Quantum teleportation through atmospheric channels
QQuantum teleportation through atmosphericchannels
K Hofmann , A A Semenov , , , W Vogel , M Bohmann , Arbeitsgruppe Theoretische Quantenoptik, Institut f¨ur Physik, Universit¨atRostock, D-18051 Rostock, Germany Institute of Physics, NAS of Ukraine, Prospect Nauky 46, UA-03028 Kiev,Ukraine Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Vul.Metrolohichna 14b, UA-03143 Kiev, Ukraine INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, ItalyE-mail: [email protected]
E-mail: [email protected]
Abstract.
We study the Kimble-Braunstein continuous-variable quantumteleportation with the quantum channel physically realized in the turbulentatmosphere. In this context, we examine the applicability of different strategiespreserving the Gaussian entanglement [Bohmann et al., Phys. Rev. A ,010302(R) (2016)] for improving the fidelity of the coherent-state teleportation.First, we demonstrate that increasing the squeezing parameter characterizing theentangled state is restricted by its optimal value, which we derive for realisticexperimentally-verified examples. Further, we consider the technique of adaptivecorrelations of losses and show its performance for channels with large squeezingparameters. Finally, we investigate the efficiencies of postselection strategies independence on the stochastic properties of the channel transmittance.27 September 2019, Submitted to:
Phys. Scr.
Keywords : Atmospheric quantum optics, quantum teleportation, fluctuating losses,free-space channels a r X i v : . [ qu a n t - ph ] S e p uantum teleportation through atmospheric channels
1. Introduction
The no-cloning theorem [1, 2] is a fundamentalconcept of quantum physics, which states that itis impossible to copy any unknown quantum statewithout destroying it at the origin. Consequently,quantum states can only be transfered from one systemto another. Such a process is usually referred to asquantum teleportation [3, 4], which is an essentialbuilding block of quantum networks. The originalprotocol [3], which has been proposed for teleportingstates of two-level systems—qubits, has first beenexperimentally implemented by two groups in 1997[5, 6]. In a recent experiment, this protocol hasbeen performed in the context of satellite mediatedquantum communications for a ground-to-satellitechannel over distances of up to 1400km [7]. Thisexperiment demonstrates the huge efforts made inorder to establish teleportation channels for quantuminformation tasks on a global scale and underlines thegeneral feasibility of such an endeavor.Continous-variable (CV) based quantum informa-tion processing [8] is an alternative approach, whosemain idea consists in using systems with infinite-dimensional Hilbert spaces instead of qubits. Remotenodes of quantum networks can also be connected byusing CV communication protocols. Quantum tele-portation in the CV regime can be realised with theBraustein-Kimble (BK) protocol [9], which has beenfirst experimentally implemented by Furusawa et al.[10]. In this case, one uses Gaussian entanglement[11, 12, 13] shared between the communication partiesas the main resource. Gaussian entanglement is alsoa recourse for other communication protocols such asCV entanglement swapping [14].Losses and other noise in optical channelsmay lead to the loss of Gaussian entanglement[15]. Particularly, Gaussian entangled states can besubdivided into classes with respect to their stabilityto constant losses [16]. Particularly, a large classof states, whose entanglement can be verified withthe Duan-Giedke-Cirac-Zoller (DGCZ) criterium [17],always preserves Gaussian entanglement under anyconstant loss conditions. An example of such DGCZ-states is the two-mode squeezed vacuum (TMSV) state,whose application has been proposed in the originalversion of the BK protocol. Consequently, the CVquantum teleportation can be performed, in principle,for entanglement shared through lossy channels [18]. However, the teleportation quality in the presence oflarge losses can be very low.In many situations, free-space channels havesome practical advantages in comparison with opticalfibers due to their mobility, possibility of satellite-mediated communications, etc. The establishment ofsuch links could facilitate global quantum networks,which eventually may lead to a quantum Internet[19]. Atmospheric links for quantum light werefirst tested in ground-to-ground experiments [20,21, 22, 23, 24, 25, 26]. Further implementationscould demonstrate the feasibility of satellite-mediatedquantum communication using small-scale experiments[27, 28, 29, 30]. Ultimately, experiments with satelliteshave been reported [31, 32, 33, 34, 35, 36, 37, 38, 7,39], which include the successful implementation ofsatellite-based quantum key distribution [36, 39] andquantum state teleportation [7].Atmospheric free-space channels differ drasticallyfrom constant loss optical fiber links. Due toatmospheric turbulent flows—causing temporal andspatial fluctuations of the optical properties of theatmosphere—the transmittance through such linksvaries in a random fashion. This effect can bedescribed by an appropriate probability distributionof the transmittance (PDT) [40]. Depending on thechannel characteristics, advanced PDT models havebeen introduced [41, 42, 43, 44], which accuratelydescribe experiments [45, 42, 43] and can account fordifferent weather conditions [43]. Note that theseatmospheric channel models can also be simulated inin-lab experiments; see, e.g., [46, 47]. Such simulationscan be an important tool for finding and testingquantum information applications for realistic free-space channels.The distribution of Gaussian entanglement throughatmospheric channels has been rigorously studied in[48]. Some conclusions of this work are of impor-tance for the present consideration of CV teleporta-tion through the atmosphere. First, the best result forthe entanglement sharing can be reached if the entan-gled state does not have coherent displacements—thisis exactly the scenario, which is used in the originalBK protocol. Second, high values of squeezing of theTMSV states are in general not useful for such channelsas they can lead to the total loss of Gaussian entangle-ment after the propagation through the atmosphere.There exists a maximal value of the squeezing param-eter beyond which Gaussian entanglement distributed uantum teleportation through atmospheric channels
2. Braunstein-Kimble protocol with losses
In this section, we will provide the framework for ouranalysis of the influence of atmospheric losses on CVteleportation. Therefore, we will briefly recall the BKprotocol [9] for the teleportation of CV quantum states.Furthermore, we show how the teleportation fidelity—the figure of merit in such a protocol—is influencedby constant losses. We demonstrate that introducingadditional losses can lead to an improved teleportationfidelity under certain circumstances such as strongsqueezing or high losses.
In figure 1, the basic scheme of the BK teleportationprotocol [9] is shown. It exists of two parties, Aliceand Bob, who want to teleport an unknown state fromAlice to Bob. The unknown input state at Alice’sside is described by the characteristic function C I ( β ).Note that the characteristic function is the Fouriertransform of the Wigner function W I ( α ) and containsall the information about the quantum state. In thereminder of this paper, we will work with characteristicfunctions rather than with Wigner functions due totechnical convenience; cf. also [53]. We denote theteleported output state at Bob’s side as C O ( β ). Toperform the teleportation Alice and Bob need to sharean entangled quantum state, which is sent along themodes A and B , cf. figure 1. In the BK protocol,the entangled state is a two-mode squeezed vacuumor Einstein-Podolsky-Rosen (EPR) state which can bedescribed by the characteristic function C EPR ( β A , β B ) = exp (cid:0) β ∗ A β A β ∗ B β B (cid:1) V β A β ∗ A β B β ∗ B (1)with the covariance matrix V = cosh (2 r ) 0 0 − sinh(2 r )0 cosh (2 r ) − sinh(2 r ) 00 − sinh(2 r ) cosh (2 r ) 0 − sinh(2 r ) 0 0 cosh (2 r ) . (2)Here, r is the squeezing parameter of the two-modesqueezed vacuum state.For our considerations, we extend the originalproposal of Braunstein and Kimble to the realisticcase of losses. The losses in the modes A and B arecharacterized via the transmission coefficients T A and T B . The (intensity) losses are then given by 1 − T .Note that the lossless case corresponds to T A = T B =1.CV teleportation protocols under constant losses havealready been studied; see, e.g., [18, 50, 51].In the first step, the input state C I ( β ) andAlice’s part of the entangled state, i.e. mode A, uantum teleportation through atmospheric channels EPRI A Ba b OS R D a D b q a p b C I ( β ) C O ( β )C EPR ( β A , β B )Alice BobClassical channel50:50 |0 T A T B |0 Figure 1.
Braunstein-Kimble teleportation scheme includinglosses. The aim is to teleport the input state C I ( β ) fromAlice to Bob. The two parties, Alice and Bob, share the twoentangled modes A and B of an two-mode squeezed vacuumstate (EPR state). Losses in mode A(B) are characterized bythe transmission coefficients T A(B) . Alice combines mode Awith the input state at a 50 : 50 beam splitter and performsa quadrature measurements of the resulting modes a and b viathe detectors D a and D b , respectively. The measurement results, q a and p b , are sent via a classical communication channel to Bob,who modifies mode B according to this results in order to obtainthe teleported output state. are superimposed with the help of a 50:50 beamsplitter. The resulting state possesses correlationsbetween the corresponding output modes a and b, andmode B; cf. figure 1. Now, Alice conducts homodynemeasurements of the quadratures q a = ( q + q A ) / √ p b = ( p − p A ) / √ α B (cid:55)→ α B + √ q a + ip b ). The output statethen reads as C O ( β ; r, T A , T B ) = C I ( β ) C G ( β ∗ , β ; r, T A , T B ) . (3)Here, C G ( β A , β B ; r, T A , T B ) is a Gaussian state of theform C G ( β A , β B ; r, T A , T B ) (4)= exp − (cid:0) β ∗ A β A β ∗ B β B (cid:1) V ( r, T A , T B ) β A β ∗ A β B β ∗ B with V ( r, T A , T B ) being a 4 × V ( r, T A , T B ) in terms of three 2 × A , B and C : V ( r, T A , T B ) = (cid:18) A CC † B (cid:19) . (5)The three 2 × A = (cid:2) T cosh (2 r ) + (cid:0) − T (cid:1)(cid:3) (cid:18) (cid:19) , (6) B = (cid:2) T cosh (2 r ) + (cid:0) − T (cid:1)(cid:3) (cid:18) (cid:19) , (7) C = − T A T B sinh (2 r ) (cid:18) (cid:19) . (8) Note that the product structure on the right-handside of (3) corresponds to a convolution of the Wignerfunction of the input state with a Gaussian function.This additional Gaussian factor resembles noise whichimpairs the teleportation and is determined by thefinite squeezing strength r and the losses in the systemcharacterized by T A and T B .In order to characterize the teleportation quality,one usually considers the fidelity [54, 55] betweenoriginal and teleported states. The fidelity quantifiesthe similarity of two quantum states and takes avalue between 0 and 1 corresponding to orthogonaland identical states, respectively. In the case ofcoherent state teleportation, teleportation fidelitieslarger than 1 / ρ I , the fidelity with the output state ˆ ρ O is definedas F = Tr (ˆ ρ I ˆ ρ O ). In terms of the correspondingcharacteristic functions this relation is given by F ( r, T A , T B ) = 1 π (cid:90) d β C I ( β ) C O ( − β ; r, T A , T B ) . (9)For the considered scenario, it depends on thesqueezing strength r and the two transmissioncoefficients T A ( B ) .As a particular case, the fidelity of coherent-stateteleportation can be directly expressed in terms of theblocks of the covariance matrix [18, 50] given in (6)-(8), F = 2 √ det E , (10)with E = 2 I + RAR + C † R + RC + B , (11)were I is the two-dimensional identity matrix, and R = (cid:18) − (cid:19) . (12)All the relevant information for the teleportationprocess about the entangled state and the channellosses are included in the covariance-matrix elements.Eventually, this yields the analytical expression for theteleportation fidelity including losses, F ( r, T B , T A ) = 24 + ( T + T ) (cosh (2 r ) − − T A T B sinh (2 r ) . (13) As long as the fidelity exceeds the classical limitof teleportation, F cl = 0 .
5, the protocol shows aquantum advantage in the considered scenario [58].As we are interested in the quantum advantage ofthe considered teleportation scenarios, we will plot thefidelities in this paper only from 0 . uantum teleportation through atmospheric channels . r and the twotransmission coefficients T A and T B . In the followingwe will analyze the dependence of the teleportationfidelity on (atmospheric) losses and develop strategiesfor minimizing the unwanted effects of these losses onthe teleportation result. Let us now discuss the influence of constant losses onthe teleportation fidelity in (13). We will analyze twocases. In the first case, we only consider losses in modeB and set the transmission coefficient T A to unity. Wewill call this the direct scheme, in which Alice maycontrol mode A in a lossless loop. In the second case,the transmission coefficient T A will be set to the valueof the transmission coefficient T B . This case will becalled the adaptive scheme as the loss in mode A isadapted to be the same loss as in mode B. r F . . . . . . . . . Figure 2.
Fidelity in dependence on the squeezing parameter.For the solid lines, T B is varied and T A is set to 1 .
0. For thedashed lines, T B is varied and T A is set to T B . The curvesare labeled with the corresponding value of the transmissioncoefficient T B . In figure 2, the teleportation fidelity is plotted independence on the squeezing parameter r , for differentlosses for the direct and adaptive schemes. For thedirect scheme (solid lines), the fidelity grows from 0 . T A = T B =1), the fidelity approachesunity with increasing squeezing parameter. Hence, weobserve that in the direct scheme with losses thereexists an optimal and finite squeezing value for whichthe best teleportation fidelity is obtained. The valueof this best teleportation fidelity and its corresponding squeezing value depend on the transmission coefficient T B . A higher transmission coefficient T B leads to ahigher possible fidelity but requires stronger squeezing.The optimal squeezing parameter which yields themaximal fidelity in the lossy case is given by r opt ( T A , T B ) = 12 arctanh (cid:18) T A T B T + T (cid:19) , (14)which reduces to r opt ( T B ) = 12 arctanh (cid:18) T B T (cid:19) , (15)in the case of losses in mode B only. From (14) weclearly see that in the uncorrelated-loss case ( T A (cid:54) = T B )there exists an optimal finite squeezing value forwhich the highest teleportation fidelity is reached.Increasing the squeezing parameter beyond r opt leadsto a reduction of the teleportation fidelity; cf. figure 2.Similar behaviors can also be observed in the case offluctuating losses (see figures 4 and 7) as we will showbelow.This can be explained by the fact that highersqueezing leads to a higher mean photon number inboth modes. Higher photon-number contributionsare, however, more sensitive towards losses as the n -th photon-number contribution scales with the n -thpower of the corresponding transmission coefficient.Therefore, increasing the squeezing parameter leadsto an overall stronger influence of the losses whichmanifests itself in a more pronounced asymmetry of thestate. The symmetry in the state is, however, essentialfor the quantum advantage in the teleportation. Thisexplains why there is a finite optimal squeezing valuein the direct scheme. More formally, this effect canbe understood by the factor C G ( β A , β B ; r, T A , T B ) inthe input-output relation (3) which is a characteristicfunction of a Gaussian distribution. The performanceof the teleportation increases when the variance of thisGaussian distribution is decreasing. Note that in theideal EPR case the distribution approaches a deltadistribution with zero variance for infinite r [9]. Forasymmetric losses ( T A (cid:54) = T B ), the variance attains itsminimum at the finite value r opt .Next, we consider the adaptive case in which thelosses in mode A are adapted to be the same as inmode B, i.e., T A = T B = T . Such an adaptive protocolcan always be realized by measuring the transmissioncoefficient in one channel and then artificially attenuatethe other channel to the measured level [48]. A similaradaptive scenarios for quantum teleportation has beenproposed in [59], however, from a different perspective.For different amounts of losses, the correspondingvalues of the teleportation fidelities are plotted in figure2 (dashed lines). In this case, the fidelity in (13)reduces to F ( r, T ) = 12 − T (1 − exp( − r )) . (16) uantum teleportation through atmospheric channels r to the upper limit F opt = 12 − T . (17)For an increasing transmission coefficient T this upperlimit also increases, but it is never higher than themaximum in the direct scheme. However, for a fixedsqueezing value the adaptive protocol can yield higherteleportation fidelities in comparison to the directscheme especially for higher losses. The crossing pointof the solid and dashed lines in figure 2, i.e., thesqueezing strength from which on the adaptive protocolperforms better than the direct teleportation, is givenby r ( T B ) = arctanh (cid:18) T B T B (cid:19) . Furthermore, the fidelity in the adaptive scenario neverdrops below the classical limit of 0 .
5, which doesnot hold for the direct scheme. Hence, we couldshow that establishing correlations in the losses canimprove the performance of the teleportation eventhough this implies to introduce additional losses. It isimportant to stress that the correlations between themodes, including correlations in the losses, can be moreimportant than the overall value of the transmittanceof the channel.
3. Braunstein-Kimble protocol withatmospheric channels
In this section, we consider the action of atmosphericlosses on the teleportation protocol. Therefore, werecall a model for the description of atmosphericlosses and apply this model to the BK teleportationprotocol. We extend the direct and adaptive schemesto the regime of atmospheric losses and analyzethe corresponding fidelities. Furthermore, we showhow postselection strategies can lead to an improvedteleportation fidelity. Finally, we study the case oftwo-way atmospheric channels in which both entangledmodes suffer from uncorrelated atmospheric losses.
In quantum optics, losses can be described by a virtualbeam splitter that superimposes the lossless state withvacuum noise. The transmission coefficient of thebeam splitter depends on the losses. Fluctuatingatmospheric channels can be modeled by such a beamsplitter for which the transmission coefficient is arandom variable [40]. The properties of the channelis then characterized by the probability distribution ofthe transmittance (PDT) [41, 42, 43, 44]. For a start, we consider atmospheric losses onlyin mode B. The output state of the quantum channelis obtained by averaging the fidelity in (13) over theatmospheric PDT P ( T B ). This means that the fidelityof the teleportation should also be averaged as¯ F ( r ) = (cid:90) dT B F ( r, T A , T B ) P ( T B ) . (18)Note that ¯ F ( r ) directly depends on the PDT P ( T B )and, hence, on the properties of the atmosphericchannel. This treatment hold for any turbulent free-space channel.In the following, we exemplarily work with thePDT of the elliptic beam model [42]. This modeltakes into account the deflection and deformation ofa Gaussian beam caused by turbulence in atmosphericchannels and shows good agreement with experimentalfree-space channels. In particular, we consider theexample of a 1 . C [60, 61, 62]. Thecorresponding distributions have average transmissioncoefficients (cid:104) T (cid:105) of 0 .
40, 0 .
70 and 0 .
84. Both distri-butions with the lower mean transmission coefficients,corresponding to stronger turbulence, have nearly thesame but displaced PDTs with a standard deviationof (cid:112) (cid:104) T (cid:105) − (cid:104) T (cid:105) = 0 . .
024 and is therefore much thinner. It is important tostress that the following considerations apply to anyPDT model.
As a first step, we consider the influence of fluctuatinglosses in atmospheric channels on the direct andadaptive teleportation schemes. We study the case inwhich mode B passes through a turbulent free-spacelink while mode A does not suffer from any losses, i.e. T A =1. For the three different turbulent strengths [cf.figure 3], the fidelity of the direct teleportation protocolis plotted in figure 4 (solid lines) in dependence on thesqueezing parameter. For this direct case, we observea similar behavior as discussed for the case of constantlosses, cf. also figure 2. For the adaptive channelprotocol, the channel transmittance in mode B has tobe monitored and the measured transmittance has tobe constantly adapted in mode A. Such an atmosphericadaptive channel protocol has been introduced in [48].The measurement of the channel transmittance can uantum teleportation through atmospheric channels T P ( T ) . . Figure 3.
Probability distribution of the atmospherictransmission coefficent given by the elliptic beam model [42], fordifferent atmospheric index-of-refraction structure constants C .The labels give the atmospheric index-of-refraction structureconstants in units of 10 − m − . Further channel parametersare given in Appendix A. be directly implemented in the procedure of balancedhomodyne detection [63, 64, 65] or can be measuredindependently with an intense reference light pulse.The average fidelity in dependence on thesqueezing parameter r for the direct and adaptiveschemes with different turbulence strengths (different C ) is shown in figure 4. A lower atmospheric index-of-refraction structure constant, C , results in a higheraverage fidelity for both schemes. Similar to the case ofconstant losses, the adaptive scheme does not improvethe maximum average fidelity. Furthermore, as seen infigure 2, the fidelity in the adaptive scheme will neverdrop below the classical limit. Therefore, the adaptivescheme is preferable in the case when the quantumchannel is realized with the relatively strong squeezing. The teleportation fidelity through free-space links canbe further improved by postselecting the events withhigh transmission coefficients. Such a procedurehas been theoretically analyzed for improving thetransmission of quadrature squeezing in [41] andexperimentally realized in [26]. Here we will study itsapplicability for the BK-CV teleportation protocol.Let us assume that we postselect only the eventswith the transmission coefficients T B ≥ T min , where T min is a certain postselection threshold. In this casethe PDT is modified to the form P ps ( T B ; T min ) = 1 E ( T min ) (cid:26) P ( T B ) T B ≥ T min T B < T min , (19)where E ( T ) = (cid:82) T dT (cid:48) P ( T (cid:48) ) is the PDT exceedance r F . . . . Figure 4.
Average fidelity ¯ F in dependence on the squeezingparameter r for the probability distributions in figure 3. Thesolid lines result from the direct scheme and the dashed linesresult from the adaptive scheme. The dashed-dotted line is theclassical limit. (complementary cumulative distribution function),which describes the total efficiency of the postselectionfor a given T min . A higher value of T min implies thatmore data is discarded. We can apply the postselectionto the direct and adaptive scheme by replacing theatmospheric probability distribution to improve theteleportation fidelity. r F . . . . Figure 5.
The average fidelity ¯ F in dependence on thesqueezing parameter r for a C of 1 . · − m − is shown.The monotone increasing curves result from the adaptive scheme.Dashed curves are with added postselection. The labelednumbers represent the chosen postselect threshold T min . Thedashed-dotted line shows the classical limit. In figure 5, we show the average fidelity independence on the squeezing parameter r for the directand adaptive schemes with postselection. For the uantum teleportation through atmospheric channels C = 1 . · − m − . For different C , the mainfeatures of the plot will not change. T min F Figure 6.
The solid (blue) line shows the postselectionefficiency, i.e., the PDT exceedance E , (left axis) in dependenceon the postselect threshold T min . The dashed (green) line andthe dashed-dotted (red) line show the fidelity (right axis) independence on T min for the adaptive and direct scheme for asqueezing parameter r = 1, respectively. We observe that the postselection improvesthe average fidelity and that it increases with thepostselection threshold T min . This improvement,however, comes with the disadvantage that thepostselection procedure implies that we have to discardpart of the data. In figure 6, the postselection efficiency E ( T min ) is shown together with the average fidelityfor the postselected direct and adaptive schemes independence on the postselection threshold T min . Weobserve that the teleportation fidelity can be improvedby means of postselection if one is willing to reducethe teleportation efficiency due to the reduction ofthe amount of teleported data. This is an importantfinding which allows to improve the teleportationperformance in free-space channels. Note that bypostselection it is even possible to increase the fidelityfrom classically achievable values ( ¯ F ≤ .
5) to valueswhich show a quantum advantage ( ¯
F > . F > .
5) in cases when alternative implementationswith constant losses fail. r F . . . . Figure 7.
Average fidelity ¯ F in dependence on the squeezingparameter r for the case of atmospheric losses in both modesA and B. The monotonously increasing curves result fromthe adaptive scheme, the others are obtained without theadaptive protocol. Dashed curves are with added postselection.The labeled numbers indicate the corresponding postselectionthreshold. The dashed-dotted line is the classical limit. We can extend the consideration of teleportationthrough atmospheric channels to the case whereboth modes, A and B, suffer from (uncorrelated)atmospheric losses. In this case, the transmissioncoefficient T A represents the atmospheric transmissioncoefficient of mode A which also fluctuates in arandom fashion. In the following, we consider the casein which both modes propagate through 1.6km-longatmospheric channels [45, 42]. Similar to the previousconsideration, we can apply an adaptive scheme to theprotocol. Here the adaptive scheme means, that weadjust the higher atmospheric transmission coefficientto the lower atmospheric transmission coefficient. Wecan furthermore apply postselection. In this case wedo not teleport if one of the atmospheric transmissionsis lower than a given limit.The joint PDT, P (cid:48) , of our adaptive schemecan be obtained straightforwardly from the initialdistribution P by mapping the random variables T a , T b (cid:55)→ min { T a , T b } [48] within the technique of orderstatistics [66], P (cid:48) ( T a , T b )= (20) δ ( T a − T b ) (cid:90) T a dT (cid:48) a P ( T (cid:48) a , T b )+ (cid:90) T b dT (cid:48) b P ( T a , T (cid:48) b ) . In figure 7, we show the average fidelity independence on the squeezing parameter r for the directand adaptive schemes, with and without postselection,for a turbulence strength characterized by C of 1 . · uantum teleportation through atmospheric channels − m − . Different from the case of atmosphericnoise in one of the entangled modes, the adaptivescheme can improve the fidelity above its maximalvalue for the uncorrelated case. Furthermore, wealso observe that postselection improves the averagefidelity. But the postselection efficiency is lowercompared to the case of a single atmospheric channel. T min F Figure 8.
The solid (blue) line shows the postselectionefficiency (left axis) in dependence on the lower limit T min foratmospheric channels A and B. The dashed (green) line andthe dashed-dotted (red) line show the fidelity (right axis) independence on the lower limit T min for the adaptive and directscheme, respectively, for a squeezing parameter of r = 1. As in the case of a single atmospheric channel,we show in figure 8 the postselection efficiency andthe average fidelity as functions of the lower limitof the transmission threshold, T min , for the directand adaptive postselection scheme. As before, anincrement of the average fidelity results in a decreaseof the postseletion efficiency. Also the postselectionefficiency for atmosphere in both channels is lowerthan for atmosphere only in one channel. Like inthe case of atmosphere only in mode B, postselectioncan increase the fidelity from classical achievablevalues (cid:0) ¯ F ≤ . (cid:1) to values which show quantumadvantages (cid:0) ¯ F > . (cid:1) . We conclude that also for two-way atmospheric channels it is possible to increasethe teleportation fidelity by means of adaptive losscorrelation and postselection.
4. Summary and outlook
We analyzed the BK-CV teleportation protocol underthe influence of constant and fluctuating losses.In particular, losses occurring in the two-modesqueezed vacuum state, used as the resource for theteleportation, were considered. We started our fullyanalytical treatment with the consideration of the influence of constant losses on such a teleportationprotocol. For this scenario, we could show thatintroducing additional losses in the system can leadto an improvement in the teleportation fidelity undercertain circumstances. This improvement stems fromthe introduced correlations in the losses which canoutstrip the negative effect of the additional loss.Analytical expressions for the teleportation fidelitydepending on the squeezing parameter and the lossparameters were derived together with the conditionsunder which the additional adaptive losses lead to animproved performance.After the consideration of the constant-loss case,we extended the treatment to fluctuating atmosphericlosses. Therefore, we first recalled the theoreticaldescription of such channels and considered threedifferent loss distributions covering both weak andstrong turbulence conditions. We could show thatthe adaptive loss-correlation technique can also beapplied in the case of fluctuating losses in atmosphericfree-space channels. Furthermore, we demonstratedthat post-selection procedures can further enhancethe teleportation fidelity under such conditions. Inthis context, we also analyzed the relation betweenthe increase of the fidelity and the amount of datawhich has to be discarded. Finally, we studiedthe case in which both entangled modes suffer formuncorrelated fluctuating losses. For this scenario, wealso demonstrated that the proposed techniques ofadaptive loss correlations and post-selection can bebeneficial for quantum state teleportation.We believe that our proposed strategies for the im-provement of CV quantum-state teleportation throughfluctuating-loss channels will help to implement suchteleportation schemes under realistic conditions, whicheventually will lead to practical applications. All pro-posed techniques can be directly implemented in com-mon teleportation experiments. It would be interestingto extend the present consideration to other telepor-tation protocols and analyze which strategies are themost practical and robust ones in the presence of atmo-spheric losses. For teleportation through atmosphericchannels, protocols relying on non-Gaussian entangle-ment might be beneficial as such state can be morerobust towards fluctuating losses. Furthermore, an ex-tension to hybrid discrete-continuous variable systemsmight be promising, as advantages of both systemscould be explored.
Acknowledgments
The authors are grateful to D. Vasylyev for enlight-ening discussions. This work has been supported byDeutsche Forschungsgemeinschaft through Grant No.VO 501/22-2. AAS also acknowledges support from uantum teleportation through atmospheric channels
Appendix A. Atmospheric channel parameters
In this Appendix, we briefly discuss how to applythe method of the elliptic-beam approximation [42] forcalculation of the mean fidelity ¯ F = (cid:82) dT P ( T ) F ( T ).Further details on the model can be found in [42].For this purpose we generate N independent Gaussianrandom vectors v i = (cid:0) x i y i Θ i Θ i (cid:1) T and randomuniformly-distributed angles χ i ∈ [0 , π/ i =1 . . . N . Here x i and y i are random coordinates ofthe beam centroid and Θ / i are related to the semi-axes, W i , of random ellipses, which model the beamprofile after transferring through the atmosphere suchthat W / i = W exp (cid:2) − Θ / i (cid:3) with W being thebeam-spot radius at the transmitter. The non-zeroelements of the covariance matrix and the means aregiven by (cid:10) Θ / i (cid:11) = ln (cid:34) (cid:16) . σ R Ω (cid:17) Ω (cid:114)(cid:16) . σ R Ω (cid:17) + 1 . σ R Ω (cid:35) , (A.1) (cid:10) ∆ x i (cid:11) = (cid:10) ∆ y i (cid:11) = 0 . W σ R Ω − , (A.2) (cid:68) ∆Θ / i (cid:69) = ln (cid:34) . σ R Ω (cid:16) . σ R Ω (cid:17) (cid:35) , (A.3) (cid:104) ∆Θ i ∆Θ i (cid:105) = ln (cid:34) − . σ R Ω (cid:16) . σ R Ω (cid:17) (cid:35) , (A.4)where σ R = 1 . C n k L is the Rytov parameter,Ω= kW / L is the Fresnel parameter, k is thewavenumber and L is the propagation distance.Based on the generated sampling data, the meanfidelity is approximated as¯ F = 1 N N (cid:88) i =1 F (cid:16)(cid:112) η m η ( v i , χ i ) (cid:17) . (A.5)Here η m is the efficiency of constant attenuation. Thefunction η ( v , χ ) reads η ( v , χ ) = η (Θ , Θ ) (A.6) × exp − r /aR (cid:16) W eff (Θ , Θ ,χ ) (cid:17) λ (cid:0) W eff(Θ1 , Θ2 ,χ ) (cid:1) . where r = (cid:112) x + y is the distance between beamand aperture centers, a is the radius of the receiver aperture. The further parameters introduced in thisfunction are given by W (Θ , Θ , χ ) =4 a (cid:104) W (cid:16) a W (Θ ) W (Θ ) × e a W
21 (Θ1) (cid:8) χ (cid:9) e a W
22 (Θ2) (cid:8) χ (cid:9)(cid:17)(cid:105) − , (A.7) η (Θ , Θ )=1 − I (cid:16) a (cid:104) W (Θ ) − W (Θ ) (cid:105)(cid:17) e − a (cid:2) W
21 (Θ1) + W
22 (Θ2) (cid:3) − (cid:20) − e − a (cid:0) W − W (cid:1) (cid:21) × exp − (cid:34) ( W (Θ )+ W (Θ )) | W (Θ ) − W (Θ ) | R (cid:16) W (Θ ) − W (Θ ) (cid:17)(cid:35) λ (cid:16) W − W (cid:17) , (A.8) R ( ξ ) = (cid:104) ln (cid:16) − exp[ − a ξ ]1 − exp[ − a ξ ]I (cid:0) a ξ (cid:1) (cid:17)(cid:105) − λ ( ξ ) , (A.9) λ ( ξ ) = 2 a ξ e − a ξ I ( a ξ )1 − exp[ − a ξ ]I (cid:0) a ξ (cid:1) × (cid:104) ln (cid:16) − exp[ − a ξ ]1 − exp[ − a ξ ]I (cid:0) a ξ (cid:1) (cid:17)(cid:105) − . (A.10)For the considered channel the following parametervalues are used: • Wavelength λ = 809nm; • Initial spot radius W = 20mm; • Propagation distance L = 1 . • Deterministic attenuation η m = 0 . . • Aperture radius a = 0 . • Atmospheric index-of-refraction structure con-stant C = (0 . , . , · − m − Note that these parameters correspond to conditions ofan experimentally implemented free-space experimentin the city of Erlangen [45]. Our model [42] showsgood agreement with the experimental distribution ofthe transmittance.
References [1] Wootters W K and Zurek W H 1982
Nature (London)
Phys. Lett. A Phys. Rev. Lett. Nat. Photonics Nature (London)
Phys. Rev. Lett. et al. Nature (London)
Rev. Mod. Phys. uantum teleportation through atmospheric channels [9] Braunstein S L and Kimble H J 1998 Phys. Rev. Lett. Science
Rev. Mod. Phys. Phys. Rev. Lett. Open Syst. Inf. Dyn. Phys. Rev. A Phys. Rev. A Phys.Rev. A Phys.Rev. Lett. Phys. Rev. A Nature (London) et al.
Nat. Phys. et al. New J. Phys. Nat. Phys. Phys. Rev. Lett. et al.
Nature (London) et al.
Nature (London)
Phys. Rev. Lett. et al.
New J. Phys. Nat. Phot. et al. Nat. Phot. Opt.Express Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. Lett.
Opt. Express Nat. Photonics , 502[36] Liao S-K et al. Nature (London) et al.
Science et al.
Optica et al. Phys. Rev. Lett.
Phys. Rev. A Phys. Rev.Lett.
Phys. Rev.Lett.
Phys. Rev.
A 96043856[44] Vasylyev D, Vogel W and Semenov A A 2018
Phys. Rev. A New J. Phys. Phys. Rev. A Phys. Rev.
A 95 063801[48] Bohmann M, Semenov A A, Sperling J and Vogel W 2016
Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Rev. A Chin. Phys. Lett. Phys. Rev. A J. Mod. Opt. Phys. Rev. A J. Mod.Opt. Phys. Rev. A Phys. Rev. Lett. Journal of Physics B: Atomic, Molecular andOptical Physics Proc. IEEE Effects of the Turbulent Atmosphere onWave Propagation (IPST, Jerusalem)[62] Andrews L C and Phillips R L 2005
Laser BeamPropagation through Random Media (SPIE Press,Bellingham)[63] Elser D, Bartley T, Heim B, Wittmann C, Sych D andLeuchs G 2009