Optimal fidelity of teleportation with continuous-variable using three-tunable parameters in realistic environment
aa r X i v : . [ qu a n t - ph ] S e p Optimal fidelity of teleportation with continuous-variable using three-tunable parametersin realistic environment
Li-Yun Hu , ∗ , Zeyang Liao , Shengli Ma and M. Suhail Zubairy ∗ Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy,Texas A&M University, College Station, TX 77843, USA Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China and*Corresponding author.
We introduce three tunable parameters to optimize the fidelity of quantum teleportation withcontinuous-variable in nonideal scheme. Using the characteristic function formalism, we present thecondition that the teleportation fidelity is independent of the amplitude of input coherent states forany entangled resource. Then we investigate the effects of tunable parameters on the fidelity withor without the presence of environment and imperfect measurements, by analytically deriving the ex-pression of fidelity for three different input coherent state distributions. It is shown that, for the lineardistribution, the optimization with three tunable parameters is the best one with respect to single- andtwo-parameter optimization. Our results reveal the usefulness of tunable parameters for improving thefidelity of teleportation and the ability against the decoherence.PACS number(s): 03.65 -a. 42.50.Dv
I. INTRODUCTION
Quantum teleportation has an indispensable role inthe manipulation of quantum states and processing ofquantum information [1–4]. Usually, the two-modesqueezed vacuum is often used as the entanglement re-source with continuous-variable (CV). However, due tothe limit of experiment, it is hard to achieve high degreeof squeezing which leads to a low effect of teleportationfidelity.In order to increase the entanglement and fidelity ofteleportation, a number of stratigies have been proposed[5–14]. Among them, non-Gaussian operations includ-ing the photon subtraction a or addition a † or the super-position of both can be used to realize this purpose forgiven Vaidman-Brauntein-Kimble (VBK) scheme. For ex-ample, the superposition operator ta † + ra is proposedfor quantum state engineering to transform a classicalstate to a nonclassical one [15], and then is performedon two-mode squeezed vacuum (TMSV) for enhancingquantum entanglement as well as the fidelity of telepor-tation [16]. It is found that the fidelity teleporting co-herent state can be further improved by optimizing thesuperposition operation compared with the other non-Gaussian states, such as photon-subtraction TMSV. Asanother example, a remarkable improvement of the tele-portation fidelity with CV can be obtained by introducinganother optimal non-Gaussian resources when given theusual and (non-)ideal VBK scheme [17–19] by consider-ing imperfect Bell measurements and damping. In Ref.18, the “shot-fidelity” and single-gain factor are used todiscuss the performance of teleportation. In fact, theseprotocols above are employed to enhance the fidelity ofteleportation by changing quantum entangled resources.Then an inverse question is that given a certain classof entangled resources with some given properties, howwe can modify the VBK scheme to improve the fidelityof teleportation. It is interesting to notice that there is an alternative method to improve the fidelity of telepor-tation by using calssical information. For instance, theteleportation fidelity of CV can be enhanced by tuningthe gain in the measurement dependent modulation onthe output field [7, 20] in Heisenberg picture and EPRresources without loss, which has been realized experi-mentally by Furusawa et al [21]. However, these two im-portant theoretical works are concerned with the studyof the ideal protocol implementation using Gaussian re-sources [7, 20]. In addition, there are some other strate-gies of gain tuning and of optimal gain [22, 23], usingthe Heisenberg picture and the Wigner function, but theycan not be directly applied to more general cases. In ad-dition, in Refs. [22, 24] the gain factor is used to max-imize the teleportation fidelity for the case of Gaussianresources, but the gain-optimized fidelity of teleporta-tion is strongly suppressed when considered the dissipa-tion. Recently, a hybrid entanglement swapping protocolis proposed experimentally to transfer discrete-variable(DV) entanglement by using continuous-variable (CV)Gaussian entangled resources and by tuning a gain fac-tor of the teleporter [23, 25, 26], which shows that DVentanglement remains present after teleportation for anysqueezing by optimal gain tuning. For more informationabout advances in quantum teleportation, we refer to arecent review paper [27] and references therein.The fidelity of teleportation, as mentioned above, canbe improved by using tunable entangled resources orclassical parameters [17, 18, 20, 23]. In Ref. [20],a three-parameter optimal stratigy is introduced to im-prove the quality of teleporttaion, including unbalancedbeam splitter (BS) and two non-unity gains. However,they only considered an ideal case. Actually, the inter-action between quantum system and environment cannot be avoided and Bell measurements are usually im-perfect. Thus, it would be interesting that whether it isstill effective to enhance the fidelity by using these tun-able parameters in realistic case. In this paper, using thecharacteristics function (CF) formalism, we shall extendthe analysis of the parameter optimization strategy forrealistic input states and non-ideal entangled resourcesand investigate the nonideal quantum teleportation byderiving an analytical expression of the teleportation fi-delity. This formalism is very convenient to discuss theteleportation for the nonideal case and any entangled re-sources. As far as we know, there is no related report upto now.This paper is arranged as following. In Sec. II, we givea description of the characteristic function formulism forthe case of nonideal parameterized teleportation with CVscheme. In this scheme, we find the condition that the fi-delity is independent of the amplitude of input coherentstates for any entangled resource. And then we presenta qualitative description about fidelity and average fi-delity. In Sec. III, we derive the analytical expression ofthe fidelity of teleportation when the TMSV and coher-ent states are used as entangled channel and teleportedstates, respectively. In Sec. IV we study the performanceof amplitude-independent optimal fidelity using the con-dition found in Sec. II. It is found that the optimal con-dition is just that the two-gain factors and θ are equal tounit and π/ , respectively. Sec. V is devoted to discussingthe optimal fidelity over these three tunable parametersand three different probability distributions for the inputcoherent states by deriving the analytical expression ofthe optimal fidelity. Our conclusions are drawn in thelast section. II. MODE AND QUANTITATIVE ANALYSIS
Here, we consider a more realistic case of teleporta-tion scheme shown in Fig. 1(a). In this scheme, thereare three tunable parameters, unbanlanced BS and twonon-unity gains ( g q and g p ). The input state (mode 1)and the entangled resources (shared by modes 2 and 3)are not limited to be pure states. Considering that themode 2 can be prepared close to the sender Alice whilethe mode 3 usually has to propagate over much longerdistance, we can assume that the mode 2 is not affectedby losses but the mode 3 is. In addition, two symmetricallossy bosonic channel have been considered before mak-ing Bell measurements, which are simulated through anextra vacuum mode and a beam splitter with transmis-sion coefficient T . The input states of modes 4 and 5 arepure vacuum states.Next, we shall give a description about the scheme inthe formalism of CF where it is very convenient to discussthe teleportation for the nonideal case and non-Gaussianentangled resourecs [18, 28]. A. The input-output relation of BS in CF formalism
In order to obtain the relation between input and out-put, we first calculate the output of a beam splitter witha vacuum and an arbitrary density operator as inputs (cid:513) (cid:1767)(cid:882) (cid:2873) (cid:882)(cid:513)(cid:513) BS (cid:2025) (cid:3042)(cid:3048)(cid:3047) (cid:2025) (cid:2869) (b) Entangled Resources D i s p l a c e mode 3noisy channel D D BS1 mode 2mode 1 BS2BS3 BobAlice q g p g p p q q out in (cid:513) (cid:1767)(cid:882) (cid:2872) (cid:513) (cid:1767)(cid:882) (cid:2873) (a) FIG. 1: (Colour online) Realistic schematic diagram for CV tele-portation. BS: Beam splitter. shown in Fig. 1(b). For simplicity, we denote the vac-uum and the input state as | i and ρ , respectively. Theoutput state (denoted as ρ ′ ) is given by ρ ′ = Tr h B ( T ) ρ ⊗ | i , h | B † ( T ) i , (1)where Tr is the partial trace over the ancilla mode 5 and B kl ( T ) = exp[ ϕ ( a k a † l − a † k a l )] is the beam splitter oper-ator describing the interaction between modes 1 and 5with cos ϕ = √ T and a k,l ( k = 1 , l = 5 ) being the pho-ton annihilation operator of the k ( l ) − modes. Using theWeyl expansion of density operator, we can express thedensity operator ρ and the vacuum projector | i h | asthe following forms: ρ = Z d απ χ ( α ) D ( − α ) , | i h | = Z d βπ e − | β | D ( − β ) , (2)where D ( α ) = exp { αa † − α ∗ a } is the displacementoperator, and χ ( α ) is the CF of ρ . On the other hand,using the following transformation relation B D ( − α ) D ( − β ) B † = D (¯ α ) D (cid:0) ¯ β (cid:1) , (3)where R = 1 − T , and ¯ α = β √ R − α √ T , ¯ β = − β √ T − α √ R , we can deriveTr h B D ( − α ) D ( − β ) B † i = D (¯ α ) Tr (cid:2) D (cid:0) ¯ β (cid:1)(cid:3) = D (¯ α ) πδ (2) ( ¯ β ) . (4)Here we have used the relation Tr D ( ¯ β ) = πδ (2) ( ¯ β ) .Then substituting Eqs. (2) and (4) into Eq. (1) yields ρ ′ = Z d αd βπ e − | β | χ ( α ) Tr (cid:2) D ( ¯ α ) D (cid:0) ¯ β (cid:1)(cid:3) = Z d αd βπ e − | β | χ ( α ) D ( ¯ α ) πδ (2) (cid:0) ¯ β (cid:1) = Z d απ e − R | α | χ (cid:16) √ T α (cid:17) D ( − α ) , (5)where e − | β | is the CF of the vacuum state, and in thesecond step in Eq. (5) the CF χ ( α ) is transformed to χ ( √ T α ) with a Gaussian term e − R | α | due to the pho-ton loss. It is then convenient to obtain the input-outputrelation of the teleportation scheme shown in Fig. 1(a)as follows. B. The input-output relation of the teleportation schemein CF formalism including photon-loss or imperfect Bellmeasurements
Now, we consider the effect of photon-loss on therelation between input and output of the teleportationscheme in CF formalism. Here we use BS2 and BS3 withvacuum inputs to simulate the photon loss or imperfectBell measurements (see Fig. 1(b)), and denote the tele-ported state, entangled resource, and auxiliary vacuumas ρ , ρ and | i , respectively. In oder to realize theteleportation, Alice shall make her Bell measurements.Before she does, the unitary state evolution can be for-mulated as ρ − = U ⊗ ρ ⊗ ρ ⊗ | i h | U † , (6)where the unitary evolution operator is defined as U = B B B and B kl are the BS operator defined be-fore. In a similar way to deriving Eq. (5), and using theWeyl expansion for the entangled resource, the reducedoutput state denoted as ρ − ≡ Tr ρ − is given by ρ − = Tr (cid:2) U ⊗ ρ ⊗ ρ ⊗ | i h | U † (cid:3) = Z d αd βd γπ χ ( α ) χ ( β, γ ) × Tr (cid:2) U D ( − α ) D ( − β ) D ( − γ ) | i h | U † (cid:3) = Z d αd βd γπ χ ( α ) χ ( β, γ ) Tr [ B × B D ( − α ) D ( − β ) D ( − γ ) | i h | B † B † ] , (7)where B D ( − α ) D ( − β ) B † = D ( − α ) D ( − β ) with α = α cos θ − β sin θ , β = β cos θ + α sin θ and cos θ being the transmission coefficient of beam splitter B . Using Eq. (1) and (4), we can obtain ρ − = Z d αd βd γπ χ ( √ T α ) χ ( √ T β, γ ) × e − R ( | α | + | β | ) D ( − α ) D ( − β ) D ( − γ ) . (8)Eq. (8) is the representation in CF of deduced density op-erator before Bell measurements but after BS2 and BS3.Then, as the first step of teleportation, Alice makesa joint measurements for modes 1 and 2 at the outputports, i.e., measures two observables corresponding to coordinate and momentum of modes 1 and 2. Thus af-ter the measurements, the outcomes ρ M ( M means mea-surement) in mode 3 are ρ M ≡ P ( q, p ) Tr [ | q i h q | ⊗ | p i h p | ρ − ] , (9)where P ( q, p ) is the probability distributionfunction of the Bell measurement outcomes, P ( q, p ) = Tr { Tr [ | q i h q | ⊗ | p i h p | ρ − ] } , and | q i and | p i are the eigenstates of coordinate and momentumoperators Q and P corresponding to modes 1 and 3,respectively.According to the definition of CF, and using the follow-ing relationsTr h | q i , h q | D ( α ) i = e i √ q Im α δ (cid:16) √ α (cid:17) , Tr h | p i , h p | D ( β ) i = e − i √ p Re β δ (cid:16) √ β (cid:17) , (10)and Tr [ D ( − γ ) D ( η )] = πδ (2) ( η − γ ) , (11)the CF of ρ M defined as χ M ( q, p ; η ) =Tr [ ρ M D ( η )] reads χ M ( q, p ; η )= P − ( q, p )sin 2 θ Z d απ exp { α ∗ ξ − αξ ∗ }× χ ( √ T α ) χ h √ T ( α cot 2 θ + α ∗ csc 2 θ ) , η i × exp (cid:26) − R α ) csc θ + (Im α ) sec θ ] (cid:27) , (12)where we have defined ξ = ( q/ cos θ + ip/ sin θ ) / √ .When T = 1 , Eq. (12) just reduces to Eq.(2.9) in Ref.[28].After Alice informs the measured results ( q, p ) toBob, Bob needs to make a unitary transformation toobtain the output state at this stage. Here, we con-sider the unitary transformation to be the displace-ment operator D ( Z ) with non-nunity and aysmetri-cal gains characteristic, where Z ≡ g q q + ig p p with g q and g p are two tunable gain parameters. Thus, afterthe displacement, the output state can be expressed as ρ D ≡ R d ηχ M ( q, p ; η ) D ( Z ) D ( − η ) D ( − Z ) /π . Usu-ally, we do not have interest in every measurement resultbut the average effect. Thus we perform an ensemble av-eraging over all measurement results, then the averageCF ¯ χ out is given by ¯ χ out ( β ) = Tr [ D ( β ) Z dqdpP ( q, p ) ρ D ]= χ ( f β − f β ∗ ) χ ( β ∗ f − βf , β ) × exp {− R [ g p (Re β ) + g q (Im β ) ] } , (13)where f = r T g q cos θ + g p sin θ ) ,f = r T g q cos θ − g p sin θ ) ,f = r T g q sin θ + g p cos θ ) ,f = r T g q sin θ − g p cos θ ) , (14)and T and R ( T + R = 1 ) denote, respectively, the trans-missivity and reflectivity of the BS2 and BS3 that stimu-late the photon losses. C. Relation between input and output including noise inmode 3
Next, we consider the effect in CF formalism of deco-herence of environment on the mode 3. Here we con-sider the case that the mode 3 propagates in a noisychannel such as photon loss, and thermal noise after Al-ice’s measurement but before its reaching Bob’s location(see Fig. 1(a)). In the interaction picture and the Born-Markov approximation, the time evolution of the densitymatrix describing the thermal environment is governedby the master equation (ME) [29]: ddt ρ ( t ) = κ ¯ n (cid:0) a † ρa − aa † ρ − ρaa † (cid:1) + κ (¯ n + 1) (cid:0) aρa † − a † aρ − ρa † a (cid:1) , (15)where κ and ¯ n are the dissipative coefficient and the av-erage thermal photon number of the environment, re-spectively. When ¯ n = 0 , Eq. (15) reduces to the onedescribing the photon-loss channel. By solving the MEin the CF form, one can find that the evolution of CFdescribed by Eq. (15) is given by χ ( γ ; t ) = χ (cid:0) γe − κt ; 0 (cid:1) exp n − Γ | γ | o , (16)where Γ = (2¯ n + 1) (cid:0) − e − κt (cid:1) / , and χ ( γ ; 0) is the CFof initial state ρ (0) . In a similar way to deriving Eq. (13),at Bob’s location, the CF ¯ χ f of final output state for theteleportation scheme can be directly given by ¯ χ f ( β ; t ) = e − Γ | β | χ ( f β − f β ∗ ) × χ (cid:0) β ∗ f − βf , βe − κt (cid:1) × exp {− R [ g p (Re β ) + g q (Im β ) ] } . (17)The form of Eq. (17) shows the different roles playedby the noise channel ( Γ , κ ) and gain factors ( g p , g q ) aswell as unbalanced BS ( θ ), the reflectivity R . The deco-herence effect from the noisy channel affects only mode3 by means of the exponentially decreasing weight e − κt in the arguments of χ . Eq. (17) is just the general description of the nonideal scheme in terms of the CF,which just reduces to the factorized form of the outputCF in Eq. (9) and Eq. (4) in Ref. [18], as expected, when κt = 0 and g q = g p = g , θ = π/ , respectively. Thus, Eq.(17) is the generalized input-output relation in the CFformalism. D. Fidelity and average fidelity
In order to measure the effectivity of the teleportationscheme, we appeal to the fidelity of teleportation, de-fined by F = Tr ( ρ in ρ out ) . Within the formalism of CF, thefidelity reads F = Z d λπ χ in ( λ ) χ out ( − λ ) , (18)where χ in and χ out are the CFs corresponding to den-sity operators ρ in and ρ out , respectively. Eq. (18) is thefundamental quantity that measures the performance ofa CV teleportation, which will be often used in the fol-lowing calculations. On the basis of Eqs. (17) and (18),we can examine the performance of teleportation for anyinput states and any entangled resources including non-Gaussian ones.In particular, when we specify the input teleportedstates at Alice’s location to be coherent states ρ = | ǫ i h ǫ | with complex amplitude ǫ , whose CF reads χ ( λ ) = e − | λ | + λǫ ∗ − ǫλ ∗ , then substituting it and Eq. (17) intoEq. (18) we can get F = Z d λπ exp (cid:26) − (cid:0) f + f + 2Γ (cid:1) | λ | (cid:27) × exp (cid:26) f f (cid:0) λ + λ ∗ (cid:1) + λ ∆ − λ ∗ ∆ ∗ (cid:27) × χ (cid:0) λf − λ ∗ f , − λe − κt (cid:1) × exp[ − R ( g p Re λ + g q Im λ )] , (19)where ∆ = (1 − f ) ǫ ∗ − ǫf . From Eq. (19) we can seethat if we choose ∆ = 0 then the fidelity will be indepen-dent of ǫ for any entangled resources. The condition of ∆ = 0 leads to g q = 1 √ T cos θ , g p = 1 √ T sin θ . (20)This is the only choice making the fidelity independentof ǫ , which allows us to have no information about theinput coherent states. The condition (20) depends on T and θ but is independent of the decoherence involvedin mode 3. This is true for any entangled resources. Inparticular, when θ = π/ , Eq. (20) just reduces to thecase in Ref. [18]; while for T = 1 and θ = π/ , thisresult is just the case discussed in Ref. [30].Generally speaking, the fidelity in Eq. (19) depends onthe teleported input states which are usually unknown bythe sender and the receiver. Here, in order to further de-scribe the fidelity, we assume a partial knowledge of theinput states about a probability distribution P ( µ ) satis-fying the normalization condition, i.e., R P ( µ ) dµ = 1 where the integral is taken over all possible values of µ .For a given distribution P ( µ ) , the average fidelity is ¯ F = Z F ( µ ) P ( µ ) dµ. (21)In the following, we will take three probability distribu-tions into account for input coherent states, such as line-,circle- and 2D-Gaussian-distribution [20]. III. TWO-MODE SQUEEZED VACUUM AS ENTANGLEDRESOURCES
In this section, we use the usual TMSV as entangledresources to analyze the performance of these three tun-able parameters for improving the fidelity of teleporta-tion. The TMSV entangled resource, most commonlyused in continuous-variable teleportation, can be gener-ated by the parametric down-conversion (PDC) process,and theoretically can be defined as | Φ i sv = S ( r ) | i = sech r exp (cid:0) a † b † tanh r (cid:1) | i , (22)where S ( r ) = exp (cid:8) r ( a † b † − ab ) (cid:9) is the two-modesqueezing opertor with r being the squeezing parameter,and a † ( a ) and b † ( b ) are photon creation (annihilation)operators. According to the definition of CF, the CF ofthe TMSV is given by χ sv ( α, β ) = exp (cid:26) − (cid:16) | α | + | β | (cid:17) cosh 2 r (cid:27) × exp (cid:26)
12 ( αβ + α ∗ β ∗ ) sinh 2 r (cid:27) . (23)In particular, for the largest entangled resource with r →∞ , and ideal measurements with T = 1 and R = 0 , aswell as banlanced BS, we have lim r →∞ χ sv ( β ∗ , β ) = 1 .Substituting it into Eq. (13) yields lim r →∞ ¯ χ out ( β ) = χ ( β ) , i.e., a perfect teleportation, as expected.When Alice use the TMSV as entangled resource toteleport the coherent states, then the fidelity in Eq. (19)can be calculated as F = 1 √ G exp ( − K | ∆ | + K (cid:0) ∆ + ∆ ∗ (cid:1) G ) , (24)where we have defined G = K − K and K = 12 (cid:0) f + f + 2Γ (cid:1) + R (cid:0) g p + g q (cid:1) + 12 (cid:0) f + f + e − κt (cid:1) cosh 2 r − f e − κt sinh 2 r,K = 12 (cid:8) f f − R (cid:0) g p − g q (cid:1) / f f cosh 2 r − f e − κt sinh 2 r (cid:9) , (25) and used the following integration formula Z d zπ exp (cid:16) ζ | z | + ξz + ηz ∗ + f z + gz ∗ (cid:17) = 1 p ζ − f g exp (cid:26) − ζξη + ξ g + η fζ − f g (cid:27) . (26)From Eq. (24) one can see that the fidelity F dependson the amplitude of the teleported coherent states. Inthe next sections, we shall consider two kinds of cases:one is independent of the amplitude by the choice in Eq.(20) and the other is not, but partial information aboutthe input state distribution is known. IV. ǫ -INDEPENDENT OPTIMAL FIDELITY In this section, we examine the fidelity for teleport-ing coherent state with two gain factors fixed to be g q = 1 / ( √ T cos θ ) , g p = 1 / ( √ T sin θ ) . This choice al-lows us to have no information about the amplitude ofcoherent states. Noticing that f = 1 , f = 0 , f = csc 2 θ and f = − cot 2 θ , then from Eq. (24) we can get F ǫ = n H [1 / ( √ T c ) , c ] H [1 / ( √ T c ) , c ] o − / , (27)where c = cos θ, c = sin θ and we defined the function H ( x, y ) as H ( x, y ) = 12 + Γ + x (cid:0) T y sinh r (cid:1) + 12 e − κt cosh 2 r − xye − κt √ T sinh 2 r. (28)It is clear that the fidelity F ǫ depends on multi-parameters such as r, κt, ¯ n, T and θ . At fixed r, κt, ¯ n and T , the optimal fidelity of teleportation is defined as F opt = max θ F ( r, θ ) . (29)In order to maximize the fidelity in Eq. (27) over θ , wecan take ∂ F ǫ /∂θ = 0 , which leads to the following con-dition cos 2 θ = 0 , (30)or csc 2 θ = 12 (cid:26) e − κt sinh 2 r /T + 2 sinh r + e − κt sinh 2 r /T + 1 + 2Γ + e − κt cosh 2 r (cid:27) . (31)It is easy to see that the first item (FI) in the right handside (RHS) of Eq. (31) is less than unit, and the seconditem (SI) satisfies (by taking T = 1 and ¯ n = 0 ) e − κt sinh 2 r /T + 1 + 2Γ + e − κt cosh 2 r e − κt sinh 2 r e − κt sinh r . (32) O p t i m a l f i de li t y Squeezing parameter r
FIG. 2: (Colour online) The optimal fidelity for teleporting co-herent states as a function of the squeezing parameter r withsome different values of T and κt as well as ¯ n = 0 . Here solidlines: the transmissivity T = 1 , . , . , . , . and κt = 0; Blue-dash lines: κt = 0 . , . , . , . and T = 1 . The corre-sponding lines are arranged from top to bottom with the in-creasing /T and κt , respectively. By numerical calculation, we can find that when thesqueezing parameter r is less than a threshold value ofabout . , the sum of ( FI + SI ) / is alway less than unitwhich will lead to an impossible case, i.e., csc 2 θ < .Thus within the region of threshold value, the optimalpoint is at θ = π/ and g q = g p = 1 / √ T which is in-dependent of ¯ n and e − κt . The threshold value of r willincrease with the increasing ¯ n and /T . Actually, thepresence of threshold value results from the decoherencethrough mode 3, since the SI is always less than unit forany squeezing r when κt = 0 .Substituting the above optimal condition into Eq.(27), we get the optimal fidelity F opt = [ 1 T + Γ + e − κt (cosh κt cosh 2 r − sinh 2 r )] − . (33)It is clear that F opt decreases with the increasing ¯ n and /T , as expected. In particular, when κt = 0 and T = 1 ,Eq. (33) just reduces to F opt = (1 + tanh r ) / , whichis the best fidelity when we use the coherent states asinputs and the TMSV as entangled resources in the BKscheme independent of teleported coherent state ampli-tude. In addition, when κt → ∞ , F opt → ( T + ¯ n +cosh r ) − , which decreases with the increasing ¯ n, r andthe decreasing T .In order to examine clearly the effects of different pa-rameters on the optimal fidelity, we plot the optimal fi-delities as a function of squeezing parameter r in Fig.2. From Fig. 2, we can see that for κt = 0 (withoutdecoherence on mode 3) the optimal fidelities increasemonotonically with the increasing squeezing parameter r and the transmissivity T . In addition, when we con-sider the effects of decoherence on mode 3, the opti-mal fidelities first increase and then decrease with theincreasing r . The maximal value and the corresponding value of r max reduces as κt increases. In fact, we cantake ∂ F opt /∂r = 0 to get a simple relation as following( cosh κt = coth 2 r max ) e r max = coth κt . (34)It is interesting to notice that r max is independent of T and ¯ n . V. AVERAGE OPTIMAL FIDELITY AND THE EFFECT OFTUNABLE PARAMETERS
In the last section, we consider the ǫ -independent op-timal fidelity. However, when g q = 1 / ( √ T cos θ ) and g p = 1 / ( √ T sin θ ) , the scenario will be changed dra-matically. In this case, the fidelity in (24) will dependon the amplitude ǫ of coherent state. In this section, weexamine the average optimal fidelity for three differentprobability distributions of the teleported input states inwhich the partial information is known by Alice and Bob.For example, they may be completely sure of phase of thestates but the amplitude is unknown [20]. A. Optimal fidelity for teleporting coherent states on aline
First, let us consider the teleportation of coherentstates on a line. Without losing the generality, here weassume that the phase of the teleported coherent statesis zero because we can always achieve this point by ro-tating frame. Then the corresponding probability distri-bution can be given by (letting ǫ = x + iy ) P ( x, y ) = 12 L δ ( y ) × (cid:26) , | x | L , else . (35)Thus substituting Eqs. (35) and (24) into Eq. (21) yieldsthe average fidelity ¯ F line = 12 L √ G Z L − L dxe − MG x = √ π L √ M Erf ( | − √ T g q cos θ | [ H ( g q , sin θ )] / L − ) , (36)where Erf { a } = 1 / √ π R a − a e − x dx is the error functionand M = (1 − √ T g q cos θ ) H ( g p , cos θ ) .Noticing the separability of g q and g p in ¯ F line , the opti-mal value of g p can be obtained by ∂ ¯ F line /∂g p = 0 equiv-alent to ∂H ( g p , cos θ ) /∂g p = 0 , which leads to g optp = e − κt √ T cos θ opt sinh 2 r T cos θ opt sinh r ) . (37)It is interesting to notice that the optimal value of g optp is related to e − κt and T but independent of the averagethermal photon-number.Next, we will maximize the fidelity by numerical cal-culation. At fixed r , κt , ¯ n and T , the optimal fidelity ofteleportation can be defined as ¯ F optline = max g q ,g p ,θ ¯ F line ( r, g q , g p , θ ) . (38)In Fig. 3 we plot the optimal fidelity as a function ofsqueezing parameter r for some different values of pa-rameters κt , ¯ n and T . In Fig. 3(a), we consider theoptimal fidelities with some different values of L and T = 1 , ¯ n = 0 as well as κt = 0 . (for comparison, thecase of κt = 0 is also plotted as dash lines). From Fig.3(a), we can see that the optimal fidelities grow withincreasing r and /L . The fidelities can be greatly opti-mized with respect to the standard teleportation schemelines (STS with g q = g p = 1 and θ = π/ , see short dash-dot-(dot) lines). Especially for a smaller L ( L = 0 . ), theoptimal fidelity can almost access to unit. While for alarger L (say L = 300 ), the fidelity achieves a limitation(still over 0.8) which is still superior to that in the STS.In Fig. 3 (b), we consider the effect of different valuesof T on the optimal fidelity at κt = 0 , . . It is shownthat the optimal values decease with the increasing /T for a given κt ; while for the case of T = 1 , by compar-ing the fidelities at κt = 0 , . for a given T , it is foundthat the optimal fidelities first increase and then decreasewith the increasing r , and it is interesting to notice thatthe optimal fidelity with κt = 0 . is superior to that with κt = 0 when r exceeds a certain cross-point.Furthermore, in order to clearly see whether there isthe ability against the decoherence by using these tun-able parameters, here we plot the optimied fidelity as afunction of the evolution time κt for some fixed values.For this purpose, we fix the squeezing parameter at theintermediate value r = 0 . with T = 0 . , ¯ n = 0 . Exper-imentally, the attainable squeezing degree is about . .Fig. 4 shows that the optimal fidelity remains above 0.8which exceeds the classical threshold up to significantlylarge values of κt . This case is true even for the limita-tion of L → ∞ . These results indicate that the optimizalfidelities by three parameters present superior behaviorto and higher ability against the decoherence than thosein the standard teleportation scheme (with g q = g p = 1 and θ = π/ , also see dash lines in Fig. 4).In addition, we make a comparison among the opti-mal effects of different optimal parameters. In Fig. 5,we plot the optimized fidelity over different tunable pa-rameters as the function of squeezing parameter r for agiven L = 1 with T = 1 , ¯ n = 0 and κt = 0 . . It is foundthat the optimization by three tunable parameters is thebest when compared to single- and two-parameter opti-mization, especially in the region of small entanglement,which indicates that the role of parameter is differentfrom each other. Thus, it is necessary to perform a simul-taneous balanced optimization over these three parame-ters to obtain a maximization of teleportation fidelity forthe probability distribution in Eq. (35). O p t i m a l f i d e li t y Squeezing parameter r (a) O p t i m a l f i de li t y Squeezing parameter r (b)
FIG. 3: The optimal fidelity for teleporting CSs as a function of r with ¯ n = 0 , κt = 0 , . corresponding to solid and dash lines,respectively. (a) L = 0 . , . , , , and T = 1; for compari-son, the teleportation in the STS is also plotted as dash-dot anddash-dot-dot lines for κt = 0 , . ; (b) T = 1 , . , . , . , . and L = 1 . The corresponding lines are arranged from top tobottom with the increasing L and /T for a given κt . B. Optimal fidelity for teleporting CSs on a circle
In this subsection, we consider the optimal fidelityfor teleporting CSs on a circle, which means ǫ = | ǫ | e iϕ ≡ Ae iϕ with a known amplitude | ǫ | = A and unknown phase ϕ . In this case, the distribu-tion function is P ( A, ϕ ) = δ ( | ǫ | − A ) / π satisfying R ∞ R π P ( A, ϕ ) d | ǫ | dϕ = 1 , then the average fidelitycan be calculated as ¯ F circle = e − R √ G ∞ X k =0 ( R ) k k ! k ! , (39)where we have set R = A { K [(1 − f ) + f ] + 4 K (1 − f ) f } /G , R = A { K (1 − f ) f + K [(1 − f ) + f ] } /G . Maximizing ¯ F circle over thesethree parameters, we can get the optimized fidelity ¯ F optcircle = max g q ,g p ,θ ¯ F circle ( r, g q , g p , θ ) . Our random nu-merical calculations show that, for the probability distri-bution, the maximum value of fidelity can be achieved atthe point with g q = g p = g and θ = π/ , which is differ-ent from the case in subsection A. Under this condition O p t i m a l f i de li t y tL=29 FIG. 4: (Colour online) The optimal fidelity for teleporting CSsas a function of κt with ¯ n = 0 , r = 0 . , T = 0 . and L =0 . , . , , , from top to bottom, respectively. O p t i m a l f i de li t y Squeezing parameter r g q =g p =1, g q =g p =g, g q ,g p , g q =g p =g, g q ,g p , FIG. 5: (Colour online) The optimal fidelity for teleporting CSsas a function of r with ¯ n = 0 , κt = 0 . , T = L = 1 for severaldifferent optimal parameters. we have f = f = g √ T and f = f = 0 , as well as R = K = 0 . Thus the optimized fidelity can be givenby ¯ F optcircle = 1Θ exp (cid:26) − A Θ (1 − g √ T ) (cid:27) , (40)where we have set Θ = Γ + [ g ( R + 1) + 1 + ( g √ T − e − κt ) cosh 2 r + 2 g √ T e − κt e − r ] / . It is obvious that Θ > . Using Eq. (39) or (40), we have plotted the optimalfidelity as a function of squeezing parameter r for somedifferent values of A and T in Fig. 6. In Fig. 6(a), weconsider the optimal fidelities with some different valuesof A with T = 1 , ¯ n = 0 as well as κt = 0 , . . FromFig. 6(a), we can see that the optimal fidelities growmonotonously with increasing r for κt = 0 , but for κt =0 . the optimal fidelities first increase and then decreasewith increasing r especially for a large A (say A = 3 ).In addition, for a small A , the optimal fidelity almostaccess to unit. In Fig. 6(b), we also examine the effect of different T on the fidelity. It is interesting to noticethat the optimal fidelity with κt = 0 . can be better thanthat with κt = 0 when the squeezing r exceeds a certainvalue. This case is similar to that in Fig. 3(b). In Fig.6, the point r max corresponding to the maximum fidelitydepends on κt, and for given κt , A and T , the value of r max can be determined by taking ∂ ¯ F optcircle /∂r = 0 , whichleads to n Θ − A (1 − g √ T ) o ∂ Θ ∂r = 0 . (41)After a straightforward calculation, we can abtain tanh 2 r max = 2 g √ T e − κt g T + e − κt . (42)or e r max − be r max + c = 0 , (43)where b = 4[ A (1 − g √ T ) − Γ] / ( g √ T − e − κt ) , and c =( g √ T + e − κt ) / ( g √ T − e − κt ) . In particular, when g =1 / √ T the fidelity is independent of the amplitude A andsince that Θ > , the value of r max in Eq. (42) reducesto that in Eq. (34).In Fig. 7, choosing the same values of parameters T,r and ¯ n as in Fig. 4, we plot the optimal fidelity as afunction of κt . For comparison, we also plot the fidelitywithout the optimization, i.e., g q = g p = 1 and θ = π/ (see dash lines). Fig. 7 shows that the teleportation fi-delity can be always above the classical limitation 0.5 upto significantly large values of κt ( when the ampli-tude A is less than about . , and while it can go belowthe limitation . for A > . when κt exceeds a certainthreshold value. The optimal fidelities with A = 15 , are indistinguishable. In the STS, the fidelity is less than0.5 when A > . Comparing the fidelities with andwithout optimization, it is shown that the former havebetter teleportation performance than the latter. How-ever, this improvement is inferior to that shown in Fig. 4where the optimal fidelities are over 0.8 for any L and κt . C. Optimal fidelity for teleporting CSs by 2D Gaussiandistribution
In the last subsection, we consider another simpleprobability distribution—–two-dimensional (2D) Gaus-sian distribution. The corresponding distribution isgiven by P ( α ) = 1 / ( πχ ) exp[ − | α | /χ ] satisfying R P ( α ) d α = 1 [18, 20, 31], where the variance param-eter χ determines the cutoff of the amplitude α . Thus,using Eqs. (21) and (24), the averaged fidelity can be O p t i m a l f i de li t y Squeezing parameter r (a) O p t i m a l f i de li t y Squeezing parameter r (b)
FIG. 6: (Colour online) The optimal fidelity for teleportingCSs as a function of r with ¯ n = 0 , κt = 0 , . (a) A =0 . , . , , , and T = 1; (b) T = 1 , . , . , . , . and A = 1 . For each optimized case (associated with a special plotstyle), the corresponding lines are arranged from top to bottomwith the increasing A and /T at the point r = 0 , respectively. calculated as ¯ F G = 1 q H ( g p , cos θ ) + χ (1 − √ T g p sin θ ) × q H ( g q , sin θ ) + χ (1 − √ T g q cos θ ) , (44)where the function H ( x, y ) is defined in Eq.(28) [ ( K + 2 K ) = H ( g q , sin θ ) , ( K − K ) = H ( g p , cos θ ) ]. Noticing that the parameters g q and g p are independent from each other in Eq. (44),thus it is not hard to obtain the optimal point by ∂ ¯ F G /∂g q = ∂ ¯ F G /∂g p = ∂ ¯ F G /∂θ = 0 , i.e., θ = π/ and g q = g p = g , where g is given by g = √ T ( e − κt sinh 2 r + 2 χ )2[1 + T (sinh r + χ )] . (45)At this optimal point, the average optimized fidelity canbe expressed as ¯ F optG = 1 H ( g, / √
2) + χ (1 − g √ T ) . (46) O p t i m a l f i de li t y tA=15 FIG. 7: (Colour online) The optimal fidelity for teleporting CSsas a function of κt with ¯ n = 0 , r = 0 . , T = 0 . and A =0 . , . , , . , , from top to bottom, respectively. It is clearly seen that the optimal factor g depends notonly on T , but also on the evolution time κt in a differentform. In particular, when T → and κt → , Eq. (45)reduces to the result in Ref. [32]. In addition, in the lim-itation case of χ → ∞ which implies that the probabilitydistribution includes the whole complex plane, then wehave g → / √ T , which just corresponds to the fidelityindependent of ǫ .Using Eq. (44) or (46), we have plotted the optimalfidelity as a function of squeezing parameter r and κt for some different values of χ in Fig. 8 and Fig. 9, re-spectively. From Fig. 8, we can see that the smaller thedistribution χ is, the higher the optimal fidelity is. As χ increases which implies that we have less knowledgeof the amplitude of the teleported states, the optimal fi-delity approaches to that in the standard scheme ( g = 1 ).In addition, as r increases, the fidelity first increases upto a κt -dependent maximum r max , and then decreasesfor a larger values of r for a given big T χ . Actually, using ∂ ¯ F optG /∂r = 0 , we can get e r max = ( e κt + 1) T χ + 1( e κt − T χ − . (47)For instance, when T = 1 , χ = 300 and κt = 0 . , then r max ≃ . , which is in agreement with the numericalresult in Fig. 8. In Fig. 9, we also consider the effect ofdecoherence on the fidelity. We can see the similar re-sults to the case of circle distribution. Among these threedistributions used above, the line distribution presentsthe most improvement for fidelity, but the Gaussian dis-tribution presents the lowest improvement. However, acommon advantage is that the fidelity with CV can beimproved by using the tunable parameters even in theenvironments when compared with the standard telepor-tation scheme.0 O p t i m a l f i de li t y Squeezing parametetr r
FIG. 8: (Colour online) The optimal fidelity for teleporting CSsas a function of r with ¯ n = 0 , T = 1 , χ = 0 . , . , , , and T = 1 . For each optimized case (associated with a special plotstyle), the corresponding lines are arranged from top to bottomwith the increasing χ . κt = 0 , . correspond to solid and dashlines, respectively. O p t i m a l f i de li t y t FIG. 9: (Colour online) The optimal fidelity for teleportingCSs as a function of κt with ¯ n = 0 , r = 0 . , T = 0 . and χ = 0 . , . , , , from top to bottom, respectively. For com-parison, the fidelities with g=1 are plotted here (dash lines). VI. CONCLUSIONS
In this paper, we have examined the performance ofthree-tunable parameters in realistic scheme of CV quan- tum teleportation with input coherent states and theTMSV entangled resources. For our purpose, we haveappealed to the input and output relation in the CF for-malism, which is convinent for nonideal inputs and anyentangled resources. In this realistic scheme, we have de-rived the condition that the fidelity is independent of theamplitude of input coherent states for any entangled re-source. In order to investigate the effect of three-tunableparameters on the fidelity of teleportation in the non-ideal scheme, we have derived the analytical expressionsof the optimal fidelity for input coherent states with threedifferent prabability distributions and investigated theperformance of optimal fidelity. It is theoretically shownthat the usefulness of tunable parameters for improvingthe fidelity of teleportation with or without the effect ofenvironment and imperfect measurements. In particu-lar, for the input coherent states with a linear distribu-tion, the optimization with three tunable parameters isthe best one with respect to single- and two-parameteroptimization, especially in the region of small squeezing.It would be interesting to extend the present analysisto teleport two-mode states (ideal or nonideal cases) us-ing multipartite (non-)Gaussian entangled resources inthe formalism of CF. In addition, a recent comparison be-tween the well-known CV VBK scheme and the recentlyproposed hybrid one by AR has been made [33, 34].It is shown that the VBK teleportation is actually infe-rior to the AR teleportation within a certain range, evenwhen considering a gain tuning and an optimized non-Gaussian resource. Thus it may be worthy of consider-ing whether these three-parameter optimization can fur-ther improve the fidelity in VBK scheme over that in ARscheme especially in the non-ideal scheme.
ACKNOWLEDGEMENTS: