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Optics Communications 279 (2007) 395–398

Re-examining generalized teleportation protocol q

Yan Xia *, Jie Song, He-Shan Song

School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China

Received 27 April 2007; received in revised form 14 July 2007; accepted 16 July 2007

Abstract

We present an explicit generalized protocol for probabilistic teleportation of an arbitrary N-qubit GHZ entangled state via only onenon-maximally two-qubit entangled state. Without entanglement concentration, using standard Bell-state measurement and classicalcommunication one cannot teleport the state with unit fidelity and unit probability. We show that by properly choosing the measurementbasis it is possible to achieve unity fidelity transfer of the state. Compared with Gordon et al’s protocol [G. Gordon, G. Rigolin, Phys.Rev. A 73 (2006) 042309], this protocol has the advantage of transmitting much less qubits and classical information for teleporting anarbitrary N-qubit GHZ state.� 2007 Elsevier B.V. All rights reserved.

PACS: 03.67.Hk; 03.65.Ud

No-cloning theorem forbids a perfect copy of an arbi-trary unknown quantum state. How to interchange differ-ent resources has ever been a question in quantumcomputation and quantum information [1–9]. Quantumteleportation, the disembodied transport of quantum statesbetween subsystems through a classical communicationchannel requiring a shared resource of entanglement, isone of the most profound results of quantum informationtheory [10]. Quantum teleportation process, originally pro-posed by Bennett et al. [11], can transmit an unknownquantum state from a sender to a spatially distant receivervia a quantum channel with the help of some classicalinformation and Boschi et al. [12] experimentally imple-mented the teleportation protocol. Their work showed inessence the interchangeability of different resources inquantum mechanics. Later, quantum teleportation hasreceived much attention [11–20] both theoretically and

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.07.010

q The project supported by National Natural Science Foundation ofChina under Grant No. 10575017.

* Corresponding author. Tel.: +86 411 84706201.E-mail addresses: xia-208@163.com (Y. Xia), hssong@dlut.edu.cn

(H.-S. Song).

experimentally in recent years due to its important applica-tions in quantum communications. For example, in 1998,Karlsson et al. [13] generalized Bennentt et al.’s teleporta-tion idea by using a 3-qubit Greenberger–Horne–Zeilinger(GHZ) state j000i + j111i instead of an EPR pair. The tele-portation of an arbitrary two-qubit state had been studiedby Lee et al. [16] and recently by Rigolin [17]. All the pre-vious proposals assume, nevertheless, that the quantumchannels used to teleport the qubits are noiseless maximallyentangled states. But in a realistic situation, however, deco-herence and noise degrade the channel and we do not havea maximally entangled state anymore. One way out of thisproblem is to employ quantum distillation protocol [24],which allows us to obtain a maximally entangled state froma large ensemble of partially states. But quantum distilla-tion only achieves a maximally entangled state asymptoti-cally. Thus, for finite runs of the distillation protocol wealways obtain an almost maximally entangled state.

In view of that we are led to ask if it is possible to imple-ment quantum teleportation by using partially entangledstates. In this scenario, Agrawal et al. [21] constructeda protocol where it is possible to achieve unity fidelityteleportation of one qubit using directly non-maximally

396 Y. Xia et al. / Optics Communications 279 (2007) 395–398

entangled channel. The price they pay to achieve unityfidelity is that the protocol is no more deterministic. Later,Gordon et al. [22] generalize Agrawal et al. [21] work andexpands it to teleport N-qubits using directly N-non-maxi-mally entangled channels.

In this paper, we consider a generalized quantum tele-portation protocol for an arbitrary N-qubit GHZ entan-gled state using only one non-maximally two-qubitentangled state channel and some operations with the aidof classical information. That is to say, there are onlytwo qubits on the quantum channel, which are much lessthan Ref. [22]. The obvious advantage in this protocol isthat the quantum channel is only one non-maximallytwo-qubit entangled state, and the two parties need nottransmit many particles for setting up the quantum chan-nel, which will reduce largely the entangled quantumresource in a noise channel. The total channel efficiencyis bounded by the entanglement of the two-qubit channel,and we can teleport an arbitrary state with unit fidelityalbeit less than unit probability, hence probabilisticteleportation.

For the sake of the clearness, let us review Gordonet al.’s [22] generalized teleportation protocol. The stateAlice wants to teleport is the most general pure state forN-qubits, j/Ai ¼

P1N

i¼0aijbðiÞi, where jb(i)i is the binaryrepresentation of the integer i with zeros padded to its leftin order to leave all binary numbers with the same amountof digits. Alice and Bob need N two-qubit channels, whichis given by N Bell-states with different degrees of entangle-ment (in general ni 5 nj, for i 5 j): j/channeli ¼ �N

i¼1j/þnii.

For each Bell-state, one qubit is with Alice and the otherone with Bob. (a) Alice performs N generalized Bell mea-surement. The states expanding each basis she projectsneed not have the same degree of entanglement (mi 5 mj

in general). (b) Alice informs Bob of the acceptable results.At most Alice transmits 2N bits of classical information toBob, two bits for each generalized Bell measurement con-sidered acceptable. (c) Bob performs unitary operationson his qubits according to the classical informationreceived from Alice. The unitary operation are{Rj}! exp(irzhj)Oj, where {Oj} = {I,rz,rx,rzrx}. I is theidentity and r are the usual Pauli matrices. The channelefficiency for the N-qubit teleportation protocol isCpro

N ¼ 22Nþ1ðPN

i¼12i�1P Ni Þ, where P N

i is the sum of all permu-tations of th product of i variables out of all {vr}, wherer = 1, . . . , N, and vr = c(nr)c(mr)/2. That is to say, if Alicewants to transmit an arbitrary N-qubit GHZ entangledstate to Bob using this protocol, they need N two-qubitchannels and cost 2N bits classical information. The proto-col efficiency are P suc ¼

QNi¼12n2

i =ð1þ n2i Þ

2.Now we propose our new protocol to realized general-

ized teleportation of an arbitrary N-qubit GHZ entangledstate from the sender to the receiver using only onenon-maximally two-qubit entangled state channel. Forconvenience, in the revised version we will quote somedescriptions and notations in the original Gordon et al.’sprotocol [22].

Let us assume Alice wants to transfer to Bob the state

jwTi ¼X1

i¼0

aijii�N1:2:...N ; ð1Þ

with ai complex andP1

i¼0jaij2 ¼ 1. The channel can beconstructed as follows. Alice shares with Bob the statej/iAB = N(j00iAB + nj11iAB), where n can be complex

and N ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jnj2

q. The first qubit A belongs to Alice,

the second one B to Bob. Note that here we allow n tobe any complex number and only for n = 1 we recoverthe Hillery et al. channel. The concurrence for this state,a well-known entanglement monotone [25], is c(n) = 2jnj/(1 + jnj2), which is a monotonically increasing function ofjnj. The initial state can be written as

jWi ¼ jwTi � j/iAB: ð2Þ

If we define the generalized Bell basis [21–23]

jw0; 0i ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jmj2q ðj00i þ mj11iÞ; ð3Þ

jw0; 1i ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jmj2q ðm�j00i � j11iÞ; ð4Þ

jw1; 0i ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jmj2q ðj01i þ mj10iÞ; ð5Þ

jw1; 1i ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jmj2q ðm�j01i � j10iÞ; ð6Þ

we introduce, as will become clear soon, a free parameter(m) in the protocol. It is a proper manipulation of thisparameter which makes the protocol work, where m* isthe complex conjugate of m. Using Eqs. (3)–(6) we can ex-press Eq. (2) as

jWi ¼ Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jmj2

q

� ½jw0; 0i1Aða0j0iN�12;3;...;N j0iB þ a1m�nj1iN�1

2;3;...;N j1iBÞþ jw0; 1i1Aða0mj0iN�1

2;3;...;N j0iB � a1nj1iN�12;3;...;N j1iBÞ

þ jw1;0i1Aða0nj0iN�12;3;...;N j1iB þ a1m�j1iN�1

2;3;...;N j0iBÞþ jw1;1i1Aða0mnj0iN�1

2;3...;N j1iB � a1j1iN�12;3;...;N j0iBÞ�: ð7Þ

In order to realize the teleportation, firstly Alice per-forms a generalized Bell measurement (BM) which isdefined to be a projective measurement onto one of thefour generalized Bell-states (Eqs. (3)–(6)). The initial statecan be projected onto generalized BM, with the appropri-ate probabilities. Alice can achieve a unity fidelity protocolby properly adjusting her measurement basis parameter m.For example, if she chooses m* = 1/n, the protocol workswhen her generalized BM gives jw0, 0i1A. There exist threeother possibilities: For m = n when she obtains jw0, 1i1A,and for m* = n when she measures jw1, 0i1A, and form = 1/n when she measures jw1, 1i1A. An interesting situa-

B

b 1

b 2

b N-1

C N O T

C N O T

C N O T

U V 23...N

Fig. 2. The schematic demonstration for reconstructing the arbitraryN-qubit GHZ entangled state.

Y. Xia et al. / Optics Communications 279 (2007) 395–398 397

tion occurs when n is real. Now, for m = n the protocolworks either if Alice measures jw0, 1i1A or jw1, 0i1A. Finally,for m = 1/n the protocol works if Alice obtains eitherjw0, 0i1A or jw1, 1i1A.

Alice transmits the acceptable results (2 bits) of her mea-surement via a classical channel to Bob, who has 16 freeparameters, four for each of Alice’s measurement results.Bob performs unitary operations like Ref. [22] on hisqubits according to the classical information received fromAlice. We restrict ourselves, however, to only one freeparameter (hj) for each result. The unitary operation are{Rj}! exp(irzh j)Oj, where {Oj} = {I, rz,rx,r zrx}. I isthe identity and r are the usual Pauli matrices. Then theunitary operations above transfer the state of the particlesof 2,3, . . . ,N and B into the following state:

jwi2;3;...N ¼ a0j0iN�12;3...N j0iB þ a1j1iN�1

2; 3...N j1iB ð8Þ

with the CproN ¼ 2

2Nþ1ðPN

i¼12i�1P Ni Þ, where P N

i is the sum ofall permutations of the product of i variables out of all{vr}, where r = 1, . . . ,N, and vr = c(nr)c(mr)/2.

Alice takes a Hadamard operation on particle 2 to sep-arate it from the entanglement with the other particles3,4, . . . ,N and B (see Fig. 1),

jwi2; 3;...;N ¼ ða0j0iN�13;4;...;N j0iB þ a1j1iN�1

3;4;...;N j1iBÞj0i2þ ða0j0iN�1

3;4;...;N j0iB � a1j1iN�13;4;...;N j1iBÞj1i2: ð9Þ

After that, Alice makes a von Neumann measurement onparticle 2 in the basis {0,1} with the result j0i2 (V2 = 0)or j1i2 (V2 = 1). Alice repeats this process for the other par-ticles 3,4, . . . ,N and obtains all the values of Vi (i 2{3,4, � � � ,N}). Alice carries out the following calculation:

V 2;3;...N ¼ V 2 � V 3 � � � � V N ; ð10Þand informs Bob of the value of V2,3, . . . , N, where the � de-notes an addition mod 2. Bob performs a single qubit oper-ation I or rz, conditioned on V = 0 or V = 1, respectively,on his particle B which will be transformed into

jwiB ¼ a0j0iB þ a1j1iB: ð11ÞThen Bob introduces N � 1 ancillary particles b1,b2, . . . ,bn�1 in the initial state j00 � � � 0ib1b2���bN�1

and entanglethem with the non-maximally two-qubit entangled state B,see Fig. 2. That is, Bob takes CNOT operation on the qubitB and an auxiliary particle bl (l = 1, 2, . . . , N � 1) by usingthe qubit B as the control qubit. After all these CONT

2

3

N

H

H

H

detector

detector

detector

Fig. 1. The schematic principle for determining the relation between thequbits 2, 3, . . . , N and B.

operations, the state of the composite quantum systemcomposed of the qubit B and bl becomes

jwiBb1b2...bN�1¼ ða0j0i�N þ a1j1i�N ÞBb1b2���bN�1

; ð12Þ

this is the state that Alice wants to send to Bob. The prob-abilistic quantum teleportation of arbitrary N-qubit GHZentangled state’s protocol efficiency are Psuc = 2n2/(1 + n2)2.

In summary, we have presented a protocol for teleporta-tion of an arbitrary N-qubit GHZ entangled state with unitfidelity but less than unit probability using only one non-maximally entangled two-qubit quantum system as thequantum channel. The generalized teleportation protocoldeveloped here shows that the protocol efficiency dependssolely on the channel and measuring-basis entanglementin a quite interesting way. Compared with the previousgeneralized teleportation protocol [22], the present onehas the following advantages. First, in our protocol, weuse only one non-maximally two-qubit entangled state asquantum channel. That is to say, there are only two-qubitson the quantum channel, which are much less than Ref.[22]. The obvious advantage is that the two parties neednot transmit many particles for setting up the quantumchannel, which will reduce largely the entangled quantumresource in a noise channel. Second, Classical communica-tion cost is greatly reduced in present one, at most, Alicetransmits 3 bits of classical information to Bob. But inRef. [22], there need cost 2N-bits classical information totransmit the same state as present protocol. So, our proto-col is more economical and consumes less classical commu-nication than Ref. [22] if N > 2. Third, for transmit N-qubitGHZ entangled state, our protocol requires only 2 + Nqubits as quantum channel and the protocol efficiency isPsuc = 2n2/(1 + n2)2, but Ref. [22] needs 2N qubits andthe protocol efficiency is P suc ¼

QNi¼12n2

i =ð1þ n2i Þ

2. Fourth,our protocol needs N � 1 single qubit measurements and 1

generalized Bell measurements but Ref. [22] needs N Bellmeasurements. Thus, we believe that this protocolmay be feasible in the realm of current experimental tech-nology though it is needs another 2N � 2 single qubitoperations (N � 1 Hadamard operations and N � 1 CNOToperations).

398 Y. Xia et al. / Optics Communications 279 (2007) 395–398

We end this contribution noting that in general decoher-ence and noise degrade the entanglement of the channel ina rather complicated way [23]. Most of the time an initiallypure state (or equivalently pure channel) evolves nonuni-tary to a mixed state. Here, however, we restricted our-selves to a ‘‘unitary loss’’ of entanglement, in which amaximally entangled pure channel evolves to a partiallyentangled one:

1ffiffiffi2p ðj00i þ j11iÞ ! 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ jmj2q ðj00i þ mj11iÞ: ð13Þ

Note that the bit flip noise j0i ! j1i is also a unitary noise,although it does not change the entanglement of the channel.Bob can easy overcome it and obtain the states, by imple-menting a proper unitary operation on his qubit at the endof the protocol. The key to this generalization is if one usesnon-maximally entangled state as a resource use non-maxi-mally entangled state measurement containing sameamount of entanglement as that of the shared resource in-stead of the Bell measurement. This also points, perhaps,to a link between global and local entanglement. In somesense ours is a generalized quantum teleportation protocolthat encompasses in a simple way probabilistic as well asdeterministic teleportation protocols. We hope that withthe existing technology it may be possible to implementthe probabilistic quantum teleportation protocol with ease.

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