Improving teleportation of continuous variables by local operations

Improving teleportation of continuous variables by local operations

Ladislav Mišta, Jr. and Radim FilipDepartment of Optics, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic

sReceived 19 March 2004; revised manuscript received 6 December 2004; published 25 March 2005d

We study a continuous-variablesCVd teleportation protocol based on a shared entangled state produced bythe quantum-nondemolitionsQNDd interaction of two vacuum states. The scheme utilizes the QND interactionor an unbalanced beam splitter in the Bell measurement. It is shown that in the nonunity gain regime the signaltransfer coefficient can be enhanced while the conditional variance product remains preserved by applyingappropriate local squeezing operation on sender’s part of the shared entangled state. In the unity gain regime,it is demonstrated that the fidelity of teleportation can be increased with the help of the local squeezingoperations on parts of the shared entangled state that effectively convert our scheme to the standard CVteleportation scheme. Further, it is proved analytically that such a choice of the local symplectic operationsminimizes the noise by which the mean number of photons in the input state is increased during the telepor-tation. Finally, our analysis reveals that the local symplectic operation on sender’s side can be integrated intothe Bell measurement if the interaction constant of the interaction in the Bell measurement can be adjustedproperly.

DOI: 10.1103/PhysRevA.71.032342 PACS numberssd: 03.67.2a, 03.65.Ud

Quantum entanglement can be used for information car-ried by quantum objects to be processed in a unique way. Forinstance, entanglement enables a disembodied transfer of anunknown state of one quantum system to another systemwith precision that cannot be achieved using only classicalresources. This so-called quantum teleportation was first pro-posed theoreticallyf1g and realized experimentallyf2g in thecontext of observables with discrete spectra. The studies ofquantum teleportation were not restricted to the discrete vari-ables, but they were also extended by Vaidmanf3g into therealm of physical quantities with continuous spectra–continuous variables. A feasible implementation of Vaid-man’s continuous-variablesCVd teleportation protocol wasthen proposed by Braunstein and KimblesBKd f4g.

In the BK protocol, the role of the CVs is played by thecanonically conjugate variablesxin and pin of the input CVsystem “in,” and the state to be teleported is an unknowncoherent state. At the beginning of the BK protocol a senderAlice sAd and a receiver BobsBd share a CV entangled stateof two CV systemsA and B described by the canonicallyconjugate variablesxA, pA, xB, andpB sfxi ,pjg= idi jd. At thefirst stage, Alice performs the so-called Bell measurement onthe systemsA and “in” by superimposing them on a balancedbeam splittersBSd and detecting at its outputs the variablesxin8 =sxin+xAd /Î2 and pA8 =spin−pAd /Î2. She obtains certainclassical valuesx̄in and p̄A and sends them via a classicalchannel to Bob. In order to recover the input variablesxinandpin in his system, Bob amplifies the measurement resultsby the gain factorÎ2 and performs the displacementsxB→xout=xB+Î2x̄in and pB→pout=pB+Î2p̄A. As a result,Bob’s conjugate variables read asxout=xin+sxA+xBd andpout=pin−spA−pBd f5g, and thus the input variables wereteleported to Bob with some added noisesxA+xB and pA−pB. Without entanglement, Alice and Bob can achieve onlylimited quality of the teleportation. In the best case, whensystemsA andB are prepared in the vacuum statesf6g, twovacuum units of noise are added into each of the input vari-

ablesxin andpin. This noise limits the quality of the telepor-tation to the classical regime. However, the noise can bereduced and therefore the so-called quantum regime of tele-portation can be achieved if Alice and Bob use a specificentangled state that possesses quantum correlations of equalstrength between canonically conjugate variables whosestrength increases with increasing entanglement; i.e.,ksxA+xBd2l=kspA−pBd2l=e−2k→0 for k→` sk.0 is a squeez-ing parameterd. Obviously, sharing this type of entanglement,Alice can teleport the input state to Bob with arbitrarily highprecision if the entanglement is sufficiently strong. This typeof entanglement is called Einstein-Podolsky-RosensEPRdentanglementf7g, which can be prepared by mixing the sys-temsA andB equally squeezed in conjugate variablesxA andpB, respectively, on a balanced BSf8g.

To date, all the ingredients of the BK scheme encompass-ing beam splitters, squeezers, and detectors of canonicallyconjugate variables were well managed only for light. There-fore, it has been possible to date to demonstrate only CVquantum teleportation of lightf9g. In these experiments, therole of the CV systems was played by the single modes of anoptical field and the CVs were realized by the quadratureamplitudes of the field. Recently, however, great attentionhas been paid to the teleportation protocols for CVs of ma-terial objectsf10,11g. These protocols involve two kinds ofCV systems: coherent linearly polarized light pulses andspin-polarized macroscopic atomic samples. The respectiveCVs xA andpA are for a light pulseA realized by the properlynormalized components of the operator of the Stokes vector;for an atomic sampleB, the respective CVsxB and pB arerealized by the properly normalized components of the col-lective spin operator of the atomic samplef12g. As in the BKscheme, the protocols exploit the EPR entanglement of twoatomic samplesf10g or the EPR entanglement of two lightbeamsf11g. However, the specific feature of the two systemsinvolved in these protocols is that they interact naturally viathe quantum-nondemolitionsQNDd interaction f11g de-scribed by the interaction Hamiltonian

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HQND = − kxApB, s1d

wherek is the coupling constant. Like the BS interaction, theQND interaction of two orthogonally squeezed vacuumstates also produces EPR entanglementf13g. More interest-ingly, the QND interaction produces the entangled state evenfrom two vacuum states, but this is no more the EPR state.Therefore, the use of such the QND entangled state in theBK protocol will not lead to its optimal performance. Forthis reason, Horoshko and KilinsHKd f14g studied the tele-portation protocol based on sharing of the QND entangle-ment and utilizing an unbalanced BS in the Bell measure-ment. They have shown that quantum regime of teleportationcan be achieved if sufficiently strong QND entanglement isavailable.

In this article we investigate a generalized CV teleporta-tion protocol that contains the HK protocol as a particularcase. Our protocol is based on the QND entanglement andutilizes either the QND interaction or an unbalanced BS inthe Bell measurement. We study both the nonunity gain andthe unity gain regimes. The nonunity gain regime is charac-terized by the conditional variance productV and the signaltransfer coefficientT f15g. It is shown in two particular ex-amples that for a fixed interaction in the Bell measurement,the parameterT can be enhanced while preserving the pa-rameterV, and thus the performance of our protocol can beimproved by applying a suitable squeezing operation on Al-ice’s part of the shared entangled state. The unity gain re-gime is characterized by the fidelityF. In contrast with theHK scheme, our scheme allows us to achieve quantum re-gime of teleportation whenF.1/2 for an arbitrarily smallamount of the shared QND entanglement. For a particularamount of the shared entanglement corresponding to the fea-sible interaction constantkt=1 f16g st is the interactiontimed, we show analytically that the use of a suitable unbal-anced BS in the Bell measurement provides a higher telepor-tation fidelity than the use of a balanced BS as proposed byHK. Recently, it was demonstrated inf17g that in the tele-portation protocol exploiting entanglement produced from asingle squeezed state and a balanced BS in the Bell measure-ment, one can increase the teleportation fidelity by applyinglocal squeezers on parts of the shared entangled state. Herewe show that the fidelity in our scheme can be increased byapplication of local squeezing operations on both parts of theshared entangled state that convert effectively the scheme tothe BK schemef4g. Moreover, it is proved analytically that ifwe characterize the unity gain teleportation by the noiseNby which the mean number of photons in the input state isincreased during the teleportation, then such a choice of thelocal operations is optimal.

The paper is organized as follows. In Sec. I we study ateleportation scheme utilizing the QND entanglement as anentanglement resource and the QND interaction or an unbal-anced BS in the Bell measurement. Section II deals with ageneric CV teleportation scheme, and the optimization of thenoiseN with respect to the local symplectic transformationsis performed. Section III contains the conclusion.

I. QUANTUM TELEPORTATION WITH QUANTUM-NONDEMOLITION INTERACTION

The scheme of our teleportation protocol is depicted inFig. 1. If not stated explicitly otherwise, an optical terminol-

ogy is used throughout the article in which the CV systemsinvolved are single modes of an optical field and the role ofthe canonically conjugate variables is played by the quadra-ture amplitudes of the optical field. We use the Heisenbergpicture description of teleportation as introduced inf5g.

At the beginning of our protocol, Alice and Bob share thestate of two modesA andB produced by the QND interac-tion s1d of two vacuum statesu0lA and u0lB, respectively. Inthe Heisenberg picture, the quadraturesxA

s0d, pAs0d, xB

s0d, andpBs0d

of the two vacuum modes transform as

xA = xAs0d, pA = pA

s0d + gpBs0d,

s2dxB = xB

s0d − gxAs0d, pB = pB

s0d,

whereg=kt st is the interaction timed is the interaction con-stant. For anyg.0, the produced state is an entangled statewhose entanglement increases with increasingg. This can beseen by noting that the purity of the reduced state of themode A given by the formula PA=1/s2ÎkxA

2lkpA2ld

=1/Î1+g2 is less than unity for anyg.0 and goes to zerowith increasingg. Here and in the rest of this section, theangle brackets denote the averaging over the initial stateualinu0lAu0lB. The produced entanglement is not the EPR en-tanglement as ksxA+xBd2l=kspA−pBd2l=fsg−1d2+1g /2ù1/2 and therefore it does not “match” the BK scheme. Forthis reason, Alice performs the Bell measurement that differsfrom the Bell measurement used in the BK scheme. She letsher modeA interact with the input mode “in” whose un-known coherent stateualin is to be teleported in another QNDinteraction HQND8 =k8xApin and subsequently measures the

FIG. 1. Schematic of the CV teleportation using the QND inter-action. A bipartite entangled state produced by the QND interactions1d of two vacuum statesu0lA and u0lB is shared by Alice and Bob.ualin, an unknown input coherent state;rout, output state;SA andSB,auxiliary squeezing operations allowing to improve the quality ofteleportation;R, the QND interaction or an unbalanced BS;Gx, Gp,electronic gains for the transformation from the classical measure-ment resultsx̄in and p̄A to Bob’s output system.

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quadraturesxin8 =xin+g8xA and pA8 =pA−g8pin sg8=k8t8d. ThequadraturesxA and pin are the so-called QND variables thatare not affected by the interactionHQND8 . This interaction,however, establishes correlations betweenpA and pin. If themode A were infinitely squeezed in the quadraturepA, themeasurement ofpA8 would provide a result proportional to theresult of the measurement ofpin. This is the principle of theQND measurement. Thus, the Bell measurement performedby Alice in fact consists of the QND measurement of thequadraturepin with the ancillary modeA being a part of anentangled system and the direct measurement of the quadra-ture xin at the output of the QND interaction. After the mea-surement, Alice has certain classical valuesx̄in and p̄A andsends them via a classical channel to Bob, who performs thedisplacementssdenotedDx,p in Fig. 1d xB→xout=xB+Gxx̄inandpB→pout=pB−sGp/g8dp̄A, where the parametersGx andGp describe normalized gains. As a result, Bob’s outputquadratures read as

xout = Gxxin + X, pout = Gppin − P, s3d

where

X = Gxg8xA + xB = sGxg8 − gdxAs0d + xB

s0d,

s4d

P =Gp

g8pA − pB =

Gp

g8pA

s0d − S1 − Gpg

g8DpB

s0d,

are the quadrature added noises. Note that, instead of usingthe QND interaction in the Bell measurement, Alice can useequally well an unbalanced BS with reflectivityR and trans-missivity T. In this case, she detects the quadraturesxin8=RxA+Txin, pA8 =−Rpin+TpA, and Bob performs the dis-placements xB→xout=xB+sGx/Tdx̄in and pB→pout=pB

−sGp/Rdp̄A. Consequently, Bob’s output quadratures aregiven by Eqs.s3d and s4d, whereg8 is replaced byR /T.

In order to quantitatively assess the quality of our telepor-tation, it is convenient to distinguish two different regimes ofteleportation.

A. Nonunity gain teleportation

The quality of teleportation is often expressed quantita-tively by the fidelityF=inka uroutualin, whererout is the out-put state. In the nonunity gain regimessGxÞ1,GpÞ1d,however, the fidelity depends strongly on gainsGx and Gpand usually it decreases very quickly with increasing devia-tion of the gains from unityf18g. Therefore, there can existnonunity gain regimes having a quantum nature that are notcaptured by the fidelity. For this reason, a more detailed char-acterization of teleportation was introduced with the help ofthe conditional variance productV and the signal transfercoefficientT f15,18g. The parameterV is defined by the for-mula

V = Voutuinx Voutuin

p , s5d

where Voutuinx =ksDxoutd2l−Gx

2ksDxind2l and Voutuinp =ksDpoutd2l

−Gp2ksDpind2l are the quadrature conditional variances be-

tween the input and output states, respectively. The condi-

tional variance productV describes the noise added into theinput state during the teleportation process. Without en-tanglement, the noise is limited from below by the fluctua-tions in Bob’s mode and thereforeVù1/4 f18g. V=0 indi-cates teleportation with no added noise.

The signal transfer coefficient is defined as a sumT=Tx+Tp of the quadrature signal transfer coefficientsTi=Sout

i /Sini si =x,pd, whereSi denotes the conventional signal-

to-noise ratio for quadraturei. For teleportation of coherentstates, the parameterT can be expressed in the formf19g

T =Gx

2

Gx2 + 2Voutuin

x +Gp

2

Gp2 + 2Voutuin

p . s6d

In the absence of entanglement, the intrinsic fluctuations ofAlice’s mode limit the transfer coefficient toTø1 f18g. Onthe other hand,T=2 corresponds to a perfect signal transfer.With respect to the properties of the parametersT andV, it isthus natural to define a quantum regime of teleportation as aregime in whichV,1/4 and simultaneouslyT.1.

In our protocol, the added noisess4d are uncorrelated withthe input state. Hence,Voutuin

x =kX2l, Voutuinp =kP2l and there-

fore

V = kX2lkP2l, s7d

T =Gx

2

Gx2 + 2kX2l

+Gp

2

Gp2 + 2kP2l

, s8d

where

kX2l = fsGxg8 − gd2 + 1g/2,s9d

kP2l = fsGp/g8d2 + s1 − Gpg/g8d2g/2

are variances calculated from the Eq.s4d.In what follows, we restrict our attention to two particular

nonunity gain regimes. For these two cases we then show,among other things, that if our scheme operates in the quan-tum regime whenV,1/4 andT.1, then the signal transfercoefficient T can be increased while preserving the condi-tional variance productV by application of a suitable localsqueezing operation on Alice’s mode.

1. The regime when the output state has minimum addi-tional noise; i.e., the gainsGx andGp are adjusted in such away thatV=Vmin;minGx,Gp

V. Deriving the Eq.s7d with re-spect toGx andGp, setting the obtained expressions equal tozero, and solving these equations, one finds that the param-eter V is minimized by Gx,min=g/g8 and Gp,min=gg8 / s1+g2d. For such gains, the parametersV andT read as

Vmin =1

4s1 + g2d, s10d

TVmin= 1 +

g82Sg2 −Î5 + 1

2DSg2 +

Î5 − 1

2D

sg2 + g82ds1 + g2 + g2g82d. s11d

Apparently, sinceVmin,1/4 for anyg.0, our scheme op-erates from the point of view of the parameterV in the quan-

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tum regime as soon as there is a nonzero shared entangle-ment. However, Eq.s11d reveals that our scheme in generaldoes not operate in the quantum regime from the point ofview of the parameter T as TVmin

.1 only if g.ÎsÎ5+1d /2<1.27. Hence, in order to fulfill both the con-ditions Vmin,1/4 andTVmin

.1 simultaneously and thus toreach in this particular case the quantum regime of telepor-tation, the shared entanglement must be sufficiently large.The parameterTVmin

depends on the asymmetry parameterg8of the Bell measurement and can be maximized with respectto it. Solving the extremal equationdTVmin

/dg8=0, one finds

that TVminhas a maximum ifg.ÎsÎ5+1d /2 and it is local-

ized in the pointgopt8 =s1+g2d1/4. Substitutinggopt8 to the Eq.s11d, we arrive at the optimal signal transfer coefficient

TVmin,opt=2g2

g2 + Î1 + g2. s12d

The latter analysis indicates that ifg8Þgopt8 , the parameterTVmin

can be increased to the optimal values12d by adjustingg8 in the Bell measurement to the optimal valuegopt8 . If theBell measurement is fixed, i.e.,g8 is fixed, the optimal valuegopt8 can be adjusted by a suitable squeezing operation onAlice’s mode. Namely, the squeezing operationxA→erAxAand pA→e−rApA srA is the squeezing parameterd effectivelytransformsg8 in Eq. s4d to g̃=erAg8. Therefore, if the squeez-ing operation is such thaterA=s1+g2d1 / 4 /g8 we attain opti-mal value of the signal transfer coefficients12d. As the pa-rameterVmin remains preserved when changingg8 sit doesnot depend ong8d, we get deeper into the quantum region ofteleportation and thus improve the quality of teleportation inour protocol. For example, if we takeg=2.5 and the Bellmeasurement is realized by the balanced BS for whichg8=R /T=1, formula s11d gives TVmin

<1.32, whereas for the

unbalanced BS withgopt8 =s1+g2d1 / 4 <1.64, it gives a highersignal transfer coefficientTVmin,opt<1.4. In addition, for thescheme withg8=1, Vmin→0, but TVmin,opt→1+g82/ s1+g82d=1.5 with increasingg, in contrast with the improvedscheme whereVmin→0 andTVmin,opt→2 with increasingg,and therefore we approach perfect teleportation from thepoint of view of the parametersT and V with increasingshared entanglement.

2. The regime when the input signals are transferred op-timally; i.e., the gainsGx andGp are adjusted in such a waythat T=Tmax;maxGx,Gp

T. Solving the extremal equations]T/]Gi =0 si =x,pd, whereT is given in Eq.s8d, one findsthat the parameterT is maximized byGx,max=s1+g2d /gg8and Gp,max=g8 /g. For such gains the parametersT and Vread as

Tmax= 1 +g2g82

s1 + g82ds1 + g2 + g82d, s13d

VTmax=

1

4S 1

g2 +1

g4D . s14d

Clearly,Tmax.1 for anyg,g8.0, and therefore our schemeoperates in the quantum regime of teleportation from the

point of view of the parameterT once there is some sharedentanglement. On the other hand, Eq.s14d reveals thatVTmax

,1/4 only if g4−g2−1=fg2−sÎ5+1d /2gfg2+sÎ5−1d /2g.0. Hence, as in the previous case, our schemeworks in the quantum regime of teleportation whenTmax

.1 and simultaneouslyVTmax,1/4 only if g.ÎsÎ5+1d /2

<1.27; i.e., only if the shared entanglement is strongenough. Further, as in the previous case, we can maximizeTmax with respect tog8. Solving the respective extremalequationdTmax/dg8=0, one finds thatTmax attains a maxi-mum at the same pointgopt8 =s1+g2d1 / 4 asTmin and it is equalto

Tmax,opt=2Î1 + g2

1 +Î1 + g2. s15d

Similar to case 1, this optimal value of the signal transfercoefficient can be reached by adjustingg8 in the Bell mea-surement to thegopt8 or, for fixedg8 in the Bell measurement,by using the same squeezing operation on Alice’s mode as incase 1. Further, forg=2.5 and for the balanced BS in the Bellmeasurementsg8=1d, Eq. s13d leads to the signal transfercoefficient Tmax<1.38, while for the unbalanced BS withgopt8 <1.64 we obtain, using Eq.s15d, a higher valueTmax,opt<1.46. Finally, in both the particular casesVTmax→0 with increasingg, whereasTmax→1+g82/ s1+g82d=1.5and Tmax,opt→2; therefore only the improved scheme ap-proaches perfect teleportation from the point of view of theparametersT andV with increasing shared entanglement.

B. Unity gain teleportation

In this regime the gains are adjusted in such a way thatGx=Gp=1. It then follows from Eqs.s3d ands4d that the firstmoments of the input state are preserved, i.e.,kxoutl=kxinland kpoutl=kpinl. This implies that in the unity gain regime,the quality of our teleportation depends only on the addednoisess4d and therefore it can be conveniently described bythe fidelity F=inkauroutualin. F=1 corresponds to a perfectteleportation whileF.1/2 indicates a quantum regime ofteleportationf6g. Since in our protocol the added noisess4dare uncorrelated both mutually and also with the input state,the fidelity can be expressed asf18g

F =1

Îs1 + kX2lds1 + kP2ld. s16d

Calculating the varianceskX2l andkP2l using Eqs.s9d whereGx=Gp=1, we finally arrive at the fidelity of teleportation ofour scheme in the form

F =2

Îf2 + s1/g8d2 + sg/g8 − 1d2gf3 + sg − g8d2g. s17d

Several important properties of our teleportation protocol canbe derived from the latter formula. First, it reveals that if theparameterg8 in the Bell measurement can be adjusted prop-erly, then the quantum regime of teleportation can be attainedfor an arbitrarily small amount of the shared entanglement.This follows from the fact that if we put, e.g.,g8=g+1 in Eq.


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s17d, thenF.1/2 for anyg.0. Second, as the denominatorin Eq. s17d is always greater or equal toÎ2Î3, the maximumachievable fidelity is bounded by the valueFmax=Î2/3<0.816 that can be achieved, e.g., in the limitg=g8→`.Finally, formulas17d also reveals that although one could betempted to think that for a giveng maximal fidelity is ob-tained forg=g8, this is not the case, and the fidelity is maxi-mized byg8 that in general differs fromg. To illustrate this,let us consider the particular value of the interaction constantg=1 that is well within reach of the current experimentf16g.Setting the derivative of Eq.s17d with respect tog8 equal tozero and solving the obtained extremal equation, one findsthat F attains a maximum forg8=4/3.g=1. On insertingg8=4/3 back into Eq.s17d for g=1, we finally obtain themaximum fidelity equal toF1=2Î6/7<0.7.

Our results can be compared with the results obtained byHoroshko and KilinsHKd f14g. In our protocol and in theHK protocol, the shared entanglement is produced by theQND interactions1d, and the Bell measurement utilizes anunbalanced BS. In contrast with our protocol, in the HKprotocol the asymmetry parameterR /T of the BS and theinteraction constantg are tied together by the relationR /T=g. Therefore, forg=g8, formula s17d reduces to the fidelityof teleportation of coherent states in the HK protocol

FHK =2

Î3s2 + 1/g2d. s18d

Clearly, as in our teleportation protocol, the maximum tele-portation fidelity isFHK,max=Î2/3, and it is achieved in thelimit g→`. However, in contrast to our protocol, quantumteleportation can be achieved only ifg.Î3/10<0.548; i.e.,FHK .1/2 only if sufficiently large shared entanglement isavailable. In addition, our results reveal that forg=1, ourscheme provides a higher teleportation fidelityF1<0.7 thanthe HK scheme, which gives onlyFHK,1=2/3<0.667.Therefore, in the teleportation scheme using the shared en-tanglement produced by the QND interactions1d with g=1of two vacua, a higher teleportation fidelity is obtained in thescheme with an unbalanced BS in the Bell measurement withT=3/5 andR=4/5 than in the HK scheme, which exploitsthe balanced BS.

The previous results reveal that no matter what the valueof the parameterg8 in the Bell measurement and how strongthe shared entanglement, the highest possible fidelityFmax=Î2/3 is always less than unity. In other words, the qualityof transfer in our teleportation protocol is limited even in thelimit of the infinitely large shared entanglement. This behav-ior is due to the added noiseX given in Eq.s4d, which alwayscontains at least one unit of the vacuum noise originatingfrom the termxB

s0d. However, this undesirable noise can bereduced and therefore the fidelity of our scheme can be in-creased if Alice and Bob transform the shared state by thelocal squeezing operationssdenoted asSA andSB in Fig. 1dxA→ sa/g8dxA, pA→ sg8 /adpA, xB→ s1/adxB, and pB→apB,wherea=s1+g2d1 / 4. These operations transform the originaladded noisesX=g8xA+xB andP=s1/g8dpA−pB into the newadded noisesX8=axA+s1/adxB and P8=s1/adpA−apB. Cal-culating the varianceskX82l and kP82l using Eq. s2d, one

finds thatkX82l=kP82l=Î1+g2−g→0 with increasingg andtherefore the new added noises vanish with increasing en-tanglement. Hence, by using the Eq.s16d we finally arrive atthe fidelity of teleportation in the improved scheme of theform

FS=1

1 +Î1 + g2 − g. s19d

Obviously, Eq.s19d demonstrates that the quantum regime ofteleportation whenFS.1/2 is again reached for anyg.0.Moreover, sinceÎ1+g2−g→0 with increasingg F→1 withincreasingg, and the output state converges to the input statewith increasing entanglement, similar to the BK scheme. Infact, formulas19d can be rewritten in the form of the fidelityof the BK scheme asFBK =1/s1+e−2kd f20g, wherek=s−1/2dlnsÎ1+g2−gd and thus the squeezing operationsSA

andSB were chosen such that they effectively transform ourscheme into the BK scheme. Characterizing our teleportationprotocol by the noise by which the mean number of photonsin the input state is increasedN=skX2l+kP2ld /2 instead ofthe fidelity s16d, such a choice ofSA and SB maximizesNand therefore our improved scheme is optimal from the pointof view of the parameterN. The analytical proof of the latterstatement is given in the next section.

We have seen that in some cases the application of suit-able local squeezing operations on parts of the shared en-tangled state can improve considerably the efficiency of theCV teleportation. Despite the fact that both the squeezedstate of a light beamf21g and a squeezed state of an atomicsamplef22g were already realized experimentally, it may bestill an experimental challenge to implement the squeezingoperations on parts of the entangled state. The obstacle canbe partially avoided and Alice’s squeezing operationSA canbe saved if one can adjust properly the interaction constantg8 sor the ratioR /Td in the Bell measurement. To be morespecific, if in our case we putg8=a=s1+g2d1 / 4 fi.e., R /T=a=s1+g2d1 / 4g, the interaction in the Bell measurement ef-fectively realizes the needed squeezing operationSA.

Contrary to the operationSA, the squeezing operationSBon Bob’s side is, in general, inevitable, in particular if oneneeds to preserve the whole quantum staterout for furtherprocessing. However, the squeezing operationSB can beavoided if the input stateualin just carries some classicalinformation encoded into its complementary quadraturesxinand pin that is read from the output staterout immediatelyafter the teleportation by homodyne measurement of eitherof the output quadraturesxout or pout. Namely, formally theoperationSB can be moved behind the displacement transfor-mation Dx,p ssee Fig. 1d that then must be changed toxB→xout=xB+ax̄in and pB→pout=pB−1/sag8dp̄A. The neededsqueezing operationSB then can be realized merely as a scal-ing x̄out→ s1/adx̄out or p̄out→ap̄out of the classical outcomesx̄out and p̄out of the homodyne measurement.

II. GENERALIZED CONTINUOUS-VARIABLETELEPORTATION PROTOCOL

In this section we will prove analytically that the im-provement of our previous unity gain teleportation scheme


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by local squeezing operations that effectively transform thescheme into the BK scheme is in a certain sense that is speci-fied below optimal. For this reason, let us consider the gen-eralization of the protocol depicted in Fig. 1. In the general-ized scheme, Alice and Bob share a Gaussian statesrABd oftwo modesA and B with vanishing first moments; i.e.,kjkl=TrsrABjkd=0 sk=1, . . . ,4d, wherej=sxA,pA,xB,pBdT is thecolumn vector of the quadraturesxA, pA, xB, andpB. Such astate is completely characterized by the variance matrixVABwith elementssVABdkl=khDjk,Djljl, whereDjk=jk−kjkl andhA,Bj;s1/2dsAB+BAd. We assume that Alice and Bob canapply locally on their parts of the shared state arbitrarysingle-mode linear transformationsSA andSB of the quadra-tures xi ,pi si =A,Bd preserving the canonical commutationrules fxi ,pjg= idi j . In matrix notation, these transformationscan be expressed by the formulas

jA8 = SAjA, jB8 = SBjB, s20d

where ji =sxi ,pidT, ji8=sxi8 ,pi8dT si =A,Bd and Si are real 2

32 matrices satisfying the so-called symplectic condition

SiJSiT = J, J = S 0 1

− 1 0D . s21d

These transformations involve single-mode squeezers andphase shifters as particular instances and are conventionallydenoted as symplectic transformations. Alice then performsthe Bell measurement. She lets her modeA and the mode“in” interact in a two-mode interactionR described by thequadratic Hamiltonian of the formHR=oi,j=1

4 aijziz j, whereaij are real coupling constants andzi si =1, . . . ,4d are com-ponents of the column vectorz=sxA8 ,pA8 ,xin ,pindT. In theHeisenberg picture the interaction is described by the two-mode linear symplectic transformation of the formf23g

z8 = Rz, s22d

wherez8=sxA9 ,pA9 ,xin8 ,pin8 dT andR is a real 434 matrix sat-isfying the two-mode symplectic condition

RVRT = V, V = J % J. s23d

The two-mode symplectic transformations involve the unbal-anced BS and QND interactions1d as particular instances.After that, Alice completes the Bell measurement by homo-dyne detection of the position quadraturexin8 on mode “in”and the momentum quadraturepA9 on modeA. Introducingthe vector of the input quadraturesjin=sxin ,pindT, the vectorof the detected quadraturesjd=sxin8 ,pA9dT can be expressed as

jd = Yjin + ZjA8 , s24d

where

Y = SR33 R34

R23 R24D, Z = SR31 R32

R21 R22D , s25d

whereRkl sk, l =1, . . . ,4d are elements of the matrixR. Alice

measures certain classical valuesx̄in and p̄A and sends themvia a classical channel to Bob, who completes the teleporta-tion by performing the displacements

jB8 → jout = jB8 + Gj̄d, s26d

where j̄d=sx̄in , p̄AdT is the c-number vector of the measure-ment results and

G = SGxx Gxp

Gpx GppD s27d

is the matrix of the unnormalized gains. Here we restrictourselves to the unity gain regime in which the first momentsof the input state are preservedskjoutl=kjinld and thereforewe chose

G = Y−1 =1

R24R33 − R23R34S R24 − R34

− R23 R33D , s28d

whereY−1 is the inverse matrix to the matrixY that is as-sumed to be regular. Hence, using Eqs.s20d, s24d, ands26d,one finally finds Bob’s output quadrature operators in theform

jout = jin + Y−1ZSAjA + SBjB. s29d

A more instructive shape can be given to the latter formula.Defining the matrixS;s3Y

−1Z, wheres3 is the Pauli diag-onal matrix s3;diags1,−1d, and taking into account thecondition s23d, it can be shown that the new matrixS satis-fies the conditions21d and it is therefore a symplectic matrix.Formulas29d can then be rewritten in the following form:

xout = xin + X, pout = pin − P, s30d

where

SX

PD = SxA9 + xB8

pA9 − pB8D = S̃AjA + s3SBjB s31d

is the column vector of the quadrature added noisesX andP

and S̃A=SSA in another symplectic transformation.The above analysis reveals that the use of a generic qua-

dratic interactionHR in the Bell measurement has two ef-fects. First, in order for our teleportation protocol to preservethe first moments of the input state, Bob has to received

classical outcomes of the Bell measurementj̄d=sx̄in , p̄AdT

transform by the matrix of gainsG=Y−1 before using themfor the displacements of his mode. For example, if the inter-actionHR is the QND interactionHQND8 =k8xA8pin consideredin the previous section, thenG is the diagonal matrixG=diags1,−1/g8d sg8=k8t8d. Second, the interactionHR per-forms effectively a symplectic transformationS on Alice’smodeA. Now, consider the case in which there are no aux-iliary symplectic transformations on Alice’s and Bob’s sidessSA=SB=1d and the state shared by Alice and Bob has theEPR entanglement. In this case, the transformationS entailsthat the actual added noisesX and P do not match with the“optimal” added noisesXEPR=xA+xB, PEPR=pA−pB andhence the amount of the noise added into the input state is


032342-6

larger in comparison with the optimal case. This undesirableeffect of the transformationS can be compensated by usingan auxiliary symplectic transformationSA on modeA of theform SA,EPR=S−1. For instance, if the Bell measurement isrealized by the QND interactionHQND8 ssee Sec. Id, thenS isthe squeezing transformationSQND=diagsg8 ,1 /g8d andthereforeSA,EPR=SQND

−1 =diags1/g8 ,g8d. If, on the other hand,Alice and Bob share a state with other than an EPR entangle-ment and the interaction constant of the interaction in theBell measurement can be controlled, it can be adjusted insuch a way that the effective transformationS matchessonAlice’s sided the actual added noises with the optimal addednoises and thus enhances the teleportation fidelity. In the caseof the entanglement produced by the QND interactions1d oftwo vacuum states, the matching is achieved ifg8=s1+g2d1 / 4 ssee Sec. Id.

Before going further, let us return for a while to the regu-larity requirement of the matrixY given in Eq. s25d thatemerged naturally when expressing the vectorjin from Eq.s24d . Obviously, this property guarantees that the measuredquadraturesxin8 andpA9 carry information about both comple-mentary quadratures of the input state and therefore the firstmomentskxinl and kpinl of the input state can be preserved,as is a natural demand for any unity gain teleportation pro-tocol. Consequently, all the interactions yielding regular ma-trix Y can be used in our scheme for the Bell measurement.For that reason, we further restrict our attention to the inter-actions for whichY is a regular matrix.

The quality of the CV teleportation protocols in a unitygain regime is well expressed quantitatively by the fidelity,which for pure input states is defined asf6g

F = 2pE−`

+`

Winsx,pdWoutsx,pddxdp, s32d

whereWinsx,pd and Woutsx,pd are Wigner functions of theinput and output state, respectively. Assuming the input state,which, similar to the shared staterAB, can be described by aGaussian Wigner function, expressing both the Wigner func-tions Win and Wout as Fourier transforms of correspondingcharacteristic functions, carrying out all integrals, and takinginto account the relationkjoutl=kjinl following from Eq.s30d,we find that

F =1

ÎdetsVout + Vind, s33d

where

Vi = S ksDxid2l khDxi,DpijlkhDpi,Dxijl ksDpid2l

D , s34d

is the single-mode variance matrix of the statei, i =out, in.Utilizing formula s30d, one finds that the output variancematrix Vout is of the form

Vout = Vin + 2N, s35d

where

2N = S kX2l − khX,Pjl− khP,Xjl kP2l

D . s36d

Substituting the last expression of the Eq.s31d for the addednoisesX and P into the matrixs36d, it can be expressed inthe following matrix form:

2N = s3S̃AAS̃ATs3 + SBBSB

T + s3S̃ACSBT + SBCTS̃A

Ts3,

s37d

whereA, B, C, andCT are 232 blocks of the variance ma-trix VAB of the shared state; i.e.,

VAB = S A C

CT BD . s38d

The fidelitys33d attains a particularly simple form in the caseof teleportation of coherent states whenVin=s1/2dI. Com-bining Eqs.s33d and s35d, the fidelity then reads as

Fcoh=1

Î1 + 2 TrN + 4 detN. s39d

The quality of our teleportation is determined by the function2 Tr N+4 detN. For a fixed shared entangled state and afixed interaction in the Bell measurement, this is a compli-cated function of six real parameters of the symplectic trans-formationsSA andSB. Hence, the maximization of the fidel-ity s39d with respect to the transformationsSA and SBamounts to finding the minimum of the function of severalreal variables, which is a task that can be hardly solved evennumerically. Instead of doing this, we will minimize anothermore simple measure of teleportation success given by thenoiseN defined through the formulaf14g

knoutl = kninl + N, s40d

whereknoutl andkninl are the mean numbers of photons in theoutput and input state, respectivelyf14g. Calculatingknoutlusing Eq.s35d, one finds that the noiseN is connected withthe matrixs36d by the formula

N = Tr N = fksxA9 + xB8d2l + kspA9 − pB8d2lg/2. s41d

To make the problem more tractable, we consider that theshared state is a fixed pure entangled two-mode Gaussianstate. In this case there exist local symplectic transformationsmA and mB that bring the variance matrixs38d to the so-called standard formf24g VTMS=smA % mBdVABsmA % mBdT,where

VTMS =1a 0 − c 0

0 a 0 c

− c 0 a 0

0 c 0 a2 , s42d

wherea=ÎdetAù1/2,c=ÎudetCu.0 anda2−c2=1/4. Ow-ing to the last equality and the inequalityc.0, we can put


032342-7

a=coshs2kd /2 andc=sinhs2kd /2, wherek.0, and thus thevariance matrix s42d describes the two-mode squeezedsTMSd vacuum state having squeezed variancesksxA

+xBd2lTMS=kspA−pBd2lTMS=2sÎdetA−ÎudetCud=e−2k fk isthe squeezing parameter andk lTMS denotes averaging overthe TMS vacuum states42dg and therefore possessing theEPR entanglement. Expressing now the symplectic matrices

S̃A and SB in Eq. s37d in the form S̃A=sAmA and SB=sBmB,wheresA,sB are some new symplectic matrices, utilizing Eq.s42d, and substituting the obtained formula for the matrixNto definition s41d, one finds that

N =a

2Trss̃As̃A

T + sBsBTd − c Trss̃AsB

Td, s43d

where s̃A=s3sAs3. Further, according to the Bloch-Messiahreductionf25g one can decompose the symplectic matricess̃AandsB as follows:

s̃A = PsadSsrAdPsbd, sB = PsgdSsrBdPsdd, s44d

where

Psud = Scosu − sinu

sinu cosuD, Ssvd = Sev 0

0 e−v D s45d

are the matrix of the phase shift byu=a ,b ,g ,d and thematrix of the single-mode squeezer with the squeezing pa-rameterv=rA,rB, respectively. On inserting expressionss44dand s45d into Eq. s43d and using the invariance of the tracewith respect to cyclic permutation of its arguments, we arriveat the formula

N = 2a coshr+ coshr− − cfscoshr+ + coshr−dcosu+

+ scoshr+ − coshr−dcosu−g, s46d

where r±=rA± rB and u±=a±b−g7d. Clearly, N is mini-mized with respect tou+ whenu+=2kp, wherek is an inte-ger. Minimization with respect tou− requires us to distin-guish three cases:sid coshr+.coshr−, sii d coshr+,coshr−,andsiii d coshr+=coshr−. The values ofu− minimizingN areequal tou−=2lp in casesid, u−=s2l +1dp in casesii d sl is aninteger in both the casesd, andu− can be arbitrary in the lastcasesiii d. In addition, in all the three cases the necessaryconditions on the extremes]N /]r±=0d are satisfied whenr±=0. Since coshr+=coshr− for r+=r−=0, the functions46dcan have an extreme only on the boundary characterized bythe conditionsiii d, where it reads

N8 = 2 coshr+sa coshr+ − c cosu+d. s47d

In order for the candidates for extrema localized in points forwhich u+=2kp andr±=0 to be minimasno further conditionis obtained asu− can be arbitraryd, the sufficient condition onthe minimum must be satisfied. It is given by the positivedefiniteness of the matrixA with elements

A11 =]2N8

]r+2 , A22 =

]2N8

]u+2 , A12 = A21 =

]2N8

]r+ ] u+. s48d

SinceA11=2s2a−cd.0, A22=2c.0, andA12=0 in the po-tential extrema as follows from the inequalitiesa−c.0 and

c.0, the matrixA is positive definite and hence the poten-tial extrema are minima all giving the same value

Nmin = 2sa − cd = 2sÎdetA − ÎudetCud

=ksxA + xBd2lTMS + kspA − pBd2lTMS

2= e−2k. s49d

The equalitiesr±=0 andu+=2kp imply that rA=rB=0 andg+d=a+b−2kp. From Eq. s44d, it then follows that thetransformations minimizings43d are the arbitrary same-phase shiftss̃A=sB=Psa+bd, where a+b is an arbitraryangle. This means that the noiseN is minimum if the trans-formationsSA and SB are such thatSA=S−1mA and SB=mBsa+b=0 was chosen for simplicityd wheremA,mB bring theshared states38d to the TMS vacuum states42d. From Eq.s37d, it then follows that 2N=NminI and therefore the fidelitys39d reads

Fcoh=1

1 +Nmin=

1

1 + e−2k s50d

and coincides with the fidelity of the BK scheme. Thus, wehave shown that if we characterize our generalized CV tele-portation scheme by the noiseN, then its performance can beimproved by the suitable auxiliary local symplectic transfor-mations on parts of the shared entangled state. The optimum,in which the noiseN attains a minimum, is achieved forthose local symplectic transformations that convert effec-tively our scheme to the BK scheme; i.e., the BK scheme isoptimal from the point of view of the noiseN. As a by-product, we have also proved that on the set of pure two-mode Gaussian states that can be produced from the TMSvacuum states42d by all local symplectic transformations,the total variancefksxA+xBd2l+kspA−pBd2lg /2 is minimizedby the TMS vacuum state. This proof generalizes for purestates the proof given recently inf26g in which the minimi-zation of the total variance was performed only with respectto a two-parameter subset of all local symplectic transforma-tions that is formed by the symplectic transformations thatcan be expressed as a product of a squeezer and a phase shift.

III. CONCLUSION

In conclusion, we have investigated the CV teleportationprotocol based on the shared entanglement produced by theQND interaction of two vacua and utilizing QND interactionor an unbalanced BS in the Bell measurement. We haveshown in two particular examples that in the nonunity gainregime the signal transfer coefficient can be increased, whilethe conditional variance product remains unchanged by asuitable squeezing operation on the sender’s side of theshared entangled state. Further, it was demonstrated that inthe unity gain regime the teleportation fidelity can be en-hanced if the shared entanglement is redistributed by localsqueezing operations converting effectively the protocol tothe standard BK teleportation protocol. Finally, it is provedthat such a choice of the local squeezing operations is opti-mal if we characterize our teleportation scheme by the noiseby which the mean number of photons in the input state isincreased in the teleportation process.


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ACKNOWLEDGMENTS

We thank G. Leuchs and N. Korolkova for the kind hos-pitality at the Erlangen University. The contributions of P.van Loock, J. Sherson, and J. Fiurášek are also gratefully

acknowledged. The research has been supported by the re-search project, Measurement and Information in OpticsMSM 6198959213, and EU under project COVAQIALsFP6-511004d. R. F. acknowledges support by Project 202/03/D239 of the Grant Agency of the Czech Republic.

f1g C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres,and W. K. Wootters, Phys. Rev. Lett.70, 1895s1993d.

f2g D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter,and A. Zeilinger, NaturesLondond 390, 575 s1997d; D. Bos-chi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Phys.Rev. Lett. 80, 1121s1998d.

f3g L. Vaidman, Phys. Rev. A49, 1473s1994d.f4g S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett.80, 869

s1998d.f5g P. van Loock and S. L. Braunstein, Phys. Rev. Lett.84, 3482

s2000d.f6g S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt.

47, 267 s2000d.f7g A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev.47, 777

s1935d.f8g D. F. Walls and G. J. Milburn,Quantum OpticssSpringer Ver-

lag, Berlin, 1994d.f9g A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H.

J. Kimble, and E. S. Polzik, Science282, 706 s1998d; W. P.Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph,H.-A. Bachor, T. Symul, and P. K. Lam, Phys. Rev. A67,032302 s2003d; T. C. Zhang, K. W. Goh, C. W. Chou, P.Lodahl, and H. J. Kimble,ibid. 67, 033802s2003d.

f10g L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, Phys. Rev.Lett. 85, 5643s2000d.

f11g A. Kuzmich and E. S. Polzik, Phys. Rev. Lett.85, 5639s2000d.

f12g B. Julsgaard, A. Kozhekin, and E. S. Polzik, NaturesLondond

413, 400 s2001d.f13g G. J. Milburn and S. L. Braunstein, Phys. Rev. A60, 937

s1999d.f14g D. B. Horoshko and S. Ya. Kilin, Phys. Rev. A61, 032304

s2000d.f15g T. C. Ralph and P. K. Lam, Phys. Rev. Lett.81, 5668s1998d.f16g J. Fiurášek, N. J. Cerf, and E. S. Polzik, Phys. Rev. Lett.93,

180501s2004d.f17g W. P. Bowen, P. K. Lam, and T. C. Ralph, J. Mod. Opt.50,

801 s2003d.f18g W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C.

Ralph, T. Symul, and P. K. Lam, IEEE J. Sel. Top. QuantumElectron. 9, 1519s2003d.

f19g J.-Ph. Poizat, J.-F. Roch, and P. Grangier, Ann. Phys.sParisd19, 265 s1994d.

f20g S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and P. van Loock,Phys. Rev. A64, 022321s2001d.

f21g R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F.Valley, Phys. Rev. Lett.55, 2409s1985d.

f22g J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, Phys. Rev.Lett. 83, 1319s1999d.

f23g R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A49, 1567s1994d.

f24g G. Vidal and R. F. Werner, Phys. Rev. A65, 032314s2002d.f25g S. L. Braunstein, e-print quant-ph/9904002.f26g G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A70,

022318s2004d.


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