Enhanced Uplink Quantum Communication with Satellites via Downlink Channels
Eduardo Villaseñor, Mingjian He, Ziqing Wang, Robert Malaney, Moe Z. Win
11 Enhanced Uplink Quantum Communication withSatellites via Downlink Channels
Eduardo Villase˜nor , Mingjian He , Ziqing Wang , Robert Malaney and Moe Z. Win . School of Electrical Engineering & Telecommunications,The University of New South Wales, Sydney, NSW 2052, Australia. Laboratory for Information and Decision Systems, Massachusetts Institute of Technology,Cambridge, MA 02139, USA.
Abstract —In developing the global Quantum Internet, quan-tum communication with low-Earth-orbit satellites will play apivotal role. Such communication will need to be two way:effective not only in the satellite-to-ground (downlink) channelbut also in the ground-to-satellite channel (uplink). Given thatlosses on this latter channel are significantly larger relative tothe former, techniques that can exploit the superior downlinkto enhance quantum communication in the uplink should beexplored. In this work we do just that - exploring how continuousvariable entanglement in the form of two-mode squeezed vacuum(TMSV) states can be used to significantly enhance the fidelityof ground-to-satellite quantum-state transfer relative to directuplink-transfer. More specifically, through detailed phase-screensimulations of beam evolution through turbulent atmospheres inboth the downlink and uplink channels, we demonstrate howa TMSV teleportation channel created by the satellite can beused to dramatically improve the fidelity of uplink coherent-state transfer relative to direct transfer. We then show how this,in turn, leads to the uplink-transmission of a higher alphabetof coherent states. Additionally, we show how non-Gaussianoperations acting on the received component of the TMSV stateat the ground station can lead to even further enhancement.Since TMSV states can be readily produced in situ on a satelliteplatform and form a reliable teleportation channel for mostquantum states, our work suggests future satellites forming partof the emerging Quantum Internet should be designed withuplink-communication via TMSV teleportation in mind.
I. I
NTRODUCTION Q UANTUM COMMUNICATIONS via low-Earth-orbit(LEO) represent a critical component of the so-calledQuantum Internet - a new heterogeneous global communica-tion system based on classical and quantum communicationtechniques whose information security will be underpinned byquantum protocols such as quantum key distribution (QKD).This new internet will also be used as the backbone commu-nication system inter-connecting future quantum computers viarouted quantum information transfer. The Quantum Internetparadigm has taken large steps forward in the past fewyears, particularly with the spectacular success of Micius -the first quantum-enabled satellite launched in 2016 [1]–[4].Building on the pioneering Micius mission, some twenty-plussatellite missions are now under development [5] - some atthe advanced design phase.The importance of satellite-based technology to the Quan-tum Internet paradigm lies in a satellite’s ability to transmit
Corresponding author: R. Malaney (email: [email protected]) quantum signals through much longer distances relative toterrestrial-only links [1], [2], [6]. Indeed, the Micius exper-iment has demonstrated quantum communication over a rangeof 7,600km [4] - a feat put into perspective by the currentterrestrial-only quantum communication record of 500km [7].The Micius experiment deployed quantum communicationprotocols via discrete variable (DV) technology where thequantum information was encoded in the polarization stateof single photons [2], [3]. Alternatively, continuous variables(CV) quantum information, where the information is encodedin the quadratures of the electromagnetic field of optical states,is widely touted as perhaps a more promising candidate totransfer quantum information [8], [9]. This is largely dueto the relative technical simplicity (and maturity) of theCV-enabled devices required to send, receive, and measurequantum signals, robustness against background noise, andthe potential of the enlarged Hilbert space associated withCV systems to lead to enhanced communication throughput inpractical settings. For these reasons, there is great interest inpursuing designs of CV-enabled quantum satellites, with manyrecent studies focusing on the more feasible satellite-to-ground(downlink) transmission of quantum signals, largely with aview to enable CV-QKD [11], [12]. As yet, there has not beenany experimental realizations of satellite-based CV quantumcommunications. In this work, we turn to a hitherto overlookedtype of satellite-based CV quantum communications, namely,the use of CV quantum downlink communications as a meansto enhance ground-to-satellite (uplink) quantum communica-tions with a LEO satellite.The main challenge faced in satellite-based quantum com-munications is the degradation of the signal as it is trans-mitted through the turbulent atmosphere of Earth [13]–[15],a degradation that is almost always larger than the noiseintroduced by the components used [16]. It is well documentedthat uplink satellite laser communications is considerablymore challenging compared to downlink satellite transmission:the turbulent eddies in the Earth’s atmosphere have a moredisruptive effect in the uplink channel. This is because the size In theory both DV and CV communication deliver the same throughput,and the reality is both systems have their pros and cons. However, therecertainly is a school-of-thought that in many pragmatic systems, the higher-dimensional encoding space directly available to CV systems will lead toenhanced outcomes. A detailed discussion of the pros and cons of both DVand CV systems is given in [10]. a r X i v : . [ qu a n t - ph ] F e b of the eddies encountered by a laser beam in the downlink atthe atmospheric entry point are significantly larger than thelaser beam’s transverse dimensions (spot size) at the entrypoint, whereas in the uplink the opposite is true [17]. Theconsequence of this is that an asymmetry in the channelsexists, with the uplink beam profile evolving in a more ran-dom fashion, especially in regard to beam wandering effects.Ultimately, this asymmetry manifests itself in higher losses inthe uplink channel [17], [18].Here, we investigate the use of quantum resources deliveredthrough the satellite downlink channel as a resource forteleportation in the uplink, and the subsequent use of thatteleportation resource to enhance quantum communicationsrelative to simple direct uplink transmission. More specifically,we consider the use of a two-mode-squeezed vacuum (TMSV)quantum teleportation channel created via the downlink chan-nel as a resource to teleport a coherent state from the groundstation to a LEO satellite. We will see that for uplink commu-nications, the use of the teleportation channel leads to signif-icantly higher fidelities compared to the direct transmission.Moreover, we find that the teleportation channel is capable oftransferring coherent states with larger amplitudes, somethingthat is very difficult via direct transmission. This latter attributeis important for many CV-based quantum protocols, such asCV-QKD, since for these protocols the capability to transmitcoherent states of different amplitudes is a key requirement.The main contributions of this work can be summarizedthus. (i) Through a series of detailed phase-screen simula-tions we quantify the asymmetric losses experienced by thedownlink and uplink channels of a LEO satellite in quantumcommunication with a terrestrial ground station. Moreover,we expand previous analyses of the uplink and downlinkchannels by quantifying and including the excess noise thatarises from each channel. This excess noise limits the accu-racy of the quadrature measurements, effectively reducing theamount of transferred quantum information. (ii) Using thesesame simulations we then determine the fidelity of coherentstate transfer through direct uplink transfer. (iii) We modelthe creation of a resource CV teleportation channel in thedownlink created by sending from the satellite one mode of anin situ produced TMSV state. (iv) We then use that resource todetermine the fidelity of coherent state transfer to the satellitevia teleportation, quantifying the gain achieved over directtransfer. (v) We then investigate a series of non-Gaussianoperations that can be invoked on the received TMSV modeat the ground station as a means to further enhance uplinkcoherent-state transfer via teleportation. Specifically, we inves-tigate photon subtraction, addition, and catalysis as the non-Gaussian operations - identifying the gains in teleportationfidelity achieved for each scheme. Sequences of these non-Gaussian operations are also investigated, and the optimalscheme amongst them identified.The remainder of this paper is as follows. In section IIwe describe CV teleportation through noisy channels. In sec-tion III we detail our phase screen simulations, comparing theirpredictions with a range of theoretical models, and discussingthe implications of our simulations in the context asymmetricdownlink/uplink channel losses. In section IV we discuss a series of non-Gaussian operations that can be applied to aTMSV state, discussing their roles in potentially enhancingCV teleportation via a noisy TMSV channel. In section V wediscuss application of our schemes to a wider range of states,and discus differences with the DV-only scheme of Micius.We draw our conclusions in section VI. Notation:
Operators are denoted by uppercase letters. Thesets of complex numbers and of positive integer numbersare denoted by C and N , respectively. For z ∈ C : | z | and arg( z ) denote the absolute value and the phase, respectively;Re ( z ) and Im ( z ) denote the real part and the imaginary part,respectively; z ∗ is the complex conjugate; and i = √− .The trace and the adjoint of an operator are denoted byTr {·} and ( · ) † , respectively. The annihilation, the creation,and the identity operators are denoted by A , A † , and I ,respectively. The displacement operator with parameter α ∈ C is D ( α ) = exp[ α A † − α ∗ A ] . II. C
ONTINUOUS VARIABLE TELEPORTATION
We consider the teleportation protocol introduced in [19].Here, we are considering the parties involved in the teleporta-tion are a ground station and a satellite in space, with thequantum channel between them corresponding to the free-space atmospheric channel as exemplified in Fig. 1a). Theteleportation protocol starts with the generation of a bipartiteentangled resource state, Ξ AB , in the satellite. Part A of Ξ AB is sent through the atmosphere to the ground station, whereit is combined with the input state using a balanced beam-splitter. Afterwards, a Bell projective measurement (using apair of homodyne detectors) on part A and the input state isperformed. The measurement result is broadcast to the satellitewhich, by doing a corrective operation on B , recovers theinput state as the final output of the protocol. To describe theteleportation protocol we follow the methodology introducedin [20]. Using this methodology the output state can becomputed by using the Wigner characteristic functions (CF)of the input state Ξ in and the entangled resource state Ξ AB .Here, we indicate CFs by χ ( ξ ) , for some complex parameter ξ . In [21] the methodology is further expanded to includeimperfect homodyne measurements, obtaining χ out ( ξ ) = χ in ( gηξ ) × χ AB ( ξ, gηξ ∗ ) e − | ξ | g (1 − η ) , (1)where g is the gain parameter, and η the efficiency of thehomodyne measurements. The CF of a generic n -mode state Ξ is obtained by taking the trace of the product of Ξ withthe displacement operator, giving χ ( ξ , ξ , ..., ξ n ) = Tr { ΞD ( ξ ) D ( ξ ) · · · D ( ξ n ) } , (2)where { ξ , ξ , ..., ξ n } ∈ C are complex arguments, each onerepresenting a mode of Ξ in the CF. We assume noiseless classical communications between satellite andground station by means of a different channel, such as radio-waves or wide-beam optical signals.
Fig. 1: a) CV teleportation of a coherent state using a bipartiteentangled resource between a satellite and a ground station.Homodyne measurement results are transmitted by the groundstation after combining the received quantum signal with thecoherent state. The satellite uses the measurement results toapply a displacement operator on the remaining mode of theentangled state to obtain the teleported state. b) In satellitecommunications the downlink channel is considerably lessnoisy than the uplink channel.In this work, we consider that the entangled resource usedis a TMSV state. The TMSV state can be considered as theapplication of the two mode squeezing operator to the vacuum | TMSV (cid:105) = S ( (cid:37) ) | , (cid:105) = e (cid:37) A A − (cid:37) ∗ A † A † | , (cid:105) , (3)where (cid:37) = re iφ is the squeezing parameter. Here, we will take φ = 0 , for simplicity. The CF of a TMSV state is χ TMSV ( ξ A , ξ B ) = exp (cid:104) − (cid:16) V ( | ξ A | + | ξ B | )+ (cid:112) V − ξ A ξ B + ξ ∗ A ξ ∗ B ) (cid:17)(cid:105) , (4)where V = cosh(2 r ) is the variance of the distribution of thequadratures. Throughout this work quadrature variances arein shot noise units (SNU), where the variance of the vacuumstate is 1 SNU ( (cid:126) = 2 ). Additionally, a coherent state (thestate we wish to transfer to the satellite) can considered as theapplication of the displacement operator to the vacuum, | α (cid:105) = D ( α ) | (cid:105) , (5)with the corresponding CF given by χ | α (cid:105) ( ξ ) = e − | ξ | +2 i Im ( ξα ∗ ) . (6)In general, we can describe the effects a noisy channel, witha given transmissivity T and excess noise (cid:15) , has on a modeof any quantum state by scaling the ξ (cid:48) s in the relevant CF by √ T , and adding a CF corresponding to a vacuum state. Fora TMSV state where only mode B is transmitted through thenoisy channel, the corresponding CF is [22], χ (cid:48) TMSV ( ξ A , ξ B ) = exp (cid:20) −
12 ( (cid:15) + 1 − T ) | ξ B | (cid:21) × χ TMSV ( ξ A , √ T ξ B ) . (7)At times, it will be convenient to refer to the transmissivity indB, as given by −
10 log T . Note, that due to the negativesign in this definition, when the transmissivity is referred to in dB a larger loss will have a higher numerical valueof the dB transmissivity. Indeed, in this work take the term“loss” to mean a transmissivity given in dB - the specifictransmissivity being referred to being clear given the context.If transmissivity is specified without reference to units thenit has its normal meaning of a ratio of energies (larger losscorresponding to lower transmissivity). A. Fidelity of teleportation
We will use the fidelity as the figure-of-merit to evaluatethe effectiveness of quantum teleportation. The fidelity, F , isa measurement of the closeness of two states Ξ and Ξ , andis given by F = 1 π (cid:90) d ξ χ Ξ ( ξ ) χ Ξ ( − ξ ) . (8)To compute the fidelity of a teleported coherent state, F T ,we first use Eq. (4), and Eq. (7) to write the CF of aTMSV state that has been transmitted through a noisy channel.Thereafter, using Eq. (1), and Eq. (6) we obtain the CF ofthe teleported state. Finally, F T is computed as in Eq. (8),resulting in F T ( V, T, (cid:15), η, g, α ) = 2∆ exp (cid:20) − | α | (1 − ˜ g ) (cid:21) , (9)where ˜ g = gη , and ∆ = V + ˜ g T ( V −
1) + ˜ g ( (cid:15) + 1) − g (cid:112) T ( V − g + 1 + g (1 − η ) . (10)Ultimately, F T depends on the characteristics of the noisychannel involved in the protocol ( T and (cid:15) ), the parameter V ,and the gain g . These last two parameters, V and g , can becontrolled to optimize the fidelity teleportation for any given T and (cid:15) .We will compare the resulting fidelity of the teleportedstates with the fidelity of states directly transmitted throughthe uplink noisy channel. The fidelity of direct transmission, F DT , is computed by first writing the CF of a coherent statethat has been transmitted through the noisy channel, as χ (cid:48)| α (cid:105) ( ξ ) = exp (cid:20) −
12 ( (cid:15) + 1 − T ) | ξ | (cid:21) χ | α (cid:105) ( √ T ξ ) . (11)Thereafter, the fidelity between the original state and thetransmitted one is computed by using Eq. (8), resulting in F DT ( V, T, (cid:15), α ) = 22 + (cid:15) exp (cid:104) − − √ T ) | α | (cid:15) (cid:105) . (12)To perform a fair assessment, it is not enough to simplyconsider a single coherent state. Instead, we must considerthe mean fidelity over an ensemble of coherent states, drawnfrom a Gaussian distribution, whose probability distribution isgiven by [21] P ( α ) = 1 σπ exp (cid:16) − | α | σ (cid:17) , (13)with σ the variance of the distribution. We can think of σ as determining the alphabet of states used when transmitting Fig. 2: Fidelities for teleportation and direct transmissiontowards the satellite, via a fixed-transmissivity channel. En-sembles of coherent states with different values of σ areconsidered. The shaded area in red marks the region where theteleportation fidelity falls below that achievable using classicalcommunications only. Recall, a higher T in dB correspondsto higher loss.quantum information, or during a protocol such as CV-QKD.We can now define the mean fidelity as ¯ F = (cid:90) dα P ( α ) F ( ..., α ) . (14)To compare the effectiveness of teleportation relative to directtransmission, we present in Fig. 2 the values of ¯ F obtainedfor transmission via a fixed noisy channel, for different valuesof σ . The excess noise in the channel is fixed as (cid:15) = 0 . .Throughout this work the efficiency of the homodyne measure-ments involved in the teleportation is fixed to η = 0 . . Addi-tionally, the values of g and V involved in the teleportationare optimized for each value of the transmissivity. When theloss is small (1 dB), the optimal value of V is approximately100, however, as the loss of the channel increases the optimalvalue of V rapidly decreases towards unity. Using purelyclassical communications, a value of F classical = 0 . can beachieved, therefore quantum state transfer is only of interestin the regime where F > F classical [23]. From the resultspresented in Fig. 2, we make two observations: First, for eachvalue of σ , there exists a threshold in the transmissivity abovewhich teleportation yields a higher mean fidelity. Second, as σ increases this threshold decreases. This second observationis important for numerous quantum communications protocols(e.g. coherent CV-QKD) in which the more states that can betransmitted, the better. These two observations indicate that thetransmission of quantum states by means of teleportation canbe a better alternative relative to simple direct transmission.In the next section we will explore this result in more detailin the context of uplink satellite communications, where weconsider teleportation from the ground station to the satellitevia a TMSV state created via the downlink channel. III. G ROUND - TO - SATELLITE STATE QUANTUMCOMMUNICATION
We consider a quantum communications setup between aground station and a satellite. In this setup, the satellite andground station have the ability to send and receive quantumoptical signals between each other. The ground station ispositioned at ground level, h = 0 km, and the satellitewhen directly overhead at an altitude H = 500 km. The totalpropagation length between the satellite and the ground stationdepends on the zenith angle, ζ , of the satellite relative to theground station. The quantum signals are in the form of shortlaser pulses with a time-bin width of τ = 100 ps, emitted froma laser with a wavelength of λ = 1550 nm. Each laser pulsehas an amplitude in the transverse plane possessing a Gaussianprofile, and with a beam waist of radius w . Although insome special configurations the beam w can be made aslarge as the transmitting aperture, without loss of generality,we will assume w is always smaller than the radius of thetransmitting aperture. As the signal propagates, its beam widthincreases due to natural diffraction as well as due to the effectsof the atmosphere. The satellite and ground station are bothequipped with a telescopic aperture to receive the quantumsignals. The radius of the aperture of the satellite is r sat , whilefor the ground station the radius is r gs . Besides the quantumsignals, the ground station and the satellite also transmit astrong optical signal which can be used as a phase referencefor performing homodyne measurements. This strong signal iscommonly called a “local oscillator” (LO). In order to studythe transmission of quantum signals through the atmosphere, itis key to have a correct model of the effects of the atmosphericturbulence on the propagating beams. Ultimately, this modelwill allow us to estimate the values T and (cid:15) of the uplink anddownlink channels. A. Modeling atmospheric channels
The effects of the atmosphere on a propagating beamare modelled using the phase screen model, based in Kol-mogorov’s theory [24]. The phase screen model is constructedby subdividing the atmosphere into regions of length ∆ h i . Foreach region the random phase changes induced to the beam bythe atmosphere are compressed into a phase screen. The phasescreen is then placed at the start of the propagation length, andthe rest of the atmosphere is taken to have a constant refractiveindex. The result at the end of the entire propagation lengthis a beam that has been deformed mimicking the effects ofthe turbulent currents in the atmosphere. Thus, this processrecreates what a receiver with an intensity detector wouldobserve. Numerically, the beam is represented by a uniformgrid of pixels, each one assigned with a complex number, andthe propagation is modelled via a Fourier algorithm [25]. Sincethe result of each beam propagation is random, the simulationsare run 10,000 times, in order to obtain a correct estimationof the properties of the channel. A detailed description of thenumerical methods used can be found elsewhere, e.g. [26]–[28]. In the phase screen model, the first requirement is a modelof the refractive index structure of the atmosphere, C n . Weuse the widely adopted H − V / model [29]: C n ( h ) = 0 . v/ (10 − h ) exp( − h/ (15) + 2 . × − exp( − h/ A exp( − h/ , where h is the altitude in meters, v = 21m / s is the rmswindspeed, and A = 1 . × − m − / the nominal value of C n at ground level. In the H − V / model, the main effectsof the turbulence are confined to an altitude of 20km, since forhigher altitudes the effects are minimal. Besides the refractiveindex, we also need the upper-bounds and lower-bounds tothe sizes of the turbulent eddies that make up the turbulentatmosphere. The upper-bounds and lower-bounds are the so-called outer-scale and inner-scale, L and l , respectively.Here, we use the empirical Coulman-Vernin profile to model L as a function of the altitude h [30] L ( h ) = 41 + (cid:0) h − (cid:1) , (16)and we set the inner-scale to be a some fraction of the outer-scale, specifically, l = δL , where δ = 0 . .With the atmospheric models specified, we now look intohow the phase screens are constructed so as to mimic theeffects of the turbulence. Each individual phase screen iscreated by performing a fast Fourier transform over a uniformsquare grid of random complex numbers, sampled from aGaussian distribution with zero mean and variance, given bythe spectral density function [27] Φ φ ( κ ) = 0 . r − / exp( − κ /κ m )( κ + κ ) / , (17)where κ is the radial spatial frequency on a plane orthogonalto the propagation direction, κ m = 5 . /l , κ = 2 π/L ,and r is the coherent length. Since the main effects inducedby the atmosphere happen between zero altitude and 20km,the uplink and downlink transmissions will possess key dif-ferences, mainly arising from the interplay between the sizesof the beam size and the turbulent eddies. During downlinktransmission the beam first encounters the atmosphere with alarge beam size - possessing essentially no curvature at thispoint. On the other hand, in the uplink channel the beamencounters the atmosphere at the start of its path where it hasa positive curvature and a small beam size. For these reasons,we expect that the loss in the downlink will be dominated byrefraction while the (higher) loss in uplink will be dominatedby beam wandering. Under the flat beam assumption, thecoherent length for the downlink can be written as r downlink0 = (cid:16) . k sec( ζ ) (cid:90) h + h − C n ( h ) dh (cid:17) − / , (18)where k = 2 π/λ , and h − and h + correspond to the lower andupper altitudes of the propagation path corresponding to therespective phase screen. For the uplink, we need to define firstsome parameters that characterize the properties of the beam,namely Θ = 1 +
LR ,
Λ = 2
Lkw , (19) where R and w , are given by R = L (cid:104) (cid:16) πw λL (cid:17)(cid:105) , w = w (cid:104) (cid:16) λLπw (cid:17)(cid:105) / , (20)where L is the total distance between satellite and groundstation (dependent on ζ ). Given these definitions, the coherentlength for the uplink channel can be written as r uplink0 = (cid:16) . k sec( ζ )( µ + 0 . µ Λ / ) (cid:17) − / , (21)where µ = (cid:90) h + h − C n ( h ) (cid:104) Θ (cid:16) H − hH − h (cid:17) + h − h H − h (cid:105) / , (22) µ = (cid:90) h + h − C n ( h ) (cid:104) − h − h H − h (cid:105) / . The position of each phase screen is determined using thecondition that the Rytov parameter, r R , is maintained constantover each length ∆ h i , specifically [31] r R = 1 . k / (cid:90) h + h − C n ( h )( h − h − ) / dh = b. (23)We set a value of b = 0 . , which corresponds to a total of 17phase screens up to 20km. In Fig. 3, we plot the H − V / model, with the positions of the phase screens set by thecondition given by Eq. (23). For comparison, we also plot thepositions of the phase screens placed at a uniform distancebetween ground level and 20km. We can see that by usingthe condition imposed by Eq. (23) the phase screens are moreadequately distributed to account for the altitude variations inthe turbulence. Finally, to account for the remaining turbulencebetween 20km and H a single phase screen is used.At the end of every beam propagation simulation we canobtain the transmissivity induced by the atmosphere by inte-grating the intensity of the beam over the receiver aperture,as T turb = (cid:82)(cid:82) D I sig dAP , (24)where I sig is the intensity (power per unit area) of the beamat the plane containing the receiver aperture, P is the initialtotal power of the beam at the point of emission, and D isthe surface area of the receiver aperture. Despite the mainsource of loss arising from the atmospheric turbulence, wealso need to account for the extinction of the signal caused byabsorption and scattering by the particles of the atmosphere, aswell as the loss due to the imperfect optical devices used. Toaccount for the extinction we adopt a transmissivity T ext =exp( − . ζ ) . For the loss due to the optical devices weconsider a transmissivity value T opt = 0 . (1dB) [32]. Thetotal transmissivity of the channel is then simply, T = T turb T ext T opt . (25) B. Excess noise
Since in CV quantum states the information is encodedin the quadratures of the states, we require an LO in or-der to extract this information via homodyne or heterodyne
Fig. 3: The H-V / model, with the positions of the phasescreens shown for two cases: equally spaced phase screens andspacing that conserves a constant value of the Rytov parameterwith b = 0 . .measurements. With this in mind, we can take the resultspresented in this work which adopt a non-zero (cid:15) , as the fidelityoutcomes expected if we were to actually measure the fidelitiesexperimentally [33]. The ideal theoretical predictions wouldcorrespond to the pure loss channel case, where (cid:15) = 0 . In[34] it is discussed that for coherent state transmission viaatmospheric channels the main components of the excess noisearise from turbulence-induced effects on the LO, in additionto time-of-arrival fluctuations caused by delays between thelaser pulses and the LO. The variations in the intensity of theLO induce an excess noise given by (cid:15) ri = σ , LO ( D ) V sig , (26)where V sig the statistical variance of the quadratures of thequantum signal, corresponding to V sig = σ for direct trans-mission, and V sig = V for the teleportation channel. For agiven aperture size, the scintillation index averaged over theaperture of the LO is σ SI , LO ( D ) = (cid:104) P LO (cid:105) / (cid:104) P LO (cid:105) − , (27)where P LO = (cid:82)(cid:82) D I LO dA is the power of the LO (withintensity given by I LO ) over the aperture. Since the uplinkchannel is more affected by beam wandering, σ SI,LO ( D ) canbe expected to be much greater for the uplink relative to thedownlink.Time-of-arrival fluctuations are caused by a broadening ofthe time-bin width of the signal pulse from τ to τ , where τ TABLE I: Satellite channel parameters.
H h λ τ is given by [34] τ = (cid:113) τ + 8 µ, (28)where µ = 0 . . δ − . δ / ) υ sec( ζ ) c , (29) υ = (cid:90) Hh C n L / dh, and where c is the speed of light in vacuum. As derived in[35], the variance of τ , is given by σ = τ / , which leadsto an excess noise [14]: (cid:15) ta = 2( kc ) (1 − ρ ta ) σ V sig , (30)where ρ ta is the timing correlation coefficient between the LOand the signal. The value of σ ta is independent of the directionof propagation of the beam. For a value of τ = 100 ps, (cid:15) ta isvirtually independent of the atmospheric turbulence, since thepulse broadening only becomes considerable for τ < . ps[34]. Therefore, considering that ρ ta = 1 − − , the noisecontribution due to the time of arrival fluctuations becomes (cid:15) ta = 0 . V sig .With the two main sources of noise outlined, we now writethe total excess noise as (cid:15) = (cid:15) ta + (cid:15) ri . The excess noisebeing directly proportional to V sig reflects the fact that dueto the fluctuating nature of atmospheric channels, the valuesof T and (cid:15) need to be estimated by repeated measurementsof the channel. This means that in a experimental setupone cannot distinguish between variations of the quadraturesdue to quantum uncertainty, or the variations induced by thefluctuating value of T . Therefore, the variations of T of thechannel effectively translate to additional excess noise. Wenote that there are additional sources of excess noise, however,their contributions are minor compared to those consideredhere [16]. C. Other channel modeling techniques
Throughout this work we use the phase screen simulationsto model the channel. Performing phase screen simulationsis essentially a numerical approach to solving the stochasticparabolic equation , and adopts a versatile technique referredto as the split-step method [27]. Despite its computationallyintensive nature, the split-step method has been widely usedto study the atmospheric optical propagation of classical lightunder a variety of conditions (see e.g. [36]–[41]). Due to itsquantitative agreement with analytical results, the split-stepmethod is also believed to be very reliable (see e.g. [42]–[44]).Other channel-modeling techniques have been proposedto simplify the description of the atmospheric propagationof quantum light under specific situations. It is worthwhileto compare their predictions with our detailed phase screensimulations. Channel modeling techniques based on the so-called elliptic-beam approximation [13] are believed to be particularly useful when the phase fluctuations of the output field amplitude can be neglected. This point is discussedfurther in [45] where it is also highlighted that homodynemeasurements can be constructed where phase fluctuationsof the output field can be neglected. Under the elliptic-beamapproximation, it is assumed that the atmospheric propagationleads to only beam wandering, beam spreading, and beamdeformation (into an elliptical form). However, the extinctionlosses due to back-scattering and absorption can also beadded phenomenologically under such an approximation [45].Although originally proposed under the assumption of a hor-izontal channel, the elliptic-beam approximation was directlyadopted in [46] to study the performance of CV-QKD in thedownlink channel. In addition, the authors of [32] proposed ageneralized channel modeling technique based on the elliptic-beam approximation, providing a comprehensive model forthe losses suffered by the quantum light in both the uplinkand downlink channels. All these works [13], [32], [45], [46]assumed an infinite outer scale (i.e. L = ∞ ) and a zero innerscale (i.e. l = 0 ), effectively neglecting the inner scale andouter scale effects.In Fig. 4, we compare the mean turbulence-induced loss ¯ T turb [ dB ] predicted by i) the phase screen simulations, andii) the channel modeling techniques (based on the elliptic-beam approximation) of [46] and [32]. Although our phasescreen simulations take into account the inner scale and outerscale effects by adopting the empirical Coulman-Vernin profile(recall Eq. (16)), for comparison we also present the resultspredicted by the phase screen simulations with L = ∞ and l = 0 . From Fig. 4 we clearly observe that the meantransmissivity in the downlink channel predicted by all theconsidered channel modeling techniques are similar. Thiscan be explained by the fact that the main source of lossin a downlink channel is diffraction loss. For the uplinkchannel, we observe that the mean transmissivities predictedby the phase screen simulations with L = ∞ and l = 0 match the mean transmissivities predicted by the generalizedchannel modeling technique. Such an observation is reasonablesince [32] indeed assumes L = ∞ and l = 0 .An interesting observation from Fig. 4 is that the meanlosses predicted with a finite outer scale and a non-zero innerscale are lower than the mean losses predicted with an infiniteouter scale and a zero inner scale. Such an observation canbe explained mainly by the fact that the presence of a finiteouter scale reduces the amount of beam wandering and long-term beam spreading [17]. This observation does not refute theconventional wisdom that the channel loss in the uplink chan-nel is higher than the channel loss in the downlink channel.However, this observation does indicate that the disadvantageof an uplink channel may be overestimated in some models.We believe that setting a finite outer scale and a non-zero innerscale (according to the empirical Coulman-Vernin profile) ismore relevant (rather than simply setting L = ∞ and l = 0 )when studying the atmospheric propagation of light through asatellite-based channel. Therefore, in the rest of this work, wewill utilize the results from the phase screen simulations thatadopted a finite outer scale and a non-zero inner scale. Fig. 4: The mean turbulence-induced loss ¯ T turb [ dB ] predictedby different channel modeling techniques. By “this work” wemean the phase screen simulations we have undertaken. Theparameters are given in Table I, with w = 15 cm and r sat = r gs = 1 m. Recall, a higher ¯ T turb in dB corresponds to higherloss. D. Ground-to-satellite state transmission
Using our phase screen simulations we model an uplinkand a downlink channel with the characteristics presented inTable I. We consider that r sat = r gs in order to focus ouranalysis in the turbulence induced loss. We do note, that in arealistic satellite communications deployment it is expectedthat the aperture of the ground station is larger than thesatellite’s aperture (see later calculations). However, settingthe apertures constant in the first instance allows for a moredirect comparison of the effects of turbulence on the links.The model returns the probability distribution function (PDF)of the loss for each channel, as seen in Fig. 5. The PDFof the downlink channel is extremely narrow compared tothe PDF corresponding to the uplink channel. This is due tothe asymmetry of the interaction between the beam and theatmosphere, as explained above. The scintillation index of theLO is computed by simulating the propagation of a strongbeam corresponding to the LO. The scintillation index valuesare several orders of magnitude larger for the uplink relativeto the downlink.Due to the fluctuating nature of the uplink and downlinkchannels we need to consider ensemble-averages when com-puting the fidelity of the transmitted states [47]. The requiredanalysis can be derived as in the non-fluctuating channel if wedefine an effective transmissivity T f , and an effective excessnoise (cid:15) f , as T f = √ T , (cid:15) f = Var( √ T ) T f V sig + (cid:15) ¯ T , (31)
Var( √ T ) = ¯ T − √ T , Fig. 5: Probability distribution functions of the transmissivityfor the satellite communications channels, uplink (red) anddownlink (blue). The parameters of the channels are given inTable I, with ζ = 0 o , w = 15 cm, and r sat = r gs = 1 m.with the mean values computed as ¯ T = (cid:90) T p ζ ( T ) dT, √ T = (cid:90) √ T p ζ ( T ) dT, (32)with p ζ ( T ) the PDF of T for a given ζ . We present in Fig. 6the properties of the downlink and uplink channels obtainedusing the phase screen simulations. Following Eqs. (26, 30),the value of (cid:15) f is proportional to the variance of the quadraturesof the quantum states transmitted through the channel. Forthis reason we show on the plot the value of (cid:15) f with a fixed V sig = 1 , to give a fair comparison between the two channels,but we emphasize that this parameter will change in ourcalculations below. We observe that, as expected, losses arehigher (i.e. larger effective transmissivity when stated in dB)for direct transmission. Moreover, the value of (cid:15) f for the directchannel is one order of magnitude greater than the value forthe teleportation channel. This is a direct consequence of thevariations in the intensity for both the quantum signal andthe LO. We do not show the results for direct transmissionmodelled for an uplink with L = ∞ and l = 0 , but we findthat (cid:15) f ≈ . for ζ = 0 o , meaning such a channel is inadequatefor the transmission of quantum states.Using the values of T f and (cid:15) f , obtained from the numericalsimulations, we then compute the fidelity of teleportation anddirect transmission of coherent states. The values of g and V are optimized relative to the loss of the teleportation channel tomaximize the mean fidelity. For the loss values anticipated forthe teleportation channel, we observe that the optimal value ofV is in the range 1 to 1.5, and the optimal value of g is in therange 1 to 1.1. The results, presented in Fig. 7, show that theteleportation channel has a significant advantage over directtransmission. We see that direct transmission is only capableof overcoming the classical limit for a reduced alphabet of σ = 2 , and low zenith angles up to o . On the other hand,the teleportation channel exceeds the classical limit for alarger range in the alphabet, and for a wide range of zenithangles. This shows that one can indeed avoid, to a significantextent, the detrimental effects of the direct uplink channel via Fig. 6: Ground-to-satellite properties for the direct transferchannel and for the teleportation channel, shown for V sig = 1 .The parameters of the channels are given in Table I, with w = 15 cm and r sat = r gs = 1 m. For the teleportationchannel the entangled resource is distributed via the downlink.The left axis (blue) corresponds to the effective transmissivity,while the right axis (red) corresponds to the effective excessnoise. Recall, a higher T f in dB corresponds to higher loss.Fig. 7: Mean fidelities for ground-to-satellite transfer via directtransmission and via teleportation, shown for different valuesof σ . The channels parameters adopted are given in Table I,with w = 15 cm and r sat = r gs = 1 m. The shaded area inred indicates the region where the fidelity can be achievedby classical communications only. The direct transmission for σ = 10 , result in mean fidelities < . for all zenithangles.a teleportation using an entangled resource distributed via thedownlink channel. We note that the values of σ considered hereencompass the ranges required to undertake high-throughputCV-QKD [16].
1) Asymmetric apertures:
A stated earlier, in the calcula-tions just described we assumed that r gs = r sat , in order tofocus our analysis in the turbulence induced loss. However,in many satellite deployments it is expected that r sat < r gs .In such a case, use of the teleportation channel in the mannerwe have described would present an even higher advantageover the direct transmission channel. For example, we findthat for a space communications setting as in Table I, with ζ = 0 o , w = 15 cm and r gs = 50 cm, the downlink channelincurs a loss of ≈ dB. Meanwhile, under the same values,but with r sat = 20 cm, the uplink channel incurs a higherloss of ≈ dB. This means that the fidelity obtained usingthe teleportation channel in this setting is approximately 0.6,while the fidelity via direct transmission would be well belowthe classical limit. It is therefore important to emphasize, thatour detailed calculations most likely represent a lower boundon the actual gain in uplink communications of many futuresatellite missions.IV. CV T ELEPORTATION WITH NON -G AUSSIAN O PERATIONS
A great deal of recent research has been focused on the pho-tonic engineering of highly non-classical, non-Gaussian statesof light, aiming to achieve enhanced entanglement and otherdesirable properties. Indeed, non-Gaussian features are essen-tial for various quantum information tasks, such as entangle-ment distillation [48]–[55], noiseless linear amplification [56]–[61], and quantum computation [62]–[65]. In entanglementdistillation and noiseless linear amplification, non-Gaussianfeatures are a requirement due to the impossibility of distilling(or amplifying) entanglement in a pure Gaussian setting [66].In universal quantum computation, non-Gaussian features areindispensable if quantum computational advantages are to beobtained [67].Non-Gaussian operations, which map Gaussian states intonon-Gaussian states, are a common approach to deliveringnon-Gaussian features into a quantum system. At the core ofnon-Gaussian operations is the application of the annihilationoperator A and the creation operator A † . There are two basictypes of these operations, namely photon subtraction (PS) andphoton addition (PA), which apply A and A † to a state,respectively. Both operations have been shown to enhance theentanglement of TMSV states (e.g., [68]–[70]). Various studieson combinations of PS and PA have also been undertaken (e.g.,[71]–[74]). A specific combination, photon catalysis (PC), isof particular research interest. Instead of subtracting or addingphotons, PC replaces photons from a state, and is known tosignificantly enhance the entanglement of TMSV states undercertain conditions (e.g., [75], [76]). If TMSV states are in factshared between a satellite and a ground station, it is naturalto ask whether non-Gaussian operations can be adopted atthe ground station to further facilitate satellite-based quantumteleportation. A. Non-Gaussian states and non-Gaussian operations
A simple experimental setup for realizing non-Gaussianoperations consists of beam-splitters and photon-number-detectors. For example, as is depicted in Fig. 8a, an inputstate interacts with an ancilla Fock state | N (cid:105) at a beam-splitter with transmissivity T b . If M photons are detected inthe ancilla output the operation has succeeded. In practice,the probability of success of a non-Gaussian operation isan important parameter to consider. In this regard, single-photon non-Gaussian operations ( M, N ∈ { , } ) usually Fig. 8: The experimental setups for (a) a wide class ofnon-Gaussian operations, (b) photon subtraction, (c) photonaddition and (d) photon catalysis.have the highest success probability for a given type of non-Gaussian operation [70], making them the best candidatesfor practical implementation. Therefore, in this work we willrestrict ourselves to the non-Gaussian operations with single-photon ancillae and single-photon detection (i.e., Fig. 8b, c,and d).In the Schrodinger picture, the transformation of the non-Gaussian operations described above can be represented by anoperator [77] O = (cid:104) M | U ( T b ) | N (cid:105) , (33)where U ( T b ) =: exp (cid:110) ( (cid:112) T b − (cid:0) A † A + B † B (cid:1) + (cid:0) AB † − A † B (cid:1) (cid:112) (1 − T b ) (cid:111) : , (34)is the beam-splitter operator, : · : means simple ordering (i.e.,normal ordering of the creation operators to the left withouttaking into account the commutation relations), and A and B are the annihilation operators of the incoming state and theancilla, respectively. Using the coherent state representationof the Fock state, | N (cid:105) = 1 √ N ! ∂ N ∂α N exp (cid:0) α B † (cid:1) | (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) α =0 , (35)we can obtain the following compact forms for the operatorsfor PS ( N = 0 , M = 1 ), PA ( N = 1 , M = 0 ), and PC( N = 1 , M = 1 ) [78], respectively, O PS = (cid:114) − T b T b A (cid:112) T b A † A , O PA = − (cid:112) − T b A † (cid:112) T b A † A , O PC = (cid:112) T b (cid:18) T b − T b A † A + 1 (cid:19) (cid:112) T b A † A . (36)Suppose a non-Gaussian operation O ∈ { O PA , O PS , O PC } isto be performed to a state. Let Ξ in be the density operator ofthe state. The resultant state after the operation can be writtenas Ξ out = 1 N OΞ in O † , (37)where N = Tr (cid:8) OΞ in O † (cid:9) is a normalization constant,which is also the probability of success of the non-Gaussianoperation. Fig. 9: CV teleportation with non-Gaussian operations per-formed at the ground station.
B. CV teleportation protocol with non-Gaussian operations
In this section, we study the use of non-Gaussian operationsin the protocol of CV quantum teleportation proposed by [19].The deployment of the protocol over satellite channels hasbeen discussed in previous sections, so we will only describeour modification in this section. Our modified protocol isillustrated in Fig. 9, where we assume the satellite and theground station already share some TMSV states that have beendistributed over the noisy channel. Before teleportation begins,the ground station will perform non-Gaussian operations to thelocal mode stored at the station. The resultant non-Gaussianstates shared between the satellite and the ground station willbe used as the entangled resource for teleportation.Similar to before, we will use the fidelity given by Eq. (8)as the metric to evaluate the effectiveness of our modifiedCV teleportation protocol. To determine the fidelity we needto derive the CFs of the non-Gaussian states. We begin thederivation with the CF of the entangled state Ξ shared betweenthe ground station and the satellite. This CF, which we repeathere for completeness, can be written as χ (cid:48) TMSV ( ξ A , ξ B ) = exp (cid:20) −
12 ( (cid:15) + 1 − T ) | ξ B | (cid:21) × χ TMSV ( ξ A , √ T ξ B ) , (38)where again (cid:15) is the channel excess noise, T is the channeltransmissivity, and χ TMSV ( ξ A , ξ B ) is the CF for the initialTMSV state prepared by the satellite – which is given byEq. (4). On performing PS to mode B of Ξ , the unnormalized CF of the resultant state is given by k PS ( ξ A , ξ B ) = Tr (cid:110) O PS ΞO † PS D ( ξ A ) D ( ξ B ) (cid:111) = T b − T b exp (cid:18) | ξ B | (cid:19) × ∂ ∂ξ B ∂ξ ∗ B (cid:20) exp (cid:18) − | ξ B | (cid:19) f ( ξ A , ξ B , (cid:112) T b ) (cid:21) , (39)where f ( ξ A , ξ B , (cid:112) T b ) = (cid:90) dξ π (1 − T b ) χ (cid:48) TMSV ( ξ A , ξ ) × exp (cid:20) T b T b −
1) ( | ξ | + | ξ B | ) (cid:21) × exp (cid:20) √ T b T b − ξ B ξ ∗ + ξ ∗ B ξ ) (cid:21) , (40) and ξ B and ξ ∗ B are independent variables.For PA and PC, the CF of the state after the non-Gaussianoperations can be obtained in a similar fashion. For PA theunnormalized CF is given by k PA ( ξ A , ξ B ) =( T b −
1) exp (cid:18) − | ξ B | (cid:19) × ∂ ∂ξ B ∂ξ ∗ B (cid:20) exp (cid:18) | ξ B | (cid:19) f ( ξ A , ξ B , (cid:112) T b ) (cid:21) . (41)For PC, the unnormalized CF is more involved, and is givenby k PC ( ξ A , ξ B ) = q exp (cid:18) | ξ B | (cid:19) ∂ ∂ξ B ∂ξ ∗ B (cid:26) exp (cid:0) −| ξ B | (cid:1) × ∂ ∂ξ B ∂ξ ∗ B (cid:20) exp (cid:18) | ξ B | (cid:19) f ( ξ A , ξ B , (cid:112) T b ) (cid:21)(cid:27) − q exp (cid:18) | ξ B | (cid:19) ∂∂ξ B (cid:26) exp (cid:0) − ξ B | (cid:1) × ∂∂ξ ∗ B (cid:20) exp (cid:18) | ξ B | (cid:19) f ( ξ A , ξ B , (cid:112) T b ) (cid:21)(cid:27) − q exp (cid:18) | ξ B | (cid:19) ∂∂ξ ∗ B (cid:26) exp (cid:0) −| ξ B | (cid:1) × ∂∂ξ B (cid:20) exp (cid:18) | ξ B | (cid:19) f ( ξ A , ξ B , (cid:112) T b ) (cid:21)(cid:27) + f ( ξ A , ξ B , (cid:112) T b ) , (42)where q = T b − T b .Additionally, we also investigate the sequential use of PSand PA. We assume the two non-Gaussian operations adoptthe same beam-splitter transmissivity. For the scenario of PSfollowed by PA (PS-PA), the unnormalized CF is given by k PS − PA ( ξ A , ξ B )= q exp (cid:18) | ξ B | (cid:19) ∂ ∂ξ B ∂ξ ∗ B (cid:26) exp (cid:0) −| ξ B | (cid:1) × ∂ ∂ξ B ∂ξ ∗ B (cid:20) exp (cid:18) | ξ B | (cid:19) f ( ξ A , ξ B , T b ) (cid:21)(cid:27) . (43)The unnormalized CF for PA followed by PS (PA-PS) is givenby k PA − PS ( ξ A , ξ B )= ( T b − exp (cid:18) − | ξ B | (cid:19) ∂ ∂ξ B ∂ξ ∗ B (cid:26) exp (cid:0) | ξ B | (cid:1) × ∂ ∂ξ B ∂ξ ∗ B (cid:20) exp (cid:18) − | ξ B | (cid:19) f ( ξ A , ξ B , T b ) (cid:21)(cid:27) . (44)The normalized CFs after the non-Gaussian operations aregiven by χ x ( ξ A , ξ B ) = 1 k x (0 , k x ( ξ A , ξ B ) , (45)where x ∈ { PS , PA , PC , PS − PA , PA − PS } . For compact-ness, the expressions for the CFs above are not shown here. Fig. 10: Mean fidelity vs. effective channel transmissivity,where r is the squeezing level (in dB) of the TMSV stateprepared by the satellite, and σ is the displacement varianceof the input coherent states. The shaded area in red indicatesthe region where the teleportation fidelity is achievable byclassical communications only. The effective excess noise isset according to Fig. 6 for T f (cid:62) dB and is . r ) × − otherwise. Recall, a higher T f in dB corresponds to higherloss. Fig. 11: Mean fidelity as a function of the displacementvariance of the coherent states σ and the squeezing parameter r of the TMSV state generated by the satellite. ζ is the satellitezenith angle. C. Results
We study the teleportation of coherent states using non-Gaussian entangled resource states, of which the CF is chosenfrom Eq. (45) depending on which non-Gaussian operation isperformed to the mode at the ground station. We use the meanfidelity ¯ F given by Eq. (14) as our performance metric. Weadopt the effective channel loss and the effective excess noiseobtained from the phase screen simulations (see Fig 6).In Fig. 10 we compare the maximized ¯ F offered by variousnon-Gaussian operations against the effective channel loss T f [dB] . At each effective channel loss level, the maximizationof ¯ F is performed on the parameter space consisting of thetransmissivity T b of the beam-splitter in the non-Gaussianoperations and the gain parameter g of the teleportationprotocol. For comparison, the case without any non-Gaussianoperation is also included (i.e., the black curve in the figure).In each sub-figure, r is the squeezing parameter of the TMSVstate generated by the satellite and σ is the variance for thedistribution of the displacement of the input coherent state(defined by Eq. (13)). For r the conversion from the lineardomain to the dB domain is given by r [dB] ≈ . r . Wesee from Fig. 10 that among the five non-Gaussian operationswe have considered, only PA-PS provides enhancement in ¯ F .PA always provides larger ¯ F than PS. When r is 5 dB, PA-PS provides the largest ¯ F over the entire range of effectivechannel loss we have considered.We next compare the teleportation scheme with the non- Gaussian operation that provides the most improvement. Thatis, teleportation with PA-PS compared to the direct transmis-sion scheme. The mean fidelity for the direct transmissionscheme is given by Eqs. (12) and (14). The results areillustrated in Fig. 11, where we compare the maximized ¯ F against r and σ for different satellite zenith angles ζ . Againthe maximization of ¯ F is performed over the parameter spaceof { T b , g } . We see in comparison to the original teleportationscheme (i.e. the TMSV case), the scheme with PA-PS canachieve the highest ¯ F for the entire range of σ we haveconsidered. PA-PS can also reduce the requirement on r of theTMSV state prepared by the satellite (to reach a certain levelof fidelity). We also notice that when σ is fixed, ¯ F providedby the original teleportation scheme decreases when r exceedsa certain value. The same trend is observed for the PA-PSscheme.In summary, we have shown in this section how non-Gaussian operations at the ground station can enhance thefidelity for teleporting coherent states by up to . Inaddition, using such non-Gaussian operations, we have shownhow the demand on the squeezing of the TMSV state preparedby the satellite can be reduced.V. D ISCUSSION
The focus of the present work is the use of CV teleportationchannels for the teleportation of coherent states, and the useof non-Gaussian operations to enhance the communicationoutcomes. However, it is perhaps worth briefly discussing theflexibility of our system in regard to the transfer of otherquantum states in the uplink, and the use of additional quantumoperations. It will also be worth discussing differences andadvantages of our system relative to DV-only systems - afterall the only currently-deployed qiantum satellite system is onesolely based on DV states [3].
A. Other Quantum States and Operations
Our scheme is actually applicable to any type of quantumstate - even DV based systems. Some DV systems, e.g.polarization, may need to be transformed first into the numberbasis. In number-basis qubit-encoding, vacuum contributionsenter directly, similar to what we have discussed earlier. Insuch schemes, the use of the TMSV entangled teleportationchannel (a CV channel) can be utilized as the resource toteleport the DV qubit state [80], and so our proposed schemeoperates directly. Our scheme also operates directly on morecomplex quantum states such as hybrid DV-CV entangledstates - even on both components of such states [81]. Thisflexibility of CV entanglement channels over DV entanglementchannels is another advantage offered by our scheme.We also note, the non-Gaussian operations we have con-sidered in this work represent a form of CV entanglementdistillation [82]. There are, of course, many other forms ofCV entanglement distillation we could have considered at theground receiver (or on-board the satellite) - we have only It is straightforward to alter polarization encoding into number-basisencoding or other forms of qubit encoding, e.g. [79]. investigated the simplest-to-deploy quantum operations. Astechnology matures (e.g. the advent of quantum memory),more sophisticated quantum operations (and entangled re-sources) will become viable as a means of further enhancingteleported uplink quantum communications; most likely out-competing any advances in the uplink-tracking technology thatcould assist direct communication. In principal, the teleporta-tion fidelity could approach one.
B. DV Polarization - Micius
We discuss now, known results from the LEO Miciussatellite in the context of teleportation of DV-polarizationstates from the ground to the satellite [3]. Different fromour system model, the teleportation experiment reported in[3] does not use the downlink to create the entanglement,but rather utilizes the uplink as a means of distributing theentanglement. Therefore, the advantage of using the superiordownlink channel is not afforded to that experiment. Fromthe aperture used in [3] - a 6.5cm radius transmitter and a15cm radius receiver telescope, a turbulence induced loss of30dB is obtained at a 500km altitude, the zenith distanceof Micius. This translates into a beam width of 10m at thereceiver plane (30m beam width and 40dB losses at 1400kmis also reported). Nonetheless, the experiment still clearlydemonstrates a fidelity of 0.8 for the teleportation of single-qubit encoded in single-polarized photons (well above theclassical fidelity limit of 2/3 for a qubit), proving the viabilityof teleportation over the large distances tested.In the context of the main idea presented in this work,use of the downlink channel (to create the entanglementchannel) in an experimental set up similar to [3] wouldbe beneficial mostly in the context of an increased rate ofteleportation, rather than an increase in fidelity. Our phase-screen simulations suggest (reversing the aperture sizes for afair comparison, that is, 6.5cm radius transmitter at the satelliteand 15cm radius on the receiving aperture) would result ina turbulence induced loss of 25dB, which would lead to afactor of ∼ − enhancement in the teleportation rate relativeto direct transmission. Of course, if we increase the groundreceiver aperture, larger enhancements could be found. Thefact that it is much easier to deploy large telescopes on theground, compared to in space, is another advantage of ourteleportation scheme.Let us briefly outline the main differences in DV-polarization teleportation relative to CV teleportation. In DV-polarization implementations the vacuum contribution does notenter the teleportation channel in the same manner it does in aCV entangled channel. In the DV-polarization channel the lossenters our calculations primarily via two avenues. One avenueis simply through the different raw detection rates set by thedifferential evolution of the beam profiles in the downlinkand uplink. As discussed, in the downlink the beam widthat the receiver will be smaller than in the uplink. For a givenreceiver aperture this translates into an increased detection rate Classical pre-processing via pre-selection based on transmissivity estima-tion using classical beams or post-processing based on measurement outcomesmay assist these operations [11]. in and by itself. We can use the phase screen calculationsdescribed earlier (e.g. Fig. 4 for equal transmit and receiveapertures of 1m) to determine this rate increase. The secondavenue is a manifestation of the vacuum through dark counts inthe photodetectors. In real-world deployments of teleportationthrough long free-space channels [3], [83], [84] a coincidencecounter is used to pair up entangled photons, typically with atime-bin width of 3ns [3]. Due to the presence of a vacuumin almost all time-bins, only of order 1 in a million events aretriggered as a photon-entangled pair. Dark counts in the bestphotodetectors are currently in the range of 20Hz. However,in orbit, and because of stray light, combined backgroundcounts are more likely to be of order 150Hz [3]. A backgroundcount in one time-bin will lead to a false identification ofan entangled pair generated between the satellite and groundstation. This is different to the CV scenario where each timebin is assumed to contain a pulse - albeit one contaminatedwith a vacuum contribution.Another major difference in DV vs. CV teleportation sys-tems is contamination caused by higher order terms in theproduction of the (single) photons that are to be teleportedin the DV systems. The optimal probability of single-photonemission (set by the user) decreases with increasing loss [85].This is due to a lower probability leading to a reduction inthe number of double pair emissions that lead to flawed Bellmeasurements. This effect is counteracted by the strength ofthe source that emits the two-photon entangled pairs (set by theuser) - the optimal value of which increases with increasingloss. These two parameters can be jointly optimised for theloss anticipated, leading to asymmetric parameter settings forthe downlink and uplink teleportation deployment [85]. Anadditional issue relevant to DV-polarization teleportation ispartial photon distinguishably at the Bell state measurementwhich leads to a drop in interference at the beam splitter, and,of course, polarization errors (in production or measurement).The relative importance of all the above terms for free spaceteleportation from ground to satellite are considered to bebackground counts (4 % ), higher-order photon emission (6 % ),polarization errors (3 % ), and photon indistinguishability (10 % )[3]. In a series of experiments over 100km [84], 143km [84]and ground-to-satellite [3] a fidelity of teleportation in therange . − . was obtained by all.Another issue in discussing DV relative to CV teleportationis the classical teleportation fidelity of both systems. That is,the fidelity that can be achieved by purely classical informationbeing communicated across the channel (e.g. the classicalinformation representing the outcome of a particular quantummeasurement). This classical information allows the receiver topartially reconstruct the desired quantum state. In the coherentstate teleportation discussed earlier this classical fidelity was1/2. However, for DV qubits it is 2/3. This fact translates intoa less useful range of teleportation fidelity for the DV scenariorelative to the CV scenario. Finally, it is worth noting that theBell state measurements used currently in DV systems are only50 % efficient. This is a consequence of the fact that Bell statemeasurements based on linear optics can only discriminatebetween two of the four Bell states. Although, in principal,full Bell state measurements in the DV basis are possible (eg via ancilla and two-qubit interactions), no real-worldimplementation of the latter exist - all current deploymentsutilize a linear-optics-only solution [3], [83], [84]. C. Future work
We recognise other input states may lead to an enhancedfidelity in both the direct uplink transmission channel and viathe resource CV teleportation channel. It is likely that in thesecircumstances we will again find some channel parametersettings where teleportation leads to better communication out-comes. However, coherent states and TMSV states are easy toproduce and are considered the “workhorses” of CV quantumcommunications, and are therefore the focus of this work. Wealso recognize more sophisticated set-ups could be considered,such as the use of classical feedback on channel conditionsto optimise the parameters of the input states (e.g. squeezinglevels and amplitudes). However, such improvements are at thecost of a considerable increase in implementation complexity.Again, it is likely that in these circumstances some channelparameter settings will provide for communication gains viateleportation relative to direct transfer. Future investigationsthat properly identify such channel settings would be useful.Our study has also been limited in terms of the aperturesettings we have adopted. We have used what we considerto be aperture settings likely deployable in next-generationsystems which take space-based quantum communication tothe production phase. Further study of possible teleportationgains for a wider range of aperture settings would also beuseful. VI. C
ONCLUSIONS
In this work, we have investigated the use of a CV teleporta-tion channel, created between a LEO satellite and a terrestrialground station, as a means to enhance quantum communicationin uplink satellite communications. Such communications areexpected to be very difficult in practice due to the severeturbulence-induced losses anticipated for uplink satellite chan-nels. Our CV teleportation channel was modelled using thesuperior (lower loss) downlink channel from the satellite as ameans to distribute one mode of an in situ satellite TMSV stateto the terrestrial station - a form of long-range entanglementdistribution that may become mainstream in coming years. Ourresults showed that use of this teleportation channel for uplinkcoherent state transfer is likely to be much superior to coherentstate transfer directly through the uplink channel. The use ofnon-Gaussian operations at the ground station was shown tofurther enhance this superiorly. Given the flexibility of CVteleportation as a means to invoke all forms of quantum statetransfer beyond just coherent state transfer, it could well bethe scheme introduced here could become the de facto choicefor all future uplink quantum communication with satellites.R
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