Gaussian Entanglement Distribution via Satellite
aa r X i v : . [ qu a n t - ph ] F e b Gaussian Entanglement Distribution via Satellite
Nedasadat Hosseinidehaj, Robert Malaney
School of Electrical Engineering & Telecommunications,The University of New South Wales,Sydney, NSW 2052, [email protected], [email protected]
Abstract —In this work we analyze three quantum communi-cation schemes for the generation of Gaussian entanglement be-tween two ground stations. Communication occurs via a satelliteover two independent atmospheric fading channels dominated byturbulence-induced beam wander. In our first scheme the engi-neering complexity remains largely on the ground transceivers,with the satellite acting simply as a reflector. Although thechannel state information of the two atmospheric channelsremains unknown in this scheme, the Gaussian entanglementgeneration between the ground stations can still be determined.On the ground, distillation and Gaussification procedures can beapplied, leading to a refined Gaussian entanglement generationrate between the ground stations. We compare the rates producedby this first scheme with two competing schemes in whichquantum complexity is added to the satellite, thereby illustratingthe trade-off between space-based engineering complexity andthe rate of ground-station entanglement generation.
I. I
NTRODUCTION
Although current quantum communication systems are lim-ited to relatively small scales, it is widely anticipated that next-generation quantum networks will in some capacity invokethe concept of free-space optical (FSO) communications (forreview see [1]) in order to extend the communication range.Coupled to this, is the growing belief that space-borne quan-tum transceivers will soon make full blown global quantumcommunications an engineering reality [2]–[12]. While itremains to be seen whether FSO quantum communicationswill be dominated by discrete single photon (qubit) technology,multi-photon continuous variable (CV) technology, or evensome hybrid of both technologies [13], it is important to fullyunderstand the capabilities of both types of technologies in thefree-space channel.Previous work in the satellite quantum communicationscenario has largely focussed on qubit technologies. In thiswork we will focus on the CV scenario, with the aim ofassessing some of the different CV quantum-communicationarchitectures that could be deployed through atmosphericfading channels. Our specific interest will be on the distri-bution of Gaussian quantum entanglement over the differentarchitectures. Gaussian entanglement between quantum stateshas been widely recognized as a basic resource for quantuminformation processing and quantum communications (forreview see [14]–[17]). Here, we will analyze three differentspace-based schemes that will allow for Gaussian entangledstates to be shared at separate ground stations.In our first scheme, referred to as direct transmissionentanglement , no quantum technology is deployed at the satellite - the satellite is utilized simply in a reflector mode[18]. The main motivation for this is that quantum engineer-ing is a highly sophisticated business, demanding leading-edge technology. Having such technology based in hard-to-reach satellite systems could potentially make global quantumcommunication systems less reliable (due to the rarity ofmaintenance), and costly to update as new quantum technologymatures. Relatively speaking, one could consider a reflectionat the satellite as a low space-based-complexity system. Inthis system a two-mode entangled squeezed state is generatedat ground station A, with one component of the beam heldat A and the other component transmitted to ground stationB via a low-earth-orbit (LEO) reflecting relay satellite. Asa proof of concept on the reflecting paradigm, we note therecent experimental tests of [11], [12] in which photons werereflected (and subsequently detected) off a LEO satellite.The other schemes we study can be considered as highspace-based-complexity in that they involve the deploymentof quantum technology at the satellite. Our second scheme,referred to as satellite-based entanglement , invokes entangle-ment generation in the satellite itself with subsequent transferto the ground stations directly. Our third scheme, referred to as entanglement swapping , utilizes on-board Gaussian entangle-ment swapping between arriving beams of photons entangledwith (and emitted by) separate ground stations. All three ofour schemes are illustrated in Fig. 1.In all schemes, the transmitted beam will encounter fluctu-ations (fading) caused by its traversal (twice) through the at-mosphere. Among the many unwanted disturbances in realisticatmospheric channels, we will concentrate here on transmis-sion fluctuations caused by beam wander, an effect anticipatedto dominate the noise contributions in many scenarios [1], [19],[20].It is the first aim of this work to provide a quantitativeassessment, in terms of resulting Gaussian entanglement, ofthe low space-based-complexity scheme in relation to the twohigh space-based-complexity schemes. A second aim of ourwork is to explore post-processing strategies that can occurat the receiving ground stations. Due to the fluctuating fadingchannels traversed by the beams, a non-Gaussian mixed stateis produced. At the receiver a post-selection strategy canbe deployed in order to distill (concentrate) the Gaussianentanglement between the two ground stations. Such post-selection strategies could be based on quantum measurementtechniques, or on classical measurements of the channelransmittance. However, such classical measurements of thechannel transmittance will require additional complexity inthe transmission/detection strategy. We will be specificallyinterested in investigating the gain in Gaussian entanglementobtained by the inclusion of this additional classical complex-ity.Note that, although largely motivated by the use of FSOin the scenario of satellite quantum communications to/fromterrestrial ground stations [2]–[12], our results are applicableto a range of FSO links such as high altitude platform (HAP)-to-satellite quantum links [21] and aircraft-to-ground quantumlinks [22]. The range of atmospheric channels we study willcover all of these different quantum communication scenarios.The structure of the remainder of this paper is as follows.In Section II, the basic concepts of Gaussian quantum statesare introduced, and our three transmission schemes overterrestrial-satellite fading channels are analyzed in terms ofoutput covariance matrices. In Section III, the performances ofthe three schemes are compared in terms of output Gaussianentanglement. In Section IV, we discuss the impact of thepost-selection strategy utilized at the ground stations. Finally,concluding remarks and future research directions are providedin Section V.II. Q
UANTUM COMMUNICATION OVER FADING CHANNELS
In the following we discuss the three quantum communica-tion schemes of Fig. 1, but first introduce some features thatwill be needed for their description.For a single bosonic mode with annihilation and creationoperators ˆ a, ˆ a † , the quadrature operators ˆ q, ˆ p are defined by ˆ q = ˆ a + ˆ a † , ˆ p = i (ˆ a † − ˆ a ) which satisfy the com-mutation relation [ˆ q, ˆ p ] = 2 i (here ~ = 2 ). The vectorof quadrature operators for a quantum state with n modescan be defined as ˆ R ,...,n = (ˆ q , ˆ p , . . . , ˆ q n , ˆ p n ) . Similarly, R ,...,n = ( q , p , . . . , q n , p n ) is defined for the real variables q, p - the eigenvalues of the quadrature operators.We will discuss both non-Gaussian and Gaussian states,the latter being states whose Wigner function is a Gaussiandistribution of the quadrature variables. Gaussian states arecompletely characterized by the first moment of the quadratureoperators D ˆ R ,...,n E and a covariance matrix (CM) M , i.e. amatrix of the second moments of the quadrature operators as M ij = 12 D ˆ R i ˆ R j + ˆ R j ˆ R i E − D ˆ R i E D ˆ R j E . (1)The CM of an n -mode quantum state is a n × n real andsymmetric matrix which must satisfy the uncertainty principle, viz. , M + i Ω ≥ , where Ω := n ⊕ k =1 ω = ω . . . ω , ω := (cid:18) − (cid:19) . (2)By local unitary operators, the first moment of every two-mode Gaussian state can be set to zero and the CM can be Fig. 1. (Color online) Entanglement generation schemes. (a) is the directtransmission scheme where reflection is used at the satellite. (b) is a schemewhere entangled photon generation takes place directly in the satellite andthen distributed in separate downlinks to the ground stations. (c) is a schemein which the Gaussian entangled states are transmitted independently throughtwo fading channels from two ground stations to the satellite. They are thenswapped via a Bell-measurement at the satellite, resulting in creation of a newentanglement between the ground stations. transformed into the following standard form M s = (cid:18) A CC T B (cid:19) , (3)where A = aI , B = bI , C = diag ( c + , c − ) , a, b, c + , c − ∈ R , and I is a × identity matrix. Considering the standardform, the symplectic spectrum of a partially transposed CM isgiven by ν ± = s ∆ ± √ ∆ − M s , (4)where ∆ = det A + det B − C . A quantitative measureof Gaussian entanglement can be derived in terms of the log-arithmic negativity E LN ( M s ) = max [0 , − log ( ν − )] , where ν − , as given above, is the smallest symplectic eigenvalue ofthe partially transposed CM [14].In free-space channels the transmittance fluctuates due toatmospheric effects. Such fading channels can be character-ized by a distribution of transmission coefficients η with aprobability density distribution p ( η ) . The main contributors totransmission losses in free-space quantum communication areatmospheric turbulence, diffraction, scattering, and absorption.iffraction, scattering, and absorption are all wavelength de-pendent, and to a large extent can be mitigated by an appropri-ate choice of communication wavelength [1], [8]. Atmosphericturbulence arises due to random fluctuations in the refractiveindex caused by stochastic temperature variations. This effectleads to beam wandering as well as beam broadening [1],[8]. In this paper, we take the usual assumption that trans-mittance fading is largely dominated by beam wander [1],[19], [20], [23]. Beam wandering causes the beam center tobe randomly displaced from the aperture center in the receiverplane. Assuming that the beam-center position is normallydistributed with variance σ b around a point at a distanceof d from the aperture center, the beam-deflection distancefluctuates according to the Rice distribution, which results inthe probability density distribution p ( η ) being given by thelog-negative generalized Rice distribution [19]. Unlike ear-lier models, e.g. the log-normal distribution, the log-negativegeneralized Rice distribution more accurately describes theoperationally-important transmission distribution tail [19]. Inthe particular case, d = 0 , when the beam spatially fluctuatesaround the center of the receiver’s aperture such fading can bedescribed by the log-negative Weibull distribution [19] [20], p ( η ) = 2 L σ b λη (cid:18) η η (cid:19) ( λ ) − exp − L σ b (cid:18) η η (cid:19) ( λ ) ! (5)for η ∈ [0 , η ] , with p ( η ) = 0 otherwise. Here, σ b is thebeam wander variance, λ is the shape parameter, L is thescale parameter, and η is the maximum transmission value.The latter three parameters are given by λ = 8 h exp( − h ) I [4 h ]1 − exp( − h ) I [4 h ] h ln (cid:16) η − exp( − h ) I [4 h ] (cid:17)i − L = β h ln (cid:16) η − exp( − h ) I [4 h ] (cid:17)i − (1/ λ ) η = 1 − exp ( − h ) , (6)where I [ . ] and I [ . ] are the modified Bessel functions, andwhere h = ( β / W ) , with β being the aperture radius and W the beam-spot radius.Note that, assuming fixed values for W and β , the trans-mittance mean value h η i always decreases with increasing σ b .Also note, that the uplink (ground-to-satellite) first traversesthe atmosphere followed by a larger-scale free-space traversal,whereas the downlink (satellite-to-ground) does the opposite.For the case of fixed fading parameters W and β , this meansthat in general the beam wander variance σ b for the uplinkis significantly larger than the downlink [1]. Finally, note thatthe rate of atmospheric fluctuations we consider are of orderkHz, which is at least a thousand times slower than typicaltransmission/detection rates [1], [20]. This means that channelmeasurements can be obtained at the cost of additional (clas-sical) transmission/receiver complexity. We will assume thatsuch measurements are in place at the ground receivers (only)in our first two schemes. As we shall see, in order to optimizeour third scheme (entanglement swapping) we will need the additional complexity of classical channel measurements at thesatellite. Channel measurements could be made via severalschemes - e.g., via coherent (classical) light pulses that areintertwined with the quantum information or via the traversalthrough the atmosphere of a local oscillator [24]. We willexplore later the cost (in terms of Gaussian entanglement) ofremoving this additional classical complexity. A. Direct Transmission Entanglement
The direct transmission scheme illustrated in Fig. 1(a) isnow analyzed and the CM of the output state calculated.Let us consider the ground station A initially possessing aGaussian two-mode entangled squeezed state. We assume onemode remains at the ground station while the other modeis transmitted over the fading uplink to the satellite, thenperfectly reflected in the satellite and sent through the fadingdownlink toward the ground station B. As a result, dependingon the initial degree of entanglement, there would exist anentangled state between the two ground stations. Note thatwe assume the separate uplink and downlink channels areindependent and non-identical.Now let us assume that the initial entangled states are two-mode squeezed vacuum states with squeezing r , then theirinitial CM can be written M i = (cid:18) v I √ v − Z √ v − Z v I (cid:19) , (7)where v = cosh (2 r ) , r ∈ [0 , ∞ ) , and Z = diag (1 , − .Note, the initial entangled states can be coherently displacedwithout changing the above CM.After transmission of the optical mode through the uplinkand then reflection through the downlink with probability den-sity distributions p AS ( η ) and p SB ( η ) , respectively, the CM ofthe two-mode state at the ground stations for two realizationof the transmission factors η (uplink) and η ′ (downlink) canbe constructed. It is straightforward to show that assuming noadditional noise sources this CM is given by M η η ′ = (cid:18) v I √ η η ′ √ v − Z √ η η ′ √ v − Z (1 + η η ′ ( v − I (cid:19) . (8)Therefore, the elements of the final CM of the resulting mixedstate are calculated by averaging the elements of M η η ′ overall possible transmission factors of the two fading channelsgiving M = (cid:18) v I c Zc Z b I (cid:19) , where b = R η R η ′ p AS ( η ) p SB ( η ′ ) (1 + η η ′ ( v − dη dη ′ c = R η R η ′ p AS ( η ) p SB ( η ′ ) √ η η ′ √ v − dη dη ′ . (9)Note that since η and η ′ are random variables, the final stateensemble is a non-Gaussian mixture of the Gaussian statesobtained for each realization of η and η ′ . Note also, in thisscheme it is only the combined channel transmissivity ηη ′ thatis measured at the ground station B. . Satellite-based Entanglement In this section the quantum communication scheme inFig. 1(b) is analyzed and CM of the output state between theterrestrial stations is computed. Here a two-mode entangledstate is directly generated within the satellite, with bothmodes then sent over separate fading downlinks to the groundstations. Again we assume that the initial entangled state is atwo-mode squeezed state described by CM M i of (7). Afterdistribution of the modes through the downlink to station Aand downlink to station B characterized by probability densitydistributions p SA ( η ) and p SB ( η ) respectively, the CM of thetwo-mode Gaussian state between the ground stations for eachrealization of η and η ′ is given by M ′ η η ′ = (cid:18) (1 + η ( v − I √ η η ′ √ v − Z √ η η ′ √ v − Z (1 + η ′ ( v − I (cid:19) . (10)Here, the two fading downlinks are independent and non-identical. The elements of the final CM are simply the av-erage of the elements of M ′ η η ′ over all possible fluctuatingtransmission factors of the two fading channels giving M ′ = (cid:18) a ′ I c ′ Zc ′ Z b ′ I (cid:19) , where a ′ = R η p SA ( η ) (1 + η ( v − dηb ′ = R η ′ p SB ( η ′ ) (1 + η ′ ( v − dη ′ c ′ = R η R η ′ p SA ( η ) p SB ( η ′ ) √ η η ′ √ v − dη dη ′ . (11)Again, the final state ensemble is a non-Gaussian mixture. Inthis scheme, the individual channel transmissivities η and η ′ are obtainable via measurements at the ground stations. C. Entanglement Swapping
The protocol of entanglement swapping as shown inFig. 1(c) is now analyzed over fading channels, and the CMof the optimal output state computed. Entanglement swapping[25] is a standard protocol to establish entanglement betweendistant quantum systems that have never interacted [26]–[28]. It is the central mechanism of quantum repeaters [29],enabling distribution of entanglement over large distances.Previously, the implementations of a swapping-based protocolin the context of CV technology has been studied mostlythrough fixed attenuation channels e.g. [27], [30]. In [30],optimal entanglement swapping with Gaussian states over alossy optical fiber with fixed attenuation has been analyzed,and we build on this analysis here in the context of twoindependent fading channels.In the entanglement swapping scheme, each ground stationinitially possesses a Gaussian two-mode entangled state. Onemode of each entangled state is kept by the ground stationand the second mode of each state is transmitted to thesatellite through a fading uplink. Here, the two fading uplinksare independent and non-identical with probability densitydistributions p AS ( η ) and p BS ( η ) for the station-A uplink andthe station-B uplink, respectively. Let us consider the entangled states initially at the groundstations to be a pair of two-mode squeezed vacuum states withthe same level of squeezing r , with modes 1 and 2 owned byground station A, and modes 3 and 4 owned by ground stationB. These pairs of entangled states will possess CMs describedby (7), that is M , = M , = (cid:18) v I √ v − Z √ v − Z v I (cid:19) . (12)After transmission of mode 2 through the uplink from stationA and transmission of mode 3 through the uplink from stationB, prior to any interaction at the satellite the transmitted statesfor each realization of η and η ′ are described by two stateswith CMs, M , η = (cid:18) v I √ η √ v − Z √ η √ v − Z (1 + η ( v − I (cid:19) M , η ′ = (cid:18) (1 + η ′ ( v − I √ η ′ √ v − Z √ η ′ √ v − Z v I (cid:19) . (13)When the two transmitted modes are received, they areswapped via a Bell measurement at the satellite. First, trans-mitted modes 2 and 3 are mixed through a balanced beam-splitter, yielding output modes u and v . Then, the newquadratures ˆ q u and ˆ p v are measured by two homodynedetectors, providing the outcomes q ′ u and p ′ v . In order tocomplete the swapping process, the satellite broadcasts theBell measurement results so that the two ground stations canproperly displace their modes according to the measurementoutcomes q ′ u and p ′ v . In practice, the displacements can beweighted by gain factors to improve the quality of the swappedentanglement. It can be shown (see Appendix A) that if thegains applied to the displacements of modes 1 and 4 are givenby g = √ η √ v − η + η ′ )( v − , g = √ η ′ √ v − η + η ′ )( v − , (14)then the CM of the conditional state of modes 1 and 4(averaged over all possible Bell measurements) at the groundstations is given by M ′′ ηη ′ = (cid:18) ( v − ηm ) I √ η η ′ m Z √ η η ′ m Z ( v − η ′ m ) I (cid:19) , where m = ( v − η + η ′ )( v − . (15)The final (ensemble averaged) swapped state shared by theground stations is the mixture of the swapped states after eachrealization of η and η ′ . The total CM of the resulting mixedswapped state is obtained by averaging elements of M ′′ ηη ′ in(15) over all possible transmission factors of the two fadinghannels, giving M ′′ = (cid:18) a ′′ I c ′′ Zc ′′ Z b ′′ I (cid:19) , where a ′′ = R η R η ′ p AS ( η ) p BS ( η ′ ) ( v − ηm ) dη dη ′ b ′′ = R η R η ′ p AS ( η ) p BS ( η ′ ) ( v − η ′ m ) dη dη ′ c ′′ = R η R η ′ p AS ( η ) p BS ( η ′ ) √ η η ′ m dη dη ′ . (16)Note, in setting the gains as described above we must assumethat the satellite itself has measured each of the transmittivitiesseparately. Again, the final state ensemble at the groundstations is a non-Gaussian mixture.III. C OMPARISON OF T HE S CHEMES
From the final CM of each scheme the logarithmic neg-ativity E LN is adopted as a measure of the entanglementbetween the two ground stations. As noted above, the resultingensemble-averaged state shared by the ground stations ineach scheme is a non-Gaussian state, and as such cannotbe described completely by its first and second moments.Therefore, the entanglement measure we compute based onthe CM of the resulting mixed state will represent only theGaussian entanglement between the terrestrial stations.We simulate the performance of each of our three schemesin terms of the Gaussian entanglement derived from theappropriate CM. For all simulations shown calculations of E LN will adopt base 2 in the logarithmic term, and thefollowing assumptions are adopted: (i) For each simulation,all initial states are two-mode squeezed states with the sameinitial squeezing r . (ii) Beam wander, as modeled by thelog-negative Weibull distribution, is used to characterize thetwo fading channels for each scheme, with β = 1 . (iii) Thetwo separate fading channels are assumed to be independent,but not necessarily identical. (iv) The beam wander stan-dard deviations σ b AS , σ b SA , σ b BS , σ b SB for the fourpossible link traversals satisfy σ b SA = k σ b AS , σ b BS = k σ b AS , σ b SB = k k σ b AS , σ b AS = σ b , where ≤ k ≤ and k ≥ . This allows us to parametrize thebeam wander dependence on geometries and communicationdirection in terms of the three independent parameters, σ b , k and k . For clarity the apertures (and beam-spot radii) will beassumed the same at satellite and ground station - we allow thebeam wander alone to model different losses at these devices(when in receive mode).Figs. 2-4 show the final Gaussian entanglement of the threecommunication schemes as a function of beam wander stan-dard deviation σ b (normalized to β ) in the uplink from stationA, and the squeezing level r of the initial entangled states. Theparameters shown in Figs. 2-4 correspond to channels withlosses of roughly 4dB through 8dB (at σ b = 1 ) in the uplink.They thus represent low-loss channels such as HAP-LEOsatellite channels where the effects of the turbulent atmosphereare relatively small [21]. Such channels are also typical of r Direct Transmission σ b E LN r Satellite−based Entanglement Generation σ b E LN r Entanglement Swapping σ b E LN Fig. 2. (Color online) Logarithmic negativity E LN of the two-mode stateat the ground stations resulting from the direct transmission (top figure),Satellite-based entanglement generation (middle figure) and the entanglementswapping (bottom figure). The results are shown with respect to the beamwander standard deviation σ b in the uplink, and the squeezing level r . Here, β/W = 1 , k = 0 . , k = 0 . . These parameters for σ b = 0 . lead toa mean loss of 3dB for the uplink from station A. short-length ( ∼ km) atmospheric FSO links as expected atground level [20].Considering each scheme, it is evident that an increase in σ b reduces entanglement while increasing the input squeezingis able to partly compensate its negative effect. For a largesqueezing level we see the output logarithmic negativitydegrades with increasing squeezing since strongly squeezedstates are more sensitive to fading. However, the main pointwe wish to draw from these results is that although thesatellite-based entanglement generation scheme is always best,its advantage over the direct transmission scheme is rathersmall in the low-loss channels considered in Figs. 2-4. Wedo note that satellite-based entanglement generation holds achannel advantage in that it does not utilize any uplink. Assuch, increases in the quality of the downlink channel relativeto an uplink channel, beyond that examined here, will leadto a corresponding increase in the entanglement advantage forthe satellite-based entanglement scheme relative to the otherschemes.We also note that the direct transmission scheme can alwaysbe configured to deliver a better entanglement outcome thanthe entanglement swapping scheme. For low values of inputsqueezing, the swapped Gaussian entanglement between thetwo terrestrial stations is smaller than the final Gaussian entan-glement of the direct transmission. However, for some channel r Direct Transmission σ b E LN r Satellite−based Entanglement Generation σ b E LN r Entanglement Swapping σ b E LN Fig. 3. (Color online) Same as Fig. 2 except here β/W = 0 . . Theseparameters for σ b = 0 . lead to a mean loss of 5.4dB for the uplink fromstation A. r Direct Transmission σ b E LN r Satellite−based Entanglement Generation σ b E LN r Entanglement Swapping σ b E LN Fig. 4. (Color online) Same as Fig. 2 except here, β/W = 0 . . Theseparameters for σ b = 0 . lead to a mean loss of 6.7dB for the uplink fromstation A. parameters ( e.g. see Fig. 2) there are values in the high- squeezing regime for which the swapping-based scheme canlead to more entanglement than direct transmission. However,for each level of σ b , if the optimal amount of squeezingfor initial entangled states is used, direct transmission canalways be configured to distribute better entanglement than theswapping scheme. Indeed, we can show that for any fadingchannel, entanglement swapping can never lead to any im-proved entanglement generation relative to direct transmission(see Appendix B). These observations on CV entanglementswapping in relation to direct transmission have been notedbefore for the case of fixed attenuation (non-fading) channels[30].Beyond noise terms introduced via atmospheric turbulence,additional losses and noise can occur at the detector includingbackground light, dark counts, and electronic noise. Losses inthe receiver components are typically well below atmosphericlosses, and can be accounted for by modification to the esti-mated transmission factors. Background light can be controlledby sufficient shielding and filtering and is not a serious issuefor homodyne detection [31], and dark counts are now at thelevel of less than 20s − in modern detectors [8]. As such,electronic noise will dominate receiver noise. Note, any othernoise (e.g., that introduced in the channel by an eavesdropper)can be included in this receiver noise so as to form the total excess noise.It is possible to accommodate the introduction of electronicnoise by the appropriate addition of extra variance terms χ ineach of our covariance matrices. In our matrix M η η ′ such anoise term appears on the lower diagonal term; in M ′ η η ′ onboth diagonal terms, in M , η in the lower diagonal term, andin M , η ′ on the upper diagonal term (all additional noise termsmultiplied by I ).To quantify the effect electronic noise can have (all resultsshown thus far have assumed zero excess noise) we havecarried out a series of additional simulations where appro-priate noise terms have been added for each scheme. Wehave assumed the same amount of noise χ in all relevantreceivers, and simulated levels of χ in the range 0.01-0.05,a range consistent with current detectors [32]. For such arange we find that the Gaussian entanglement for the directtransmission scheme is reduced by approximately 2-9 percent,with the entanglement reduction for the other two schemesboth approximately 4-16 percent. Such results are consistentwith the fact the simulations of the direct transmission schemeincludes noise at only one receiver.Ground-to-satellite communications are anticipated to un-dergo much stronger losses than those illustrated in Figs. 2-4,with single FSO uplink channels anticipated to have losses oforder 25dB and beyond [1], [2]. Under such losses, distributionof entanglement between the ground stations will be a fruitlessendeavor without the intervention of a highly-selective post-selection strategy. IV. P OST - SELECTION
Here, processing strategies which enhance the Gaussianentanglement of the non-Gaussian mixed state between theround stations are investigated. The post-selection strategieswhich occur at the receiving ground station can be based onclassical measurements of the channel transmittance, or onquantum measurements. We are interested in quantifying theperformance of these two different measurement strategies.Note that in both post-selection strategies Gaussification oc-curs in the sense that the conditioned states are more Gaussianin nature due to the enhanced concentration of low-loss statesin the final ensemble. For clarity we will study post-selectionstrategies in the context of our lowest complexity scheme,namely, the direct transmission scheme.
A. Classical Post-selection
Although fading noise diminishes Gaussian entanglement,it also provides the possibility to recover it. Post-selection oflarge transmission windows, as introduced in [20] for the caseof a single fading channel, offers a possibility for improvingthe Gaussian entanglement in cases where it was stronglydiminished by the wider fading. In this scenario, a subset ofthe channel transmittance distribution, with high transmittivity,is selected to contribute to the resulting post-selected state.For this form of post-selection to operate in our directtransmission scheme, coherent (classical) light pulses arereflected of the satellite in order to measure the transmittanceof the combined channel ζ = η η ′ at the receiving groundstation, where again η and η ′ are random variables describingtransmission factors of the uplink and downlink, respectively.The received quantum state is kept or discarded, conditionedon the classical measurement outcome being larger or smallerthan the post-selection threshold ζ th . Providing we have a formfor the probability density distribution p ( ζ ) , the resulting post-selected CM can be calculated as M ps = (cid:18) v I c ps Zc ps Z b ps I (cid:19) , where b ps = P s R η η ′ ζ th p ( ζ ) (1 + ζ ( v − dζc ps = P s R η η ′ ζ th p ( ζ ) √ ζ √ v − dζ . (17)Here, P s is the total probability for the combined channeltransmission to fall within the post-selected region, and isgiven by P s = Z η η ′ ζ th p ( ζ ) dζ . (18)Using M ps , the Gaussian entanglement in terms of the log-arithmic negativity of the post-selected state can be computed.This is illustrated in Fig. 5 with respect to the post-selectionthreshold ζ th and success probability P s , respectively (solidlines). Note that in these calculations no closed-form solutionfor p ( ζ ) could be used, so a numerically determined formwas utilized. Fig. 5 explicitly shows for this specific fadingchannel the trade-off in increased Gaussian entanglement (asthe threshold value increases) at the cost of lower successprobability. The other curves (dashed) in this figure relate toquantum post-selection, which is discussed next. B. Quantum Post-selection
As we have just seen, classical post-selection offers thepossibility of concentrating the Gaussian entanglement at theground station. However, this comes at additional complexityin the transmission and detection strategy at the groundstations, due to the requirement for ongoing reliable channelestimation. As such, it is useful to explore how Gaussianentanglement concentration may be possible without suchchannel estimation. To investigate this we will generalize tothe combined fading channel, the distillation scheme recentlyproposed by [33] for the single fading channel.Recalling that in the direct transmission scheme, one mode(beam A ) from the initial two-mode entangled state is atground station A and the other mode (beam B ) is transmittedto ground station B via a relaying satellite. From Eq.(8), theCM of the two-mode Gaussian state between the terrestrialstations after each realization of η and η ′ can be re-written as M ηη ′ = v c q v c p c q b q c p b p ,b q = b p = 1 + η η ′ ( v − , c q = − c p = √ η η ′ √ v − . (19)Entanglement distillation is implemented at the receivingground station by extracting a small portion (beam t ) of thereceived mixed state using a tap beam splitter with transmit-tivity of T and reflectivity of R = 1 − T . A single quadrature(for instance, the amplitude quadrature, ˆ q t ) is then measuredon the tapped beam. If the measurement outcome is abovethe threshold value q th , then the remaining state (beam B ′ )is kept, otherwise it is discarded. The Wigner function of thestate before the beam splitter for each realization of η and η ′ is given by W ηη ′ ( q A , p A , q B , p B ) = exp (cid:16) − R AB M − ηη ′ R TAB (cid:17) π p det M ηη ′ , (20)where R AB = ( q A , p A , q B , p B ) . Given the Wigner functionfor the vacuum state as W v ( q v , p v ) = π exp (cid:0) − (cid:0) q v + p v (cid:1)(cid:1) ,the conditional Wigner function of the output state afterdistillation for each realization of η and η ′ is given by W dηη ′ ( q A , p A , q B ′ , p B ′ ) = R ∞ q th dq t R ∞−∞ dp t W ηη ′ ( q A , p A , ˜ q B , ˜ p B ) W v (˜ q v , ˜ p v ) , (21)where ˜ q B = √ T q B ′ + √ Rq t , ˜ p B = √ T p B ′ + √ Rp t , ˜ q v = √ T q t − √ Rq B ′ , and ˜ p v = √ T p t − √ Rp B ′ . Fromthe resultant Wigner function, W dηη ′ , the moments of thequadrature operators after the distillation for each realizationof η and η ′ can be calculated [33]. Since W dηη ′ is a Gaussiandistribution of the quadrature variables, these moments can beritten as h q A i ηη ′ = √ R c q √ πV t,q exp (cid:16) − q th V t,q (cid:17) h q B ′ i ηη ′ = √ T R ( b q − √ πV t,q exp (cid:16) − q th V t,q (cid:17)(cid:10) q A (cid:11) ηη ′ = R c q q th √ πV t,q exp (cid:16) − q th V t,q (cid:17) + v Erfc (cid:18) q th √ V t,q (cid:19)(cid:10) q B ′ (cid:11) ηη ′ = R T ( b q − q th √ πV t,q exp (cid:16) − q th V t,q (cid:17) + R T ( b q − + b q V t,q Erfc (cid:18) q th √ V t,q (cid:19) h q A q B ′ i ηη ′ = √ T R ( b q − c q q th √ πV t,q exp (cid:16) − q th V t,q (cid:17) + √ T c q Erfc (cid:18) q th √ V t,q (cid:19) , (22)where h . i denotes the expectation value and V t,q = Rb q + T .The elements of the total CM of the resulting distilled state arecalculated by averaging over all possible transmission factorsof the two fading channels giving the final distilled CM M d = a dq c dq a dp c dp c dq b dq c dp b dp , where a dq = (cid:10) q A (cid:11) − h q A i b dq = (cid:10) q B ′ (cid:11) − h q B ′ i c dq = h q A i h q B ′ i − h q A q B ′ i a dp = P s R η R η ′ p AS ( η ) p SB ( η ′ ) P ηη ′ v dη dη ′ b dp = P s R η R η ′ p AS ( η ) p SB ( η ′ ) P ηη ′ ( T b p + R ) dη dη ′ c dp = P s R η R η ′ p AS ( η ) p SB ( η ′ ) P ηη ′ √ T c p dη, dη ′ h q A i = P s R η R η ′ p AS ( η ) p SB ( η ′ ) h q A i ηη ′ dη dη ′ h q B ′ i = P s R η R η ′ p AS ( η ) p SB ( η ′ ) h q B ′ i ηη ′ dη dη ′ (cid:10) q A (cid:11) = P s R η R η ′ p AS ( η ) p SB ( η ′ ) (cid:10) q A (cid:11) ηη ′ dη dη ′ (cid:10) q B ′ (cid:11) = P s R η R η ′ p AS ( η ) p SB ( η ′ ) (cid:10) q B ′ (cid:11) ηη ′ dη dη ′ h q A q B ′ i = P s R η R η ′ p AS ( η ) p SB ( η ′ ) h q A q B ′ i ηη ′ dη dη ′ , (23) ζ th0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.60.70.80.911.1 E LN q th−6 −5 −4 −3 −2 −1 00.60.70.80.911.1 log P s E LN Fig. 5. (Color online) Logarithmic negativity E LN of the two-mode statesat the ground stations (for the direct transmission scheme) in terms of theclassical post-selection threshold ζ th (solid line in top figure), quantum post-selection threshold q th (dashed line in top figure), and success probabilityof classical/quantum post-selection P s (bottom figure). Here, T = 0 . , r = 1 . , β/W = 0 . , σ b = β , k = 0 . , k = 0 . . This channelcorresponds to a mean loss of 6.4dB in the uplink, and 4.4dB in the downlink. E LN ζ th−7 −6 −5 −4 −3 −2 −1 000.511.5 log P s E LN Fig. 6. Logarithmic negativity E LN of the two-mode states at the groundstations (for the direct transmission scheme) in terms of the classical post-selection threshold ζ th (top figure), and success probability of classical post-selection P s (bottom figure). Here, r = 1 . , β/W = 0 . , σ b AS =22 β, σ b SB = 2 β . This channel corresponds to a mean loss of 30dB inthe uplink, and 10dB in the downlink. and where P ηη ′ = Erfc (cid:18) q th √ V t,q (cid:19) ,P s = R η R η ′ p AS ( η ) p SB ( η ′ ) P ηη ′ dη dη ′ . (24)Note that here, P s is now the total success probability ofdistilling the mixed state, and P ηη ′ is implicitly dependent on η and η ′ through the Wigner function W dηη ′ ( q A , p A , q B ′ , p B ′ ) - but is not to be confused with p ( ζ ) defined for the classicalpost-selection.sing M d , the Gaussian entanglement in terms of the log-arithmic negativity of the quantum post-selected state can becomputed. This is illustrated in Fig. 5 with respect to the post-selection threshold q th and success probability P s , respectively(dashed lines). Similar to the classical post-selection, we seethe amount of Gaussian entanglement is increased by theaction of the quantum distillation. However, it is evident thatthe improvement in Gaussian entanglement is more probableby the classical post-selection. Furthermore, considering thesame success probability of each strategy, the classical post-selection is able to generate more Gaussian entanglementcompared to the quantum post-selection protocol. Althoughimprovements in the quantum post-selection strategy can bemade, due to its direct selection of better channels the classicalpost-selection scheme will always provide a better result.The results described here illustrate the price to be paidfor deploying simpler transmission/detection strategies (nochannel estimation) at the ground stations.Our final result is to look at the entanglement generationrates in the high-loss scenario where the direct transmissionscheme is utilized with terrestrial ground stations. In suchscenarios one could expect typically 25-30dB loss in the uplinkand 5-10dB in the downlink. Fig. 6 shows an example of sucha link scenario. Here the quantum post-selection strategy isnot shown, as its success probability is found to be too smallin such high-loss scenarios. We can see from Fig. 6 that forthe specific channel shown, levels of Gaussian entanglementat E LN > can be found for success probabilities < − .These success probabilities can be multiplied by the trans-mission rates (currently of order Hz) in order to obtain amode (pair) generation rate at E LN > of Hz. Note, thisshould only be considered as a typical rate for the duration ofa single pass of a LEO satellite - as the channel characteristicswill vary during the actual LEO pass-over timescale (which isof order a few hundred seconds [8]).V. C
ONCLUSIONS AND F UTURE D IRECTIONS
In deploying quantum communications, we are largely facedwith three options, the use of fibers, the use of free-spacechannels, or the use of satellite-based communications. Thesetechnologies are complementary and all will likely play a rolein the emerging global quantum communication infrastructure.Fiber technology has the key advantage that once in place, anundisturbed channel from A to B exists. However, fiber suffersfrom large losses which therefore limit its distance - althoughsuch distance limitations may be overcome by the developmentof suitable quantum repeaters [29]. Replacing the fiber channelwith a free-space channel has the immediate advantage offewer losses [2], but such a channel is subject to potentialground-dwelling line-of-sight (LoS) blockages, and is alsoultimately distance-limited by the visible horizon. Nonethe-less, free-space optical communication has a role to play inmany scenarios [21], [22]. Free-space quantum communicationvia satellite has the additional advantage that communicationcan take place when there is no direct free-space LoS fromA to B in place. Assuming LoS from a satellite to the two ground stations exists, satellite-based communication canproceed. The range of this communication is also potentiallymuch larger than that allowed for by a direct ground-basedfree-space connection (no terrestrial horizon limit and lowerlosses at high altitudes). Use of satellites also allows forfundamental studies on the impact of relativity on quantumcommunications [34]. The key disadvantage of satellite-basedquantum communications is turbulence induced losses, thesubject of this work.In this work we have explored a range of quantum com-munication architectures anticipated to play a role in nextgeneration satellite-based communication systems and quan-tified the expected entanglement generation rates they giverise to. We have focussed on the trade-off between the quan-tum complexity introduced at the satellite and the resultantGaussian quantum entanglement between two ground stations(or HAPs). We have found that for low-loss fading channelcharacteristics a low-complexity direct transmission scheme(reflection at the satellite) will produce CV entanglementgeneration rates at the ground stations not too dissimilarfrom those anticipated for a scheme based on entanglementgenerated at the satellite itself. For high-loss channels we findthat a direct transmission scheme can provide for useful levelsof entanglement generation. When the downlink channels canbe assured to be significantly better than uplink channels,entanglement generation within the satellite will provide for acorresponding significant improvement in entanglement ratesat the ground stations - albeit at the cost of embedding quan-tum systems in the satellite. In all cases we find entanglementswapping at the satellite to be an inferior solution.We have also investigated the role played by post-selectionin concentrating the entanglement between the ground stations.More specifically, we have investigated the price to be paidif simple transmission and detection strategies are adopted atthe ground station in which no channel estimation is required.The quantum post-selection techniques can be utilized in suchscenarios, but in general will provide reduced entanglementoutcomes relative to classical post-selection techniques basedon channel estimation. In high-loss channels classical post-selection is required.Given the losses anticipated in satellite-based communi-cations, future work should focus on additional effects thatlead to enhanced protection of entangled modes transmittedthrough a turbulent atmosphere. Of particular value would bethe use of coding techniques applied to the CV states, useof non-Gaussian states as initial transmission modes, and useof quantum feedback control between the two ground stations.The payoff of such techniques would likely be of most value ina direct transmission scheme. The reliability of CV versions ofquantum applications such as quantum key distribution (QKD)[35] and quantum location verification [36] over high-lossatmospheric fading channels are also worthy of investigation.Consideration of hybrid CV/single-photon architectures in thedeployment of such techniques and applications would be ofparticular interest. Finally, we note the application of spatial-diversity techniques as applied to FSO communications, andhe ability of full diversity to be achieved even for transmittersonly a few cm apart [37]. The role of such diversity techniquesin compensating the large losses in uplink ground-satellitechannels is also worth exploring in the context of Gaussianentanglement distribution.VI. A
CKNOWLEDGMENTS
This work has been funded by the University of NewSouth Wales (Australia). The authors gratefully acknowledgevaluable discussions with Hendra Nurdin.R
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For completeness we detail the analysis that leads to thesolutions in our adopted entanglement swapping scheme.Here, we follow closely the fixed-attenuation analysis of [30],generalizing it to the case of combined fading channels.
A. Entanglement Swapping Covariance Matrices
Here we wish to highlight how (14) and (15) of the main textare derived. Let us consider entanglement swapping involvingtwo pairs of entangled modes, one pair consists of modes 1 and2 and the second pair consists of modes 3 and 4. We assumethat the two pairs are described by two Gaussian states, havingdifferent CMs and zero first moments, i.e. , M , = (cid:18) aI CC T bI (cid:19) , C = diag ( c + , c − ) M , = (cid:18) dI FF T eI (cid:19) , F = diag ( f + , f − ) D ˆ R , E = D ˆ R , E = 0 . (25)In the Wigner function formalism, the initial 4-mode state isdescribed by the product of the Wigner function of two inputstates W in ( R , , , ) = W in ( R , ) W in ( R , ) W in ( R i,j ) ∝ exp( − R i,j M − i,j R Ti,j ) , (26)here i, j ∈ { , , , } . The inverse of CMs, M , and M , can be computed as M − , = ( λ ij ) = bab − c − c + ab − c bab − c − − c − ab − c − − c + ab − c aab − c − c − ab − c − aab − c − M − , = ( γ ij ) = ede − f − f + de − f ede − f − − f − de − f − − f + de − f dde − f − f − de − f − dde − f − . (27)Thus, the Wigner function of the 4-mode state before theswapping can be given as W in ( R , , , ) ∝ exp (cid:0) − (cid:0) R , M − , R T , + R , M − , R T , (cid:1)(cid:1) =exp (cid:8) − (cid:0) λ q + λ q + λ p + λ p γ q + γ q + γ p + γ p +2 λ q q + 2 λ p p + 2 γ q q + 2 γ p p } ) . (28)The swapping is first performed by mixing two modes 2 and3 through a balanced beam splitter, yielding output modes u and v which at the level of quadrature variables are describedby q u = √ ( q − q ) , p u = √ ( p − p ) q v = √ ( q + q ) , p v = √ ( p + p ) . (29)With these relations, the Wigner function of the new 4-mode state after the beam-splitter, W BS ( R , ,u,v ) , can thenbe obtained from the Wigner function (28), namely W BS ( R , ,u,v ) ∝ exp( − R , ,u,v M − , ,u,v R T , ,u,v ) , (30) where R , ,u,v = ( q , p , q , p , q u , p v , q v , p u ) M − , ,u,v = λ λ γ
00 0 0 γ kλ − kγ kλ kγ kλ kγ kλ − kγ kλ kλ kλ kλ − kγ kγ kγ − kγ δ δ δ δ δ δ δ δ , (31)and where δ = λ + γ , δ = λ + γ ,δ = λ − γ , δ = λ − γ , k = √ . (32)Then, the new quadratures ˆ q u and ˆ p v are measured with twohomodyne detectors, providing the outcomes q ′ u and p ′ v withprobability P ( q ′ u , p ′ v ) . As a result of this measurement, theinitial 4-mode state conditionally collapses into a 2-modestate consisting of modes 1 and 4. The Wigner functionof this conditional output state is obtained by integrating W BS ( R , ,u,v ) over the unmeasured quadratures q v , p u , giving W cond ( R , ) ∝ Z Z W BS ( R , ,u,v ) dq v dp u (cid:12)(cid:12) q u = q ′ u , p v = p ′ v (33)To make progress we use the partial Gaussian integral formu-lation for n variables for the case where we wish to integrateover the last n − m of them, viz ., R . . . R exp (cid:2) − q T Qq (cid:3) dq m +1 . . . dq n ∝ exp (cid:8) − u T U u (cid:9) , (34)where Q = (cid:18) U VV T W (cid:19) , U = U − V W − V T ,q = (cid:18) uw (cid:19) , u = q ... q m , w = q m +1 ... q n . (35)omparing (34) with our problem for integrating W BS ( R , ,u,v ) over the quadratures q v , p u , we will have U = λ kλ λ kλ γ − kγ
00 0 0 γ kγ kλ − kγ δ kλ kγ δ V = kλ kλ kγ − kγ δ δ , W = (cid:18) δ δ (cid:19) q = R , ,u,v , u = q p q p q u p v , w = (cid:18) q v p u (cid:19) . (36)The Wigner function for the conditional state of modes 1 and4 is then given by W cond ( R , ) ∝ exp (cid:8) − u T U u (cid:9) U = U U U U U U U U U U U U U U U U U U (37)where U = e ( b + d ) − f a ( de − f )+ e ( ab − c ) , U = − c + f + a ( de − f )+ e ( ab − c ) U = −√ ec + a ( de − f )+ e ( ab − c ) , U = e ( b + d ) − f − a ( de − f − )+ e ( ab − c − ) U = c − f − a ( de − f − )+ e ( ab − c − ) , U = −√ ec − a ( de − f − )+ e ( ab − c − ) U = a ( b + d ) − c a ( de − f )+ e ( ab − c ) , U = √ af + a ( de − f )+ e ( ab − c ) U = a ( b + d ) − c − a ( de − f − )+ e ( ab − c − ) , U = −√ af − a ( de − f − )+ e ( ab − c − ) U = aea ( de − f )+ e ( ab − c ) , U = aea ( de − f − )+ e ( ab − c − ) . (38)From the above we can see that the first moments of theconditional output state depends on the measurement results q ′ u and p ′ v and each of the four quadratures will be proportional to (cid:0) − q ′ u U , − p ′ v U , − q ′ u U , − p ′ v U (cid:1) T . Let us consider U ′ as U ′ = U U U U U U U U . (39)Thus, the CM of the conditional state of modes 1 and 4 canbe obtained by inverting U ′ to give M , = a − c b + d c + f + b + d a − c − b + d − c − f − b + dc + f + b + d e − f b + d − c − f − b + d e − f − b + d . (40)By setting the matrices of (13) to those of (25), we find(40) leads to (15). However, the protocol is not complete. Afinal subtlety is that as it stands this matrix represents theoutcome for a one-shot Bell measurement. We still have toaverage over all Bell measurement results. But as we nowshow if we optimize our choice of gains in the displacementprocedure of the protocol, we will arrive at (15) as the finalCM averaged over all Bell measurement results (for a specificchannel realization).In order to complete the swapping process, the measurementresults are broadcast so that modes 1 and 4 can properlybe displaced according to the measurement outcomes q ′ u and p ′ v . In practice, the displacements should be weighted bygain factors so as to improve the quality of the swappedentanglement. In terms of the quadrature operators, theseconditional displacements can be expressed as (cid:26) ˆ q → ˆ q − g √ q ′ u ˆ p → ˆ p + g √ p ′ v , (cid:26) ˆ q → ˆ q + g √ q ′ u ˆ p → ˆ p + g √ p ′ v (41)where g and g are the gain factors for the displacement ofmodes 1 and 4, respectively.Using (41) the first moments of the four quadratures of thedisplaced conditional state W dis ( R , ) are then proportionalto √ q ′ u − g ( e ( b + d ) − f ) − g c + f + + ec + a ( de − f )+ e ( ab − c ) p ′ v g ( e ( b + d ) − f − ) + g c − f − + ec − a ( de − f − )+ e ( ab − c − ) q ′ u g ( a ( b + d ) − c ) + g c + f + − af + a ( de − f )+ e ( ab − c ) p ′ v g ( a ( b + d ) − c − ) + g c − f − + af − a ( de − f − )+ e ( ab − c − ) . (42)The Wigner function of the output state averaged over allpossible Bell measurements is therefore given by W ens ( R , ) = Z Z P ( q ′ u , p ′ v ) W dis ( R , ) dq ′ u dp ′ v , (43)where P ( q ′ u , p ′ v ) is the probability density of the Bell measure-ment outcomes. This average leads to a zero-mean two-modeaussian state with the following CM M ens = m m m m m m m m m = a + ( b + d ) g − c + g m = a + ( b + d ) g + 2 c − g m = e + ( b + d ) g − f + g m = e + ( b + d ) g + 2 f − g m = c + g + f + g − g g ( b + d ) m = c − g + f − g + g g ( b + d ) . (44)The optimal choice of gains are those for which all terms of(42) equal zero. In such a case the CM of the averaged state(44) is equal to that of the conditional state in (40). Assumingphase-independent gains, this optimal point is obtained for c + = − c − =: c and f + = − f − =: f , and g = cb + d , g = fb + d , (45)and the CM of (40) is obtained. Again by inspecting matrices(13) and (25), we find (45) leads to (14). B. Effective Loss Channels
As shown by [38], any CM of the standard form M s = (cid:18) a I c Zc Z b I (cid:19) (46)( a, b, c ∈ R ) which satisfies the uncertainty principle and isentangled (the positive partial transpose (PPT) criterion forseparability is violated [39]), is equivalent to the CM of alossy two-mode squeezed state with effective squeezing r e and effective channel transmissions η ae and η be for the firstand second modes, respectively. These effective parametersare given by cosh(2 r e ) = c +( a − b − c − ( a − b − ,η ae = a − r e ) − , η be = b − r e ) − . (47)Therefore, the CM for each realization of η and η ′ which aregiven by (8), (10) and (15) for direct transmission, satellite-based entanglement generation and swapping, respectively,can all be re-written in the context of lossy two-modesqueezed states (of course the first two schemes can bedirectly seen as loss channels). Averaging over all possiblevalues of η and η ′ , total effective transmittivities and totaleffective squeezing can be computed for all three schemes asfollows. (i) Direct transmission: cosh(2 r ) = v , η a = 1 η b = R η R η ′ p AS ( η ) p SB ( η ′ ) η η ′ dη dη ′ . (48) (ii) Satellite-based entanglement generation: cosh(2 r ′ ) = vη a ′ = R η p SA ( η ) η dη , η b ′ = R η ′ p SB ( η ) η ′ dη ′ . (49) (iii) Entanglement swapping: cosh(2 r ′′ ) = R η R η ′ p AS ( η ) p BS ( η ′ ) cosh(2 r ′′ η η ′ ) dη dη ′ cosh(2 r ′′ η η ′ ) = ( η + η ′ ) (1 − v )+ ηη ′ ( v +3 ) + ( η + η ′ ) ( v − η + η ′ − η + η ′ )( v − η a ′′ = R η R η ′ p AS ( η ) p BS ( η ′ ) − ( η + η ′ − ) ( v − η (1 − v )+2( η ′ − dη dη ′ η b ′′ = R η R η ′ p AS ( η ) p BS ( η ′ ) − ( η + η ′ − ) ( v − η ′ (1 − v )+2( η − dη dη ′ . (50)Given the following constraints; < v < ∞ , < cosh(2 r ′′ ) < ∞ , and ≤ η a ≤ (likewise η b , η a ′ , η b ′ , η a ′′ , η b ′′ ) it is straightforward to show that thetotal effective transmittivity η a ′′ η b ′′ for the swapping schemeis always less than or equal to η a η b for direct transmission.In addition, we know that in practice, the overall effectivetransmittivity η a ′ η b ′ for the satellite-based entanglement gen-eration is larger than the overall transmittivity η a η bb