Optics Communications 282 (2009) 670–673

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Optics Communications

journal homepage: www.elsevier .com/locate/optcom

Scheme for implementing controlled teleportation and dense codingwith genuine pentaqubit entangled state in cavity QED

Xue-Wen Wang, Zhao-Hui Peng *, Chun-Xia Jia, Yan-Hui Wang, Xiao-Juan LiuSchool of Physics, Hunan University of Science and Technology, Xiangtan 411201, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 September 2008Accepted 3 October 2008

PACS:03.67.�a42.50.Dv

Keywords:Controlled teleportationControlled dense codingCavity QED

0030-4018/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.optcom.2008.10.051

* Corresponding author. Tel./fax: +86 0732 829143E-mail address: raul121991@126.com (Z.-H. Peng)

We present a simple scheme for implementing perfect controlled teleportation and dense coding with thegenuine pentaqubit entangled state proposed by [I.D.K. Brown, S. Stepney, A. Sudbery, S.L. Braunstein, J.Phys. A: Math. Gen. 38 (2005) 1119]. In cavity QED, we have proposed to prepare the Brown state anddemonstrated the feasibility of this scheme. The distinct feature of this scheme is that it is insensitiveto both the cavity decay and the thermal field, which is of importance in view of experiment.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction On the other hand, Hao et al. [8] have proposed a controlled

In 1993, Bennett et al. [1] first presented the quantum telepor-tation scheme where an arbitrary unknown quantum state can betransferred from the sender (Alice) to a distant receiver (Bob) withthe help of an Einstein–Podolsky–Rosen (EPR) pair. In 1998, Karls-son et al. [2] generalized the teleportation scheme by using a tri-partite Greenberger–Horne–Zeilinger (GHZ) state instead of anEPR pair. In their scheme, conditioned on one receiver’s measure-ment outcome, the other receiver can recover the unknown quan-tum state which is initially in the sender’s qubit. The essential ideaof this controlled teleportation (CT) scheme is to let an unknownquantum state be recovered by a remote receiver only when hecooperates with the controller. In general, CT of a single-qubit statecan be done via a single-GHZ trio [2]. A trivial way for CT of N-qubitstate is to use N-GHZ trios [3,4]. However, Man et al. [5] haveshown that this can be done with less consumed quantum re-sources via just N � 1 EPR pairs plus one GHZ trio or a genuine(2N þ 1)-qubit entangled state. In the case of N ¼ 2, they have pre-sented a genuine pentaqubit entangled state [5], which is notreducible to a tensor product of a GHZ trio and one EPR pair. Yanet al. [6] and Gao [7] have also presented the probabilistic CT of un-known two-qubit state with a pentaqubit quantum channel. But itis easy to demonstrate that their quantum channel is just the ten-sor product of the GHZ trio and one EPR pair.

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3..

dense coding (CDC) scheme by using the tripartite GHZ state. Inthis scheme, one party (Alice) can transmit information to the sec-ond party (Bob) whereas the local measurement of the third party(Charlie) serves as quantum erasure. Charlie can control the quan-tum channel between Alice and Bob via a local measurement torealize CDC between Alice and Bob. Chen and Kuang [9] have gen-eralized the CDC scheme of the tripartite GHZ quantum channel tothe case of an (N þ 2)-qubit GHZ quantum channel via a series oflocal measurements. In the practical system Ye and Yu [10] pro-posed to implement CDC with tripartite GHZ state and W state.Chen et al. [11] chose the tripartite even parity state, which is lo-cally equivalent to the GHZ state, to implement CDC in the cavityquantum electrodynamics (QED) system.

Recently, Brown et al. [12] have present a highly entangled pen-taqubit state as follows:

jW5i12345 ¼12ðjggei123jw

�i45 þ jgegi123j/�i45 þ jeggi123jw

þi45

þ jeeei123j/þi45Þ; ð1Þ

where

jw�i45 ¼1ffiffiffi2p ðjgei45 � jegi45Þ; ð2Þ

j/�i45 ¼1ffiffiffi2p ðjggi45 � jeei45Þ: ð3Þ

Muralidharan and Panigraphi [13] have shown that the Brownstate is genuinely entangled and it is also robust against loss of

X.-W. Wang et al. / Optics Communications 282 (2009) 670–673 671

qubits. Then they have proposed to implement perfect teleporta-tion, quantum state sharing and superdense coding with the Brownstate. As we know, the EPR state can be used to teleport an un-known quantum state [1] and we can obtain arbitrary two-qubitEPR state from the tripartite GHZ state by a single-qubit operation.Thus, we can also use the tripartite GHZ state as the quantumchannel to accomplish teleportation of an unknown quantum state.If we make a single-qubit measurement on qubit 3 in the basisj þ i3 ¼ 1ffiffi

2p ðjgi3 þ jei3Þ or j � i3 ¼ 1ffiffi

2p ðjgi3 � jei3Þ, the rest qubits in

the Brown state will collapse into

jp004 i1245 ¼

12ðjggijw�i þ jgeij/�i þ jegijwþi þ jeeij/þiÞ1245; ð4Þ

or

j-004 i1245 ¼

12ð�jggijw�i þ jgeij/�i þ jegijwþi � jeeij/þiÞ1245: ð5Þ

Now, let us analyze the entanglement properties of the entan-gled states jp00

4 i1245 and j-004 i1245. We can imagine qubits 1, 2, 4

and 5 as a bipartite system where qubits 1 and 2 belong to Alice,while qubits 4 and 5 belong to Bob. Thus, the amount of entangle-ment (the von Neumann entropy) between Alice and Bob is two,which is the maximal entanglement between qubits 1, 2 and 4,5. Similarly, it is easy to demonstrate that qubits 1, 5 and 2, 4 aremaximally entangled too. However, the amount of entanglementbetween qubits 1, 4 and 2, 5 is one, which is not maximally entan-gled. From these analysis we can conclude that both of the entan-gled states jp00

4 i1245 and j-004 i1245 have the similar entanglement

properties to the quadqubit entangled state jv00iwhich can be em-ployed to implement teleportation of arbitrary two-qubit state andquantum dense coding [14]. Similar to the relation between the tri-partite GHZ state and EPR state, it is easy to understand why theBrown state can also be used to implement perfect teleportationand dense coding. On the other hand, the tripartite GHZ state hasone more qubit than the EPR state and it can also be used to imple-ment CT [2] and CDC [8]. Compared with the quadqubit entangledstate, the Brown state has one more qubit. In principle, it can com-plete the tasks that the quadqubit entangled state can not. Thus, itmay be very interesting to investigate how to implement CT andCDC with the Brown state.

It is known that the preparation and measurement of Bell stateor multiqubit entangled state is essential for quantum dense cod-ing and quantum teleportation. In the early experiment four Bellstates could not been distinguished completely [15,16]. Up to2001, Kim et al. [17] had implemented the complete Bell statemeasurement with the nonlinear interaction in optical system. Re-cently, Schuck et al. [18] have implemented the complete deter-ministic Bell-state measurement with only linear opticalelements. For multiqubit entangled state, Pan and Zeilinger [19]proposed to identify only two of the GHZ states in optical system.Yang and Han [20] proposed to implement local measurement for aset of n-qubit maximally entangled GHZ states in cavity QED. InRefs. [13 and 14] the authors had not discussed how to prepareand distinguish the quadqubit entangled state jv00i and the Brownstate in the practical system. In this paper we will propose to pre-pare the Brown state, distinguish the quadqubit entangled statesjp00

4 i1245 and j-004 i1245, and then implement perfect CT and CDC in

cavity QED. It is noted that we need make an additional two-qubitunitary transformation and a CNOT operation in order to completethe CT Scheme, which is different from the usual CT scheme.

2. Controlled teleportation of an arbitrary two-qubit state

In Ref. [5] Man et al. have shown that one can implement CT ofan arbitrary two-qubit state via a EPR pair plus one GHZ trio or agenuinely entangled pentaqubit state. In Refs. [6,7] the authors

also proposed the probabilistic CT scheme with a pentaqubitentangled state. Inspired by these schemes we investigate how toimplement CT of an arbitrary two-qubit state with the Brown state.We assume that Alice has an arbitrary two-qubit state as follows:

juiab ¼ ajggiab þ bjgeiab þ cjegiab þ djeeiab; ð6Þ

where a, b, c and d are the unknown complex coefficients such thatjaj2 þ jbj2 þ jcj2 þ jdj2 ¼ 1. She wants to teleport this state to Bob viathe help of Charlie. We choose the Brown state as the quantumchannel, where qubits 1 and 2, qubits 4 and 5, qubit 3 belong toAlice, Bob and Charlie respectively. The initial combined state ofthe whole system can be rewritten as

juiab � jW5i12345

¼ 14j/�ia1j/

�ib2ðajew�i345�2bjg/�i345�1cjgwþi345�1

��2dje/þi345Þ þ j/

�ia1jw�ib2ðajg/�i345�2bjew�i345�1cje/þi345

�1�2djgwþi345Þ þ jw�ia1j/

�ib2ðajgwþi345�2bje/þi345

�1cjew�i345�1�2djg/�i345Þ þ jw�ia1jw

�ib2ðaje/þi345

�2bjgwþi345�1cjg/�i345�1�2djew�i345Þ�; ð7Þ

where �1 and �2 correspond to the Bell-state measurements on qu-bits a, 1 and b, 2 respectively. Now Alice can make two joint Bell-statemeasurements on qubits a, 1 and b, 2 respectively. Then the rest qu-bits belong to Bob and Charlie will collapse into the following states:

ajew�i345�2bjg/�i345�1cjgwþi345�1�2dje/þi345; ð8Þajg/�i345�2bjew�i345�1cje/þi345�1�2djgwþi345; ð9Þajgwþi345�2bje/þi345�1cjew�i345�1�2djg/�i345; ð10Þaje/þi345�2bjgwþi345�1cjg/�i345�1�2djew�i345: ð11Þ

Then Alice informs Bob and Charlie the results of the measure-ments through the classical communication. For assisting Bob,Charlie should make a single-qubit measurement on qubit 3 inthe basis j � i3 ¼ 1ffiffi

2p ðjgi3 � jei3Þ and then transmits the outcome

to Bob over a classical communication channel. If the outcome ofCharie’s measurement on qubit 3 is j þ i3, the qubits belong toBob will collapse into

ajw�i45�2bj/�i45�1cjwþi45�1�2dj/þi45; ð12Þaj/�i45�2bjw�i45�1cj/þi45�1�2djwþi45; ð13Þajwþi45�2bj/þi45�1cjw�i45�1�2dj/�i45; ð14Þaj/þi45�2bjwþi45�1cj/�i45�1�2djw�i45: ð15Þ

Otherwise, if the outcome of Charie’s measurement on qubit 3 isj � i3, the qubits belong to Bob will collapse into the followingstates

� ajw�i45�2bj/�i45�1cjwþi45 ��1�2dj/þi45; ð16Þaj/�i45 ��2bjw�i45 ��1cj/þi45�1�2djwþi45; ð17Þajwþi45 ��2bj/þi45 ��1cjw�i45�1�2dj/�i45; ð18Þ� aj/þi45�2bjwþi45�1cj/�i45 ��1�2djw�i45: ð19Þ

In order to reconstruct the initial state on his own qubits, Bobcan apply the appropriate unitary transformation according tothe outcomes of Alice’s Bell-state measurements and Charlie’s con-trol operation. Without loss of generality we assume that Alice’smeasurements on qubits a, 1 and b, 2 are j/þia1 and jw�ib2, andCharlie’s measurement on qubit 3 is j þ i3. Then the state of qubits4 and 5 is just as shown in Eq. (13). In order to recover the initialstate, firstly Bob can make the unitary transformations as follows:

j/�i45 ! jggi45; ð20Þj/þi45 ! jeei45; ð21Þjw�i45 ! jgei45; ð22Þjwþi45 ! jegi45: ð23Þ

672 X.-W. Wang et al. / Optics Communications 282 (2009) 670–673

Then he can make a CNOT operation (where qubit 4 is the con-trol bit and qubit 5 is the target bit), which will lead to

ajggi45 � bjgei45 þ cjegi45 � djeei45: ð24Þ

Now Bob can make the local operation rz on qubit 5 and thenreconstruct the initial state jui on his own qubits perfectly. Forthe other measurement results of Alice and Charlie, Bob can alsoreconstruct the initial state perfectly with the correspondingoperations.

3. Controlled dense coding

In this section we will propose to implement CDC with theBrown state. In most of the previous schemes the sender and thereceiver only have one fixed qubit. But there are more than onecontrollers. In our scheme, it is different that we assume that thesender and the receiver have two qubits respectively. Suppose thatAlice, Bob and Charlie share the Brown state, where qubits 1 and 2,qubits 4 and 5, qubit 3 belong to Alice, Bob and Charlie respec-tively. In order to encode four bits of classical information, Alicenow can apply one of four local operators ri or rj ði; j ¼ 0;1;2;3Þ(where r0 ¼ I is the identity operator and r1 ¼ rx, r2 ¼ rz,r3 ¼ �iry are the Pauli operators) on her two qubits and thentransform the Brown state into the following 16 orthogonal entan-gled states

jWij5i12345 ¼ r1

i � r2j

� �jW00

5 i12345 ði; j ¼ 0;1;2;3Þ; ð25Þ

where jW005 i12345 is just the Brown state as shown in Eq. (1). Then

Alice sends qubits 1 and 2 to Bob. For assisting Bob to extract theencoded information, Charlie should make a measurement on qubit3 in the basis j � i3 ¼ 1ffiffi

2p ðjgi3 � jei3Þ and then transmits the outcome

to Bob over a classical communication channel. If the outcome ofCharie’s measurement on qubit 3 is j þ i3, the qubits belong toBob will collapse into the following states:

jpij4i1245 ¼ r1

i � r2j

� �jp00

4 i1245 ði; j ¼ 0;1;2;3Þ: ð26Þ

Otherwise, if the outcome of Charie’s measurement on qubit 3 isj � i3, the qubits belong to Bob will collapse into

j-ij4i1245 ¼ r1

i � r2j

� �j-00

4 i1245 ði; j ¼ 0;1;2;3Þ: ð27Þ

Now Bob need to distinguish the orthogonal entangled states asshown in Eqs. (26) and (27). Then he can distinguish what opera-tions Alice has done on his qubits and extract four bits of classicalinformation after receiving two qubits and a classical bit from Aliceand Charlie respectively. In the following section we will considerhow to implement the CT and CDC in the cavity QED system.

4. Physical realization

The cavity QED system is one of the possible candidates forengineering quantum entanglement in quantum information pro-cessing (QIP). In most previous schemes the cavity is used to storethe quantum information and transfer it back to the atomic sys-tem, thus the cavity decay is one of the main obstacles to imple-ment QIP in cavity QED. Recently Zheng [21] have proposed anovel scheme in which two identical atoms simultaneously inter-act with a nonresonant cavity field and a strong classical field.The photon-number dependent parts in the interaction Hamilto-nian of the system are canceled and thus the scheme is insensitiveto both the cavity decay and thermal field, which is of importancein view of experiment. Following this idea, we consider N identicaltwo-level atoms simultaneously interacting with a single-modecavity and driven by a strong classical field. In the rotating waveapproximation, the Hamiltonian of the system is

H ¼x0

XN

j¼1

Sjz þxaayaþ

XN

j¼1

g ayS�j þ aSþj� �

þX Sþj e�ixt þ S�j eixt� �h i

;

ð28Þwhere Sj

z ¼ 12 ðjejihejj � jgjihgjjÞ, Sþj ¼ jejihgjj, S�j ¼ jgjihejj with jgji and

jeji being the ground and excited states of the jth atom, ay and a arethe creation and annihilation operators of the cavity mode, g is theatom-cavity coupling strength, X is the Rabi frequency, x0, xa andx are the atomic transition frequency, the cavity frequency and thefrequency of the classical field. Supposing x0 ¼ x, in the interactionpicture the interaction Hamiltonian is

Hi ¼XN

j¼1

g e�idtayS�j þ eidtaSþj� �

þX Sþj e�ixt þ S�j eixt� �h i

; ð29Þ

where d ¼ x0 �xa is the detuning between the atomic transitionfrequency and the cavity frequency. In the strong driving regimeX� d� g, there is no energy exchange between the atomic systemand the cavity. Then in the interaction picture, the effective interac-tion Hamiltonian is [21]

He ¼k2

XN

j¼1

ðjejihejj þ jgjihgjjÞ þXN

j;k¼1;j–k

Sþj S�k þ Sþj Sþk þ H � C�� �" #

;

ð30Þwhere k ¼ g2

2d. It is noted that the effective Hamiltonian is indepen-dent of the cavity field state, allowing it to be in a thermal state.Then the evolution operator of the system is

UðtÞ ¼ e�iH0te�iHet ; ð31Þ

where

H0 ¼ XXN

j¼1

Sþj þ S�j� �

: ð32Þ

In order to prepare the Brown state, we assume that five atomsare initially in the state jggegei12345. Firstly we let atoms 2, 3 and 5interact simultaneously with a single-mode cavity and driven by astrong classical field. Choose the interaction time and the Rabi fre-quency appropriately so that kt1 ¼ ð2kþ 1=4Þp and Xt1 ¼ ð2mþ3=4Þp. Secondly, we let atoms 1, 2 and 4 undergo the same evolu-tions. Finally, we let atoms 4 and 5 interact with the single-modecavity and driven by the classical field. Choose the interaction timeappropriately so that kt2 ¼ p=4 and Xt2 ¼ np and then the stateevolution of the whole system is

jggegei12345 !12jggei123

1ffiffiffi2p ðjgei45 � ijegi45Þ þ ijgegi123

�

� 1ffiffiffi2p ðjggi45 � ijeei45Þ � jeggi123

1ffiffiffi2p ðjegi45 � ijgei45Þ

� þijeeei1231ffiffiffi2p ðjeei45 � ijggi45Þ

�: ð33Þ

After making the transformations j gi3 ! �i j gi3 and j ei4 !�i j ei4 we can obtain the Brown state as shown in Eq. (1).

In the controlled teleportation, Alice must make two joint Bell-state measurements and Bob need make the transformations asshown in Eqs. (20)–(23). It is noted that we can complete thesein the cavity QED system. Firstly we perform the rotationjei ! ijei on atom i, leading to

jw�iij !1ffiffiffi2p ðjgei � ijegiÞij; ð34Þ

j/�iij !1ffiffiffi2p ðjggi � ijeeiÞij: ð35Þ

Then we let atoms i and j interact with a single-mode cavity anddriven by the classical field. After the interaction time t2 we obtainthe following evolutions:

X.-W. Wang et al. / Optics Communications 282 (2009) 670–673 673

j/�iij ! jggiij; ð36Þj/þiij ! �ijeeiij; ð37Þjw�iij ! jgeiij; ð38Þjwþiij ! �ijegiij: ð39Þ

Hence, the joint Bell-state measurement can be achieved bydetecting atoms i and j separately. After making the local operationjeii ! ijeii we can also implement the transformations in Eqs.(20)–(23). It is noted that we can adopt the method in Ref. [22]to implement the CNOT gate in cavity QED. So we can completethe CT of arbitrary two-qubit state in cavity QED.

In the CDC scheme we need distinguish the orthogonal entan-gled states jpij

4i1245 and j-ij4i1245. It is easy to verify that the quadqu-

bit entangled states jp004 i1245 and j-00

4 i1245 are locally equivalent.Without loss of generality, we only need distinguish the entangledstate jp00

4 i1245 in cavity QED. Firstly we perform the rotationjgi ! ijgi on atom 4. Then we let atoms 4 and 5 interact with thesingle-mode cavity and classical field for the interaction time t2

and it will lead to

jp004 i1245 !

12ðijgggei þ ijgeggi þ jegegi þ jeeeeiÞ1245: ð40Þ

Then we let atoms 1, 4 and 5 interact with the single-mode cav-ity and classical field for the interaction time t1 and obtain the evo-lution as follows:

jp004 i1245 !

1ffiffiffi2p ðjegegi þ jeeeeiÞ1245: ð41Þ

After performing the rotation jgi ! ijgi on qubit 2, we then letqubits 2 and 5 undergo the same evolution as qubits 4 and 5,and then we can obtain the following evolution:

jp004 i1245 ! jeeeei1245: ð42Þ

It is noted that we have discarded the global phase factor duringthe evolution process for simplicity. With the same procedure, wecan also transform the other orthogonal entangled states as shownin Eqs. (26) and (27) into the product states fjggggi1245; � � �;jeeeei1245g. Hence, we can distinguish the quadqubit entangledstates jpij

4i1245 and j-ij4i1245 by detecting four atoms separately. So

we can also implement the CDC scheme in cavity QED.

5. Conclusion

In conclusion, we have presented a simple and efficient schemeof CT and CDC with the Brown state. In cavity QED, we have pre-pared the Brown state, distinguish the quadqubit entangled statesjpij

4i1245 and j-ij4i1245, and then demonstrated the feasibility of our

scheme. The distinct advantage of this scheme is that it is insensi-tive to both the cavity decay and the thermal field and based onpresent cavity QED techniques as shown in Ref. [23] our scheme

might be realizable in the future. In the CT and CDC, the senderand the receiver must be under the supervision of the third user.Thus, in the future our scheme might be applied in the communi-cation of the confidential information that must be kept by severalusers. In Ref. [13], in order to teleport an arbitrary two-qubit stateto Bob Alice must make a joint five-qubit measurement which is achallenge for the practical realization with present experimentaltechnