} is given by^{36}
where
is the beam waist at a distance L from the focal region (L = 0), \({L}_{{\rm{R}}}:=\pi {w}_{0}^{2}{\lambda }^{1}\) is the Rayleigh range, λ is the wavelength of the optical mode, and w_{0} is the initial beamwaist radius.
We model the atmosphere as a homogeneous absorptive layer of finite thickness in order to characterize η_{atm}. Uniformity of the atmospheric layer then implies uniform absorption (at a given wavelength), such that η_{atm} depends only on the optical path traversed through the atmosphere. Under these assumptions, and using the BeerLambert law^{107}, for small zenith angles we have that
with \({\eta }_{\,\text{atm}}^{\text{zen}\,}\) the transmittance at zenith (ζ = 0). For \( \zeta \, >\,\frac{\pi }{2}\), we set η_{atm} = 0, because the satellite is over the horizon and thus out of sight. The zenith angle ζ is given by
for a circular orbit of altitude h, with R_{E} ≈ 6378 km being the earth’s radius.
Note that the model of atmospheric transmittance given by eqs. (11) and (12) is quite accurate for small zenith angles^{37}. However, for spacebased quantum communication at or near the horizon (i.e., for ζ = ±π/2), more exact methods relying on the standard atmospheric model must be used^{106}. In practice, it makes sense to set η_{atm} = 0 at large zenith angles, effectively severing the quantum channel, because the loss will typically be too high for the link to be practically useful.
To summarize, the following parameters characterize the total loss η_{sg} = η_{fs}η_{atm}: the initial beam waist w_{0}, the receiving aperture radius r, the wavelength λ of the satellitetoground signals, and the atmospheric transmittance \({\eta }_{\,\text{atm}}^{\text{zen}\,}\) at zenith. See Table 3 for the values that we take for these parameters in our simulations.
Using the values in Table 3, we plot in Fig. 8 (bottom) the total transmittance as a function of the ground distance d between two ground stations with a satellite at the midpoint; see Fig. 8 (top). We observe that for larger ground separations the total transmittance \({\eta }_{\,\text{sg}\,}^{2}\) is actually larger for a higher altitude than for a lower altitude; for example, beyond approximately d = 1600 km the transmittance for h = 1000 km is larger than for h = 500 km. We also observe that there are altitudes at which the transmittance is maximal. Intuitively, beyond the maximum point, the atmospheric contribution to the loss is less dominant, while below the maximum (i.e., for lower altitudes) the atmosphere is the dominant source of loss. This feature is unique for optical transmission from satellite to ground.
Noise model
We now consider photon transmission in the presence of background photons. We analyze the scenario in which a source generates an entangled photon pair and distributes the individual photons to two parties, Alice (A) and Bob (B). We allow the distributed photons to mix with spurious photons (noise) from an uncorrelated thermal source, assuming a low thermal background (which can be ensured via stringent filtering). We then determine, in the high loss and low noise regime, the fidelity of the distributed entangled photon pair.
First, consider a tensor product of thermal states for the horizontal and vertical polarization modes:
where \({\bar{n}}_{k}\) is the average number of photons in the thermal state for the polarization mode k. We assume this state comes from an incoherent source with no polarization preference (e.g., the sun), such that \({\bar{n}}_{{\rm{H}}}={\bar{n}}_{{\rm{V}}}=:\bar{n}/2\). Furthermore, we assume some (nonpolarization) filtering procedure, which reduces the number of background thermal photons, such that \(\bar{n}\ll 1\). We then rewrite the above state to first order in the small parameter \(\bar{n}\):
where \(\left\,\text{vac}\,\right\rangle =\left0\right\rangle \otimes \left0\right\rangle\), and
We thus define our approximate thermal background state as
which serves as a good approximation to a low thermal background. The transmission channel from the source to the ground is then approximately
where \({U}^{{\eta }_{\text{sg}}}\) is the beamsplitter unitary (see, for e.g., ref. ^{104}), and A_{1} and A_{2} refer to the horizontal and vertical polarization modes, respectively, of the dualrail quantum system being transmitted; similarly for E_{1} and E_{2}. Note that for \(\bar{n}=0\), the transformation given by eq. (18) is equal to the transformation in (7). For a source state \({\rho }_{AB}^{S}\), with A ≡ A_{1}A_{2} and B ≡ B_{1}B_{2}, the quantum state shared by Alice and Bob after transmission of the state \({\rho }_{AB}^{S}\) from the satellite to the ground stations is
Let us first assume that we have an ideal twophoton source, which generates one of the four twophoton polarizationentangled Bell states, i.e., a state of the form \({\rho }^{S}={\Phi }^{\pm }:=\left{\Phi }^{\pm }\right\rangle\!\left\langle {\Phi }^{\pm }\right\) or \({\rho }^{S}={\Psi }^{\pm }:=\left{\Psi }^{\pm }\right\rangle\!\left\langle {\Psi }^{\pm }\right\), where
After transmission, we assume postselection on coincident events, along with high loss and low noise (\({\eta }_{\,\text{sg}\,}^{(1)},{\eta }_{\,\text{sg}\,}^{(2)},\bar{n}\ll 1\)). The postselection allows one to discard any occurrence in which one site registers a photon and the other does not. Furthermore, under the highloss and lownoise assumptions, we can discard potential fourphoton and threephoton occurrences, as these occur with negligible probability compared to the twophoton events. We thus focus our full attention on the twophoton state corresponding to one photon received at Alice’s site and one photon received at Bob’s site. Mathematically, this (unnormalized) state is given by
where
is the projection onto the twophotoncoincidence subspace. With \({\rho }_{AB}^{S}={\Phi }_{AB}^{\pm }\), it is straightforward to show that
where
with analogous definitions for x_{2}, y_{2}, z_{2}. The fidelity of this quantum state conditioned on one photon received by Alice and one photon received by Bob is therefore
Assuming that \({\eta }_{\,\text{sg}\,}^{(1)}={\eta }_{\,\text{sg}\,}^{(2)}={\eta }_{\text{sg}}\) and \({\bar{n}}_{1}={\bar{n}}_{2}=\bar{n}\), so that x_{1} = x_{2}, y_{1} = y_{2}, and z_{1} = z_{2}, and under the highloss and lownoise assumption, for ρ^{S} = Φ^{+} this reduces to
The ratio \(\frac{{\eta }_{\text{sg}}}{\bar{n}}\) is just the local signaltonoise ratio (SNR). Thus, assuming a fidelity constraint F ≳ F^{⋆}, we obtain the following bound on the SNR needed at each site in order to maintain a fidelity of F^{⋆} during operation:
Here, we have assumed that the fidelity lies within some small range close to one (e.g., 0.95 ≤ F^{⋆} ≤ 1) and expanded to first order in 1 − F^{⋆}. As an example, consider F^{⋆} = 0.99. Then, we must have SNR ≳ 150 at each site. Given that η_{sg} ~ 10^{−3}, this implies a constraint on the number of background photons per detection window of \(\bar{n}\,\lesssim \,7\times 1{0}^{6}\).
Let us now consider an initially imperfect Bell state generated by a nonideal entangled photonpair source. Specifically, we consider the state
where f_{0} is the initial fidelity. Using the fact that
in the highloss lownoise regime, and in the symmetric case \({\eta }_{\,\text{sg}\,}^{(1)}={\eta }_{\,\text{sg}\,}^{(2)}={\eta }_{\text{sg}}\) and \({\bar{n}}_{1}={\bar{n}}_{2}=\bar{n}\), we obtain
Note that 1/4 ≤ F ≤ f_{0}. See Fig. 9 for a plot of this fidelity as a function of the signaltonoise ratio.
The background photon number \(\bar{n}\) can be expressed in terms of the photon flux/rate at a receiving site. Let \({\mathcal{R}}\) be the number of background photons/s detected at a receiving site and ΔT be the coincidence timewindow. Then, \(\bar{n}={\mathcal{R}}\Delta T\). Assuming background photons collected from, e.g., moonlight or sunlight, are the dominant source of noise, we have the following expression for the background photon rate^{109,110}:
where hc/λ is the photon energy at mean wavelength λ (h is Planck’s constant and c is the speed of light), Δλ is the filter bandwidth, Ω_{fov} is the field of view of a receiving telescope (in steradians, sr) with radius r, and H is the total spectral irradiance in units Wm^{−2} μm^{−1} sr^{−1}. In the case of daytime operating conditions, the total spectral irradiance includes direct solar irradiance as well as diffuse sky radiation, with the latter consisting mainly of solar light scattered by atmospheric constituents.
The spectral irradiance is generally a complicated function of atmospheric conditions, the sun/moon sky position relative to the telescope pointing angle, time of day and year, etc. Thus, for simplicity, in what follows we keep H as an open parameter but consider it to fall roughly within a typical range of H ∈ [10^{−5}, 25] (in units Wm^{−2} μm^{−1} sr^{−1}), associated with clearsky conditions, with the lower value corresponding to a moonless clear night and the upper value corresponding to clear daytime conditions, when the sun is in nearview of the optical receiver (see, for e.g., refs ^{109,110}).
Using the relation \(\bar{n}={\mathcal{R}}\Delta T\), with \({\mathcal{R}}\) given by eq. (34), in Fig. 10 we plot the fidelity in eq. (29) as a function of the spectral irradiance H for several orbital altitudes h and groundstation separation distances d. To make the plot, we consider the situation depicted in Fig. 8, in which the satellite passes over the zenith of two ground stations and is at the midpoint between them. Note that spectral irradiance values on the order of 1 Wm^{−2} μm^{−1} sr^{−1} (and above) correspond to clear daytime conditions^{61,109,110}. Thus, for our chosen filter parameters, we see that entanglement distribution across, e.g., a groundstation separation distance of more than 2000 km, only seems feasible during the night (H ≲ 10^{−2} Wm^{−2} μm^{−1} sr^{−1}). We note, however, that these results are quite sensitive to the filtering parameters, owing to the steep slope of the fidelity in its midregion.
An interesting extension of these results would be to consider a dynamic model, in which one parameterizes the satellitetoground transmittance and background photon rate in time. We do such a parameterization for the transmittance in this work; however, parameterizing the background photon rate requires realtime modeling of, e.g., the sun position relative to the satellite orbit, modeling diffuse sky radiation, etc. Work along these lines has already been done for satellitetoground quantum key distribution between a satellite and a lone ground station (see, for e.g., ref. ^{110}). A full, dynamical analysis of the fidelity for a noisy, globalscale satellitetoground entanglement distribution protocol—utilizing, e.g., the asymmetric noise model derived above—is an interesting direction for future research.
Simulation details
In order to perform our simulations and to obtain optimal satellite configurations, we used satellite constellations corresponding to the 42 pairs (N_{R}, N_{S}) shown in Table 4, where N_{R} is the number of rings in the constellation and N_{S} is the number of satellites per ring (see Fig. 2).
Let \(\{{{\boldsymbol{r}}}_{i}(t)\in {{\mathbb{R}}}^{3}:1\le i\le {N}_{R}{N}_{S}\}\) be the positions of the satellites relative to the center of the earth at time t. If the satellites are at an altitude of h, then ∣∣r_{i}(t)∣∣_{2} = R_{E} + h for all t, where R_{E} is the radius of the earth and ∣∣ ⋅ ∣∣_{2} denotes the Euclidean norm. Let \({{\boldsymbol{g}}}_{j}(t)\in {{\mathbb{R}}}^{3}\) be the position of the jth ground station relative to the center of the earth at time t.
The distance between the ith satellite and the jth ground station at time t is given by L_{i,j}(t) = ∣∣r_{i}(t) − g_{j}(t)∣∣_{2}. Then, the satellitetoground transmittance between the ith satellite, at altitude h, and the jth ground station is given at time t by
with η_{fs}(L_{i,j}(t)) given by eq. (9) and η_{atm}(L_{i,j}(t), h) given by eq. (11). The total transmittance \({\eta }_{\,\text{tot}\,}^{(i,{j}_{1},{j}_{2})}(t;h)\) at time t corresponding to the ith satellite, at altitude h, transmitting one of a pair of entangled photons to ground station j_{1} and the other photon to ground station j_{2}, is
In order for a satellite to be considered within range of a given ground station pair, we require two conditions to be satisfied: (1) the satellite is visible to both ground stations; and (2) the total loss (given via eq. (36)) is less than 90 dB. If at least one of these two conditions is not satisfied, then the satellite is considered to be not within range of the ground station pair. Any interval of time in which at least one of the conditions is not satisfied is called a “time gap”. We define the function
which tells us whether the ground station pair (j_{1}, j_{2}) is within range of the ith satellite, at altitude h, at time t.
When performing our simulations, we find that at some times a satellite is within range of multiple ground station pairs. In other words, it can happen that a particular ground station pair is within range of multiple satellites at the same time. We anticipate that, in the near future, satellites will only have one entanglement source on board, so we impose the requirement that at any given time a satellite can distribute entanglement to only one groundstation pair. This requirement makes it necessary to uniquely assign a satellite to a groundstation pair at all times during the simulation. We assign a satellite to the groundstation pair that has the lowest loss among all groundstation pairs within range of that satellite. This type of assignment strategy means that, depending on the total number of satellites, there are times at which groundstation pairs do not receive any entangled photon pairs even though they are within range of a satellite (perhaps several), simply because the loss would be too high. More sophisticated timesharing assignment strategies are possible, in which higher loss assignments are taken at certain times for the purpose of distributing entanglement to as many different groundstation pairs as possible. We do not consider such an assignment strategy here (see ref. ^{98} for work in this direction), except for when there is a groundstation pair that has only one satellite in view, but that satellite is in range of several other ground stations. In this case, we assign that satellite to the “lone” groundstation pair even if the loss is higher than another possible assignment of that satellite. We let s_{t}(j_{1}, j_{2}) ∈ {1, 2, …, N_{R}N_{S}} denote the satellite assigned to the ground station pair (j_{1}, j_{2}) at time t. For brevity, we write s_{t} ≡ s_{t}(j_{1}, j_{2}) if the two ground stations being considered is clear from the context. If no satellite assignment exists for the pair (j_{1}, j_{2}) at time t, then we set s_{t}(j_{1}, j_{2}) = 0.
For two ground stations j_{1} and j_{2} separated by a distance d, the average loss over a time T for the satellite configuration given by (N_{R}, N_{S}, h) is
If no satellite assignment exists for the pair (j_{1}, j_{2}) at time t, then we set \({\eta }_{\,\text{tot}\,}^{({s}_{t},{j}_{1},{j}_{2})}(t;h)=0\). We write \({\overline{\eta }}_{\text{dB}}({N}_{R},{N}_{S},h,d)\) when the ground station pair (j_{1}, j_{2}) and the simulation time T are understood from the context.
We also define the average rate over time T for two ground stations j_{1} and j_{2} separated by a distance d for the satellite configuration given by (N_{R}, N_{S}, h) as follows:
where
is the average number of entangled pairs received by the ground stations j_{1} and j_{2} at time t, with \({R}_{\,\text{source}\,}^{{s}_{t}}\) being the source rate of the satellite s_{t}. For the simulations, we estimate \({\overline{P}}^{({s}_{t},{j}_{1},{j}_{2})}(t)\) in a single shot by taking a sample from the binomial distribution Bin(n, p) with \(n={R}_{\,\text{source}\,}^{{s}_{t}}\) trials and success probability \(p={\eta }_{\,\text{tot}\,}^{({s}_{t},{j}_{1},{j}_{2})}(t;h)\) per trial. Throughout this work, we assume that \({R}_{\,\text{source}\,}^{{s}_{t}}=1{0}^{9}\) ebits/s for all satellites^{100}. We write \(\overline{R}({N}_{{\rm{R}}},{N}_{{\rm{S}}},h,d)\) when the ground station pair (j_{1}, j_{2}) and the simulation time T are understood from the context.
Figures of merit
Our figures of merit are based on entanglement distribution to a particular pair (j_{1}, j_{2}) of ground stations. The goal is to determine the optimal satellite configuration such that there are no time gaps in a period of time T for the pair (j_{1}, j_{2}). The first three figures of merit separately optimize the total number of satellites, the average loss over the time T, and the average rate over the time T:
In all three cases, we optimize over the pairs (N_{R}, N_{S}) shown in Table 4. We write N_{opt}(h, d), \({\overline{\eta }}_{\,\text{dB}}^{\text{opt}\,}(h,d)\), and \({\overline{R}}^{\text{opt}}(h,d)\) when both the ground station pair (j_{1}, j_{2}) and the simulation time T are understood from the context.
In order to obtain a satellite configuration that balances both the total number of satellites and the average rate, we define the following figure of merit:
which has an intuitive interpretation as the average number of ebits/s per satellite in the network. From this, we define
which is simply the figure of merit \({c}_{T}^{({j}_{1},{j}_{2})}({N}_{{\rm{R}}},{N}_{{\rm{S}}},h,d)\) optimized over the pairs (N_{R}, N_{S}) in Table 4. We write C(h, d) when both the ground station pair (j_{1}, j_{2}) and the simulation time T are understood from the context.
All of the quantities defined above can be defined in an analogous fashion for multiple ground station pairs (instead of just one ground station pair). For example, suppose that we have a set \({\mathcal{V}}\) of ground stations and a set \({\mathcal{E}}=\{(v,v^{\prime} ,{d}_{v,v^{\prime} }):v,v^{\prime} \in {\mathcal{V}},{d}_{v,v^{\prime} }\in {\mathbb{R}}\}\) telling us how the ground stations are connected pairwise to each other, with \({d}_{v,v^{\prime} }\) being the physical distance between ground stations v and \(v^{\prime}\). (Note that \(({\mathcal{V}},{\mathcal{E}})\) corresponds to a weighted graph, with \({\mathcal{V}}\) being the nodes of the graph and \({\mathcal{E}}\) being the weighed edges.) Then,
Letting
we can define the following quantities:
Quantum repeater rates
In order to compare the satellitebased entanglementdistribution rates obtained in this work with rates that can be achieved using groundbased quantum repeater schemes, we consider a chain of quantum repeaters of total length d in which there are M elementary links and each repeater “halfnode” has N_{mem} quantum memories; see Fig. 11 for an example. This results in N_{mem} parallel quantum repeater chains between the end nodes. If we allow entanglement distribution to occur independently for each of the parallel chains, and we assume that the quantum memories have infinite coherence time, then the expected number of time steps until one endtoend pair is obtained, i.e., the expected waiting time, has been shown in [ref. ^{111}, Appendix B] to be
Now, the duration of each time step, i.e., the repetition rate, is limited by the classical communication time between neighboring nodes for heralding of the signals. (This is the bestcase scenario. We do not consider other factors that affect the repetition rate, such as the memory readwrite time.) The classical communication time is given by \(\frac{2(d/M)}{c}\), resulting in a repetition rate of \(\frac{c}{2(d/M)}\) for each of the N_{mem} parallel links of an elementary link. The total repetition rate is therefore \(\frac{c{N}_{\text{mem}}}{2(d/M)}\). The formula in eq. (5) for the rate then follows.
A higher rate than the one in eq. (5) can be achieved by allowing for spatial multiplexing, i.e., by allowing cross connections between the different parallel chains^{112}. An analytic expression for the waiting time in this scenario, in the case of M = 2 elementary links, has been derived in ref. ^{113}. A general formula for the waiting time for an arbitrary number M of elementary links appears to be unknown.
Data availability
The datasets generated during the current study are available in the arXiv repository, https://arxiv.org/abs/1912.06678.
Code availability
All code used to obtain the data for the current study are available in the arXiv repository, https://arxiv.org/abs/1912.06678.
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Acknowledgements
We dedicate this work to the memory of our mentor and guide Jonathan P. Dowling. Jon’s unflinching support, his enthusiasm for research, and his inspiring vision for the future of quantum technologies and the quantum internet, made this work possible. S.K. acknowledges support from the National Science Foundation and the Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship. A.J.B. acknowledges support from the National Science Foundation. R.A.D., M.P.B., and J.P.D. acknowledge support from the Army Research Office, Air Force Office of Scientific Research and the National Science Foundation.
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S.K., A.J.B., and J.P.D. developed the main ideas behind the project. S.K., R.A.D., and M.P.B. developed and ran the simulations. S.K. and A.J.B. wrote the manuscript. S.K. and A.J.B. contributed equally to this work.
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Khatri, S., Brady, A.J., Desporte, R.A. et al. Spooky action at a global distance: analysis of spacebased entanglement distribution for the quantum internet. npj Quantum Inf 7, 4 (2021). https://doi.org/10.1038/s41534020003275
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DOI: https://doi.org/10.1038/s41534020003275
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