Teleportation of two-mode squeezed states Satyabrata Adhikari,∗ A. S. Majumdar,† and N. Nayak‡ S. N. Bose National Centre for Basic Sciences, Salt Lake, Kolkata 700 098, India (Dated: February 2, 2008) We consider two-mode squeezed states which are parametrized by the squeezing parameter and the phase. We present a scheme for teleporting such entangled states of continuous variables from Alice to Bob. Our protocol is operationalized through the creation of a four-mode entangled state shared by Alice and Bob using linear amplifiers and beam splitters. Teleportation of theentangled state proceeds with local operations and theclassical communication of four bits. Wecompute the fidelity of teleportation and find that it exhibits a trade-off with the magnitudeof entanglement of theresultant teleported state. 8 0 PACSnumbers: 03.67.Mn,42.50.Dv 0 2 I. INTRODUCTION Since quantum entanglement is fragile and is easily n destroyed in distribution, establishing entanglement be- a J tween quantum systems at distant locations, and trans- Quantumteleportationisanimportantandvitalquan- porting entanglement from one location to another are 1 tum information processing task where an arbitrary un- 2 ratherchallengingtasks. Variousingeniousmethodshave knownquantumstatecanbereplicatedatadistantloca- been proposed to accomplish these, such as by using en- tion using previously shared entanglement and classical ] tanglement swapping protocols[14], quantum repeaters h communication between the sender and the receiver. A by combining operations of swapping with entanglement p remarkable application of entangled states having many purification[15], and by continuous measurements[16]. - ramifications in information technology, quantum tele- t For continuous variable systems, some protocols for en- n portationcan also be combined with other operations to a tanglement swapping[17], establishing entanglement be- construct advanced quantum circuits useful for informa- u tween distant stations through teleportation[18], testing tion processing[1]. The originalteleportationprotocolof q the efficiency of teleportation with the aid of a third [ Bennett et al.[2] for an unknown qubit using an EPR party[19], and combining teleportation with cloning[20] pair has been generalized to the case of non-maximally 2 havebeenproposed. Butnoprotocolfortheexplicttele- entangled or a noisy channel between the sender and v portation of an entangled continuous variable state ex- the receiver[3]. The loss of fidelity for teleportation us- 7 istsintheliterature,akintoasimilarschemefordiscrete ing non-maximallyentangledchannelscouldbe compen- 7 variablesforteleportingatwo-qubitentangledstate,that 7 sated by schemes for probabilistic teleportation[4]. The has been presented recently[21]. 2 first experimental demonstration of quantum teleporta- . tion was reported by Bouwmeester et al.[5]. 0 1 Quantumteleportationisalsopossibleforsystemscor- 7 responding to infinite dimensional Hilbert spaces[6, 7, 8, 0 The aim of this work is to propose an explicit scheme 9, 10, 18, 19, 20]. The teleportation process for con- : forthe teleportationofanunknowntwo-modeentangled v tinuous variables was originally formulated in terms of state of continuous variables from one party (Alice) to i Wignerfunctions[7]andhasalsobeenextended interms X the other distant party (Bob). For this purpose we first of characteristic functions[8] of the quantum systems in- r showhowanentangledstateoffourmodescanbegener- volved. Schemes for obtaining optimal fidelity of tele- a atedandsharedbyAliceandBobwiththehelpoflinear portation using Gaussian[9] as well as non-Gaussian[10] amplifiersandbeam-spitters. Ourprotocolforteleporta- resource states have been devised. The first experiment tioncanthenproceedintheusualwaywithAlicemaking of continuous variable teleportation was performed by measurements on her side and communicating their re- Furusawa et al.[11]. Since then there have been fur- sults classically to Bob who in turns makes a local oper- ther improvements in the fidelity of teleportation ob- ationtoobtainthe teleportedentangledtwo-modestate. tained in experiments[12]. Recently, an experimental The communication of four bits of information from Al- characterization of continuous variable quantum com- ice to Bob is required,similar to the case of the protocol munication channels has been established by shared en- for teleporting entangled states of two qubits[21]. We tanglement together with local operations and classical compute the entanglement of the teleported state with communications[13]. Bob,andalso the fidelity ofteleportationas functions of the squeezing parameters of the states generated by the source and the teleportation amplifiers. A trade-off be- tween the entanglement of the teleported state and the ∗[email protected] †[email protected] fidelity of teleportation is observed with respect to the ‡[email protected] squeezing parameters. 2 II. THE TELEPORTATION PROTOCOL where c = Cosh(2r),s = Sinh(2r),k = Sin(2φ),h = Cos(2φ) with r being the squeezingparameterandφ the amplifier phase. Bob uses two beam-splitters (BS1 and BS2) represented by the matrix B as 1 TA2 4 2 9 BS3 I2/√2 0 0 I2/√2 15 10 B = 0 I2/√2 I2/√2 0  (3) 1 BS1 7 1  0 I2/√2 −I2/√2 0  TA1 SA3 I2/√2 0 0 −I2/√2 3 8 where, I is a 2 2 identity matrix, to mix the modes BS2 6 BS4 2 × 16 (x ,x ) and (x ,x ), respectively. The CM correspond- 1 4 2 3 12 D 5 11 ing to the four output modes (x5,x6,x15,x16) of the two beam-splitters are given by D 14 13 σ(5)(6)(15)(16) =B σ(1)(3)(2)(4)B† (4) 1 1 FIG. 1: Schematic diagram for the teleportation protocol. Bob then supplies the modes x and x to Alice and 15 6 Alice’samplifierSA3generatesthetwo-modeentangledstate keeps the remaining modes x and x with himself. (x7,x8) to be teleported. Bob has two amplifiers TA1 and 5 16 Hence, Alice and Bob share a four-mode entangledstate TA2 and two beam-splitters BS1 and BS2 using which he to be used for the teleportation protocol. generatesafour-modestate(x5,x6,x15,x16). Hekeepstwoof thesemodesx5andx16withhimself,andsendstheremaining Alice hasthe twoentangledmodesx7 andx8 originat- two modes x6 and x15 to Alice. Using the beam-splitters ingfromhersourceamplifierSA3toteleport,inaddition BS3andBS4Alicecombineshermodesx7 andx8 withthose tothetwomodesx andx whichBobhassenther. The 15 6 sent by Bob, and performs four measurements on theoutput combinedsix-modestate(fourmodeswithAliceandtwo modes x9,x10,x11,x12. She then communicates her results with Bob) are represented by the CM to Bob who uses these to apply a unitary transformation to displace the modes x5 and x16. The final teleported state is found in themodes x13 and x14. σ(5)(6)(15)(16)(7)(8) =σ(5)(6)(15)(16) σ(7)(8) ⊕ x uy 0 vy 0 − 0 x+uy 0 vy σ(7)(8) = −  (5) Our protocol is as follows. Let Alice hold the source vy 0 x+uy 0 parametric amplifier SA3 whose two output entangled  0 vy 0 x uy  − −  modesaretobeteleported. Bobpossessestwoteleporta- tion amplifiers TA1 andTA2 whichare requiredas com- where x = Cosh(2q), y = Sinh(2q), u = Cos(2η), v = pulsory accessories for our protocol of teleportation of Sin(2η),withqbeingthesqueezingparameterofthetwo- two-mode entangled states. Alice’s task is to teleport modestate(x andx )withAlice,andηbeingthephase 7 8 the entangledstate ofthe modesx7 andx8 assignaland of amplifier SA3. idler originating from her SA3 to Bob. But prior to this Toproceedfurtherwiththeteleportationprotocol,Al- one requires to set up a four-mode entangled state to be ice uses two beam-spiltters BS3 and BS4 represented by shared by Alice and Bob. For this purpose, consider the modes x and x coming out of the amplifier TA1, and 1 3 I /√2 0 0 0 I /√2 0 the modes x and x coming out of TA2. The covari- 2 2 2 4 0 I 0 0 0 0 ance matrix of the four modes x ,x ,x ,x is given by  2  σ(1)(2)(3)(4) =σ(1)(3) σ(2)(4),with1 xi2 3(Xi4,Pi), andwe B = 0 0 I2/√2 0 0 I2/√2 ⊕ ≡ 2  0 0 0 I 0 0  assume that the two amplifiers are similar, i.e.,  2  I /√2 0 0 0 I /√2 0   2 − 2  α γ  0 0 I /√2 0 0 I /√2 σ(1)(3) =σ(2)(4) =(cid:18)γT β(cid:19) (1)  2 − 2 (6) to combine the modes (x and x ) and (x and x ), 15 7 6 8 with respectively in order to obtain the four output modes (x ,x ,x ,x ). Alice then makes measurements on c hs 0 9 10 11 12 α = − these four modes. Without loss of generality, let us as- (cid:18) 0 c+hs(cid:19) sume that her choice of measurements leads to the re- β = c+hs 0 (2) sults (X9,P10,X11,P12), respectively, whichshe commu- (cid:18) 0 c hs(cid:19) nicates to Bob. − ks 0 Bob then uses these measurement results to displace γ = (cid:18) 0 ks(cid:19) the state of the modes x5 and x16 with him, by applying − 3 the unitary transformation The CM σ(13)(14) can also be expressed as 2 1 0 0 0 0 0 0 σ(13)(14) =(σ′)(7)(8)+2(c+ks)I (12) −q3 −q3  0 0 1 0 0 0 2 0 where U = −q3 q3   0 0 0 q23 q13 0 0 0  (σ′)(7)(8) =x−0uy x+0uy −0vy v0y   0 0 0 0 0 1 0 2 vy 0 x+uy 0  q3 (7−)q3  −0 vy 0 x−uy to obtain finally the modes x and x . The gains √2 (13) 13 14 andI isthe4 4identitymatrix. Fork = 1,andinthe on the classical measurements have been chosen such that the resultant matrix √3UKB (with K defined be- limit r ,×one obtains σ(13)(14) =(σ′)(7−)(8). It can be 2 shownt→hat∞theGaussianstatewithCM(σ′)(7)(8) andthe low) has all the elements either 0, 1 or 1. The com- − Gaussian state with CM σ(7)(8) are equivalent under lo- bined physical processes (the beam splitters used by Al- cal linear unitary Bogoliubov operations (LLUBOs)[22]. ice, the measurements performed by Alice, and the uni- Thus,inthelimitofidealinputsqueezing,thetwo-mode tary transformation performed by Bob) takes the CM σ(5)(6)(15)(16)(7)(8) to eintangled state is teleported perfectly. Further, if the output states from the amplifiers TA1 and TA2 are co- σ(13)(14) =(UKB )σ(5)(6)(15)(16)(7)(8)(UKB )† (8) herentstates,i.e.,r =0,thenthevariancesofthemodes 2 2 x and x are increasedby twice the levelof the vacuum 7 8 where noise. Thusourteleportationprotocolinthisspecialcase reproduces the results obtained by Tan[18] for a scheme 1 0 0 0 0 0 0 0 0 0 0 0 of teleporting a single-mode state. 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 III. ENTANGLEMENT AND FIDELITY OF K =  (9) 0 0 0 0 0 0 1 0 0 0 0 0 THE OUTPUT MODES   0 0 0 0 0 0 0 1 0 0 0 0   0 0 0 0 0 0 0 0 0 1 0 0   0 0 0 0 0 0 0 0 0 0 0 1   incorporatesthemeasurementsmadebyAlice. Notethat the measurements on the four modes (x ,x ,x ,x ) 9 10 11 12 with the choice of outputs (X ,P ,X ,P ) that we 9 10 11 12 have made, corresponds to integrating out these vari- ables, and hence in the CM (9) the corresponding rows are deleted. Thus, the teleported signal and idler can be found as the modesx andx withBob,representedby theCM 13 14 σ 0 σ 0 11 13 0 σ 0 σ σ(13)(14) = 22 24 (10) σ 0 σ 0 31 33  0 σ 0 σ   42 44 where 2c+2ks+x uy σ = − FIG. 2: (Coloronline) The logarithmic negativity En for the 11 3 final teleported state is plotted versus the squeezing q of the vy two-mode state generated from the source amplifier SA3 (x- σ = σ = 13 31 − 3 axis),andthesqueezingrofthetwotwo-modestatesfromthe 2c+2ks+x+uy teleportation amplifiers TA1 and TA2, respectively (y-axis). σ = (11) 22 3 vy σ = σ = 24 42 3 In order to check that the teleported modes x13 and 2c+2ks+x+uy x indeed represent an entangled pair, we compute the 14 σ = 33 3 symplecticeigenvaluesofthepartialtransposeoftheCM 2c+2ks+x uy σ(13)(14) given by Eq.(10). For the modes x13 and x14 σ44 = − to be entangled, the smallest symplectic eigenvalue of 3 4 σ˜(13)(14) has to be less than one[23], i.e., ν˜− < 1. For with σin = σ(7)(8) and σout = σ(13)(14) in the present simplicity,weassumethatthephaseoftheamplifierSA3 case given by Eqs.(5) and (10) respectively. We plot the is chosen such that u = 0 and v = 1, and similarly, the fidelity of teleportation versus the squeezing parameters phases of TA1 and TA2 are such that h = k = 1/√2. in Fig.3. We notice that a maximum fidelity of 0.38 is Then ν˜ is given by possible in this scheme for an initial two-mode coher- − ent state generated by the source amplifier. The fidelity 1 ν˜ = (2c+x y)2 2s2 (14) stays nearly constant with variation of the squeezing of − 3 − − the states from the teleportation amplifiers. Thus co- p herent states generated by the teleportation amplifiers The magnitude of entanglement in the teleported state can also be used to implement this protocol. However, is given by the logarithmic negativity defined as increased squeezing of the initial state from the source amplifier leads to the loss of fidelity. The latter feature EN =max[0,−log2ν˜−] (15) is in sharp contrast to the magnitude of entanglement EN is plottedin Fig.2versusthe squeezingparametersq of the final two-mode state with Bob, which increases corresponding to the two-mode state (x ,x ) to be tele- with the increase of squeezing of Alice’s two-mode state. 7 8 ported, and r corresponding to the states (x ,x ) and This result seems to support the contention of Johnson 1 3 (x ,x )originatingfromthe amplifiersTA1 andTA2 re- et al.[19] that the averagefidelity may not be a good in- 2 4 spectively. Oneseesthatthe magnitudeofentanglement dicator of the quality of quantum teleportation. Though oftheteleportedoutputtwo-modestateobtainedbyBob theirresultswereobtainedinthecontextofteleportation goes up with increased squeezing of the states coming of single modes together with the physical transport of from both the source and the teleportation amplifiers. modesbetweenthreeparties[19],itmightbemoreappro- priatetousethemeasureof‘entanglementfidelity’[7,25] to quantify, where possible, the ability of a process to preserve entanglement. IV. CONCLUSIONS Tosummarize,wehavepresentedthefirstexplicitpro- tocol for teleportation of two-mode entangled squeezed states. Our scheme is accomplished by the creation of a four-mode state shared initially by two distant parties throughbeamsgeneratedbytwoteleportationamplifiers FIG.3: (Coloronline)ThefidelityofteleportationF isplotted andcombinedbytwobeamsplitters. Teleportationtakes versus the squeezing q of the state to be teleported (x-axis), andthesqueezingroftheoutputmodesoftheamplifiersTA1 place with the usual local measurements, unitary oper- and TA2 (y-axis). ations and the classical communication of four bits par- allel to the case involving entangled states of discrete variables[21]. If coherent states from teleportation am- We finally compute the fidelity of the teleported en- plifiers are used to create the four-mode state shared by tangled state. The fidelity can be obtained from the the two parties for enabling teleportation, the variance expression[24] given by of the entangled two-mode teleported state increases by twicethelevelofthevacuumnoise. Ingeneral,theentan- 1 glement obtained for the teleported state increases with F = (16) Det.[σin+σout]+δ √δ the squeezing of the initial two-mode state which how- − p ever leads to loss of fidelity of teleportation. 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