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VOLUME 80, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JANUARY 1998

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Teleportation of Continuous Quantum Variables

Samuel L. BraunsteinSEECS, University of Wales, Bangor LL57 1UT, United Kingdom

H. J. KimbleNorman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 911

(Received 8 September 1997)

Quantum teleportation is analyzed for states of dynamical variables with continuous spectra, icontrast to previous work with discrete (spin) variables. The entanglement fidelity of the schemis computed, including the roles of finite quantum correlation and nonideal detection efficiency. Aprotocol is presented for teleporting the wave function of a single mode of the electromagnetifield with high fidelity using squeezed-state entanglement and current experimental capability[S0031-9007(97)05114-4]

PACS numbers: 03.67.–a, 03.65.Bz, 42.50.Dv

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Quantum mechanics offers certain unique capabilitifor the processing of information, whether for computation or communication [1]. A particularly startling dis-covery by Bennettet al. is the possibility for teleportationof a quantum state, whereby anunknownstate of a spin-12particle is transported by “Alice” from a sending stationto “Bob” at a receiving terminal by conveying 2 bits ofclassical information [2]. The enabling capability for thisremarkable process is what Bell termed the irreducibnonlocal content of quantum mechanics, namely that Aice and Bob share an entangled quantum state and expits nonlocal characteristics for the teleportation procesFor spin-12 particles, this entangled state is a pair of spinin a Bell state as in Bohm’s version of the Einstein, Podosky, and Rosen (EPR) paradox [3] and for which Bell fomulated his famous inequalities [4].

Beyond the context of dichotomic variables, Vaidmahas analyzed teleportation of the wave function of a ondimensional particle in a beautiful variation of the original EPR paradox [5]. In this case, the nonlocal resourshared by Alice and Bob is the EPR state with perfecorrelations in both position and momentum. The goalthis Letter is to extend Vaidman’s analysis to incorporafinite (nonsingular) degrees of correlation among the relvant particles and to include inefficiencies in the measurment process. The “quality” of the resulting protocol foteleportation is quantified with the first explicit computation of the fidelity of entanglement for a process acting oan infinite dimensional Hilbert space. We further describa realistic implementation for the quantum teleportationstates of continuous variables, where now the entang

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state shared by Alice and Bob is a highly squeezed twmode state of the electromagnetic field, with the quadture amplitudes of the field playing the roles of positioand momentum. Indeed, an experimental demonstratof the original EPR paradox for variables with a continuous spectrum has previously been carried out [6,7], whwhen combined with our analysis, forms the basis of aalizable experiment to teleport the complete quantum stof a single mode of the electromagnetic field.

Note that up until now, all experimental proposafor teleportation have involved dichotomic variablesSUs2d [2,8–11], with optical schemes accomplishing thBell-operator measurement with low efficiency. Indeethe recent report of teleportation via parametric dowconversion [12] succeeds onlya posteriori with rarepost-selected detection events. By contrast, our scheemploys linear elements corresponding to operationsSU(1,1) [13] for Bell-state detection and thus shouoperate at near unit absolute efficiency, enablinga prioriteleportation as originally envisionaged in Ref. [2].

As shown schematically in Fig. 1, an unknown inpustate described by the Wigner functionWinsad is to beteleported to a remote station, with the teleported (outpstate denoted byWoutsad. In analogy with the previouslyproposed scheme for teleportation of the state of a spin1

2particle, Alice (at the sending station) and Bob (at threceiving terminal) have previously arranged to shareentangled state which is sent along paths 1 and 2. Witthe context of our scheme in SU(1,1), the entangled stdistributed to Alice and Bob is described by the WignfunctionWEPRsa1, a2d [4]

ants for the

WEPRsa1; a2d 4

p2exph2e22rfsx1 2 x2d2 1 s p1 1 p2d2g 2 e12rfsx1 1 x2d2 1 s p1 2 p2d2gj

! Cdsx1 1 x2dds p1 2 p2d , (1)

whereaj xj 1 ipj . Here, the real quantitiessxj , pjd correspond to canonically conjugate variables for the relevpathways and describe, for example, position and momentum for a massive particle, and quadrature amplitude

© 1998 The American Physical Society 869


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FIG. 1. Scheme for quantum teleportation of an (unknowinput stateWinsad from Alice’s sending stationS to Bob’sremote receiving terminalR, resulting in the teleported outputstateWoutsad.

electromagnetic field. Note that forr ! `, the statedescribed by Eq. (1) becomes precisely the EPR stateRef. [3] employed by Vaidman [5] and provides an ideentangled “pair” shared between the teleportation sendand receiving stations, albeit with divergent energy in thlimit.

As for the protocol itself, the first step in teleportingthe (unknown) stateWinsaind is to form new variablesba,b along pathssa, bd which are linear superpositionsof those of the initially independent pathwaysin and1 at the sending stationS of Fig. 1, namely ba,b

1p2

sa1 6 aind. The resulting Wigner function in thevariables sba; bb; a2d exhibits “entanglement” betweenthe paths sa, bd and the remote path 2. Step 2 aS is then to measure the observables correspondto Reba 1p

2sx1 1 xind ; xa and Imbb 1p

2s p1 2

pind ; pb at the detectorssDa, Dbd shown in Fig. 1,with the resulting classical outcomes denoted bysixz

, ipbd,

respectively. We define ideal measurement ofsxa, pbd tobe that for which the distributionPabsixa ; ipb d is identicalto the associated Wigner functionWabsxa; pbd. With theentangled state of paths (1,2) given by Eq. (1), we find

Pabsixa ; ipb d 2Z

d2aWinsadGnfp

2sixa 2 iipb d 2 ag

; 2fWin ± Gng fp

2sixa 2 iipb dg , (2)

with ± denoting convolution andGn as a complex Gauss-ian distribution with variancen cosh2ry2. Notethat such ideal detectors provide “perfect” informatioabout sxa, pbd via sixa

, ipbd, while all information about

spa, xbd ; sssIm ba 1p2

s p1 1 pind, Rebb 1p2

sx1 2

xindddd is lost. Furthermore, althoughsixa , ipb d containsa small amount of information about the fiducial staWinsad Winsxin, pind, this information goes to zero

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for r ! `. Nonetheless, the third and final step at thsending station is to transmit thisclassicalinformation tothe receiving terminal.

As illustrated in Fig. 1, receipt ofsixa , ipb d allows Bobto construct the teleported stateWoutsa2d from compo-nent 2 of the EPR state. That this resurrection is posible can be understood by examining the (unnormalizeWigner function for the system obtained by integratinout s pa, xbd in correspondence to Alice’s detection osxa, pbd, namely

Gnsa2d fWin ± Gtg sp

2sixa 2 iipb d 1 tanh2ra2d , (3)

where the variancet sech2ry2. Note that asr !

`, Gtsad quickly approaches a delta function, whileGnsad describes a broad background state. Thus, flarge r, the reduced state of mode 2 is described bybroad pedestal with negligible probability upon which sita randomly located peak ata2 ø

p2 sixa 2 iipb d closely

mimicing the incoming stateWinsad. The location of thisrandom “displacement” is distributed according to Eq. (2and is the classical information that Alice sends to Bob.

By way of the actuatorAx,p shown in Fig. 1, Bobthus performs linear displacements of the real and imainery components of the complex amplitudea2 to pro-duce aout a2 1

p2 sixa 2 iipb d, where the quantities

sixa , ipb d are scaled tosxa, pbd. Integrating outixa andipb

yields the ensemble description of states produced atoutput of the teleportation device on an ensemble of inpstatesWin, namely

Wout Win ± Gs , (4)

wheres e22r is the variance of the complex GaussiaGs , thus completing the teleportation process.

Clearly, for r ! ` the teleported state of Eq. (4)reproduces the original unknown stateWin [5]. However,note that asr ! 0, Wout also mimicsWin, now with twoextra units of vacuum noise (i.e.,s 1

2 1 12 ). One of

these noise contributions arises from Alice’s attemptmeasure bothsxin, pind [14], while the second comes fromBob’s use of this necessarily noisy information to generaa coherent state at

p2 sixa 2 iipb d. In this wayquantum

mechanics extracts two tariffs(one at each instance of theborder crossing between quantum and classical domaineach of which we term the quantum duty orquduty).Note that the limitr 0 corresponds to what might beconsidered “classical” teleportation for which the “besmeasurement” of the coherent amplitude of the unknowstate is made [14] and sent to the receiving station, wheit is used to produce a coherent state of that classicamplitude. For anyr . 0, our quantum teleportationprotocol beats this classical scheme.

Before calculating an actual figure of merit for our protocol, we now specialize from general continuous varables to the case of a single mode of the electromagnefield and thereby to actual physical implementations othe various transformations shown in Fig. 1. Beginnin


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with the EPR state itself, we note that such a state cangenerated by nondegenerate parametric amplification wthe quantitiessxj , pjd as the quadrature-phase amplitudeof the field [6], as has been experimentally confirmed vtype-II down-conversion [7]. The linear transformationba,b 1p

2sa1 6 aind is accomplished by the simple su-

perposition of modesin and 1 at a50y50 beam splitter.The detectorssDa, Dbd of Fig. 1 are now just balanced ho-modyne detectors with the phases of their respective looscillators set to recordsxa, pbd in the observed photocur-rents sixa

, ipbd. Note that for unit efficiency, homodyne

detection provides an ideal quantum measurement ofquadrature amplitudes required for our protocol [15–17

Nonideal detectors, each having (amplitude) efficiench, may be modeled by using a pair of auxiliary beamsplitters at sDa, Dbd to introduce noise from a pairof vacuum modes described by annihilation operatosca,b, da,bd [15,18]. It is then convenient to introduceannihiliation operators corresponding to the “modes” othe photocurrents described by

ia,b hba,b 1

s1 2 h2

2sca,b 1 da,bd , (5)

where these fictitious objects allow us to apply an analoof the Wigner-function formalism to the photocurrentand to incorporate the effects of nonideal photodetetion in a straightforward fashion. For example, loss ithe response of Alice’s detectors [Eq. (2)] leads to thconvolution

Pabsixa , ipb d 1

h2 fPab ± Gz g fsixa 1 iipb dyhg , (6)

whereGz has variancez s1 2 h2dy2h2, which goes tozero forh ! 1 in correspondence with the ideal characteof homodyne detection. Substituting forPab from Eq. (2)then gives

Pabsixa, ipb

d 2

h2fWin ± Gng

∑p2

hsixa

2 iipbd∏

, (7)

wheren 12 cosh2r 1 s1 2 h2dyh2.

Within the context of the electromagnetic field, Bob caefficiently perform the required phase-space displacemof mode 2 based upon the classical informationsixa

, ipbd

received from Alice by combining the field of mode 2with a (classical) coherent state of mean amplitudEyt, whereE

p2 sixa 2 iipb dyh, at a highly reflecting

mirror of transmissivityt ! 0. The mean state after thisshift is the final teleported state, namely

Wout Win ± Gs , (8)

whereGssad 1

ps exps 2jaj2

s d with s e22r 112h2

h2 .The teleportation evolution described by Eq. (8) may b

written in density matrix form as

rout Z

d2jGssjdDsijdrinDysijd , (9)

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cal

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rs

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whererin is the original state being teleported andDsadis the displacement operator. The dynamics associatwith Eq. (9) were first studied by Glauber [19] and Lachs[20] for an “incoming” vacuum stater j0l k0j and forsqueezed vacuum by Vourdas and Weiner [21]. The dtailed behavior of the photocount statistics under this dynamics was investigated by Musslimaniet al. [22]. Thesereferences also relate the development of the convolutionformalism used here (see also Refs. [23,24]).

To illustrate the protocol, consider teleportation of thecoherent superposition state

jcl ~ j 1 al 1 eifj 2 al , (10)

with corresponding Wigner functionWinsad illustratedin Fig. 2(a). The teleported Wigner functionWoutsad ascomputed from Eq. (8) is shown for Fig. 2(b) for parameters corresponding to210 dB of squeezing (i.e.,r 1.15) with efficiency h2 0.99, which should be com-pared to the parameters of Ref. [25] [namely squeezinr 0.69 (i.e., 6 dB of squeezing), and detectors with absolute quantum efficiencyh2 0.99 6 0.02]. Note thatthe quantum character of the state survives teleportatioincluding negative values forWout associated with quan-tum interference for the off-diagonal components ofrin.For comparison, note that for classical teleportation (i.er 0), Wcl

out consists of the (incoherent) superposition otwo distributions centered at6a, each of which is broad-ened by thequduty.

To provide a quantitative measure of the “quality” ofthe output state, we note that the strongest measurefidelity of a teleported state relative to the input state igiven by theentanglement fidelity[26]. For processesdescribed by Eq. (9), it is given by

Fe Z

d2jGssjdjxWinsjdj2 , (11)

FIG. 2. (a) Wigner functionWinsad for the input state ofEq. (10) with a 1.5i and f p. (b) Teleported outputstateWoutsad for r 1.15 andh2 0.99.

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where xWinsjd tr Dsijdrin is the characteristic func-

tion for the incoming state’s Wigner function.For the coherent superposition of Eq. (10) direct subs

tution yields a fidelity of entanglementFe of

11 1 s

21 1 e24jaj2

2 exps24sjaj2

11s d 2 exps 24jaj2

11s d2s1 1 sd s1 1 e22jaj2 cosfd2

.

(12)

For the state shown in Fig. 2(b) this fidelity is 0.6285 for 1.15 and h2 0.99 compared to 0.2487 forr 0and the same detector efficiency. This latter fidelitprecludes observation of any quantum features in tclassically teleported state, while the former case yielobservable quantum characteristics as seen in Fig. 2.

Beyond any one particular state, let us now concentraon high fidelity teleportation in general. In this casthe Gaussian weighting described byGs is sufficientlynarrow so that only the lowest terms in an expansioaboutj 0 of xWin will contribute. That is,jxWinsjdj2may be approximated by

1 2 jp2sDad2 2 j2sDapd2 2 2jjj2jDaj2 , (13)

where jDaj2 ; kjaj2l 2 jkalj2 averaged overWinsad.Thus, the condition for high fidelity teleportation (i.e.1 2 Fe ø 1) becomes1yjDaj2 ¿ s. Now jDaj2 isjust the number of photons (plus12 ) in the incomingstateafter it has been shifted so as to haveno coherentamplitude. Roughly speaking it is the maximal rmspread of the Wigner function of the unknown quantumstate being teleported, and so its reciprocal bounds the sof “important” small scale features in that state, thougthere can indeed be smaller features. Apparently thencondition for high entanglement fidelity says that featurein the Wigner function smaller than1yjDaj do not give asignificant contribution to the state’s identity.

In conclusion, our analysis suggests that existinexperimental capabilities should suffice to telepomanifestly quantum or nonclassical states of the eletromagnetic field with reasonable fidelity. For sucexperiments, extensions of our analysis to the teleportion of broad bandwidth information must be made anwill be discussed elsewhere. In qualitative terms, oscheme should allow efficient teleportation every inversbandwidth, in sharp contrast to relatively rare transfefor proposals involving weak down conversion for spidegrees of freedom. Although our analysis is the firto obtain explicitly the fidelity of entanglement on aninfinite dimensional Hilbert space, an unresolved issuis whether or not our protocol is “optimum,” either withrespect to this measure or with regard to other criteriathe area of quantum communication (e.g., the abilityteleport optimally an “alphabet”hjj of orthogonal statesW

jin). More generally, the work presented here is part

a larger program to extend classical communication wicomplex amplitutes into the quantum domain.

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S. L. B. was funded in part by EPSRC Grant No. GRL91344 and by a Humboldt Fellowship. H. J. K. acknowl-edges support from DARPA via the QUIC Institute ad-ministered by ARO, from the Office of Naval Researchand from the National Science Foundation. Both apprecate the hospitality of the Institute for Theoretical Physicsunder National Science Foundation Grant No. PHY9407194.

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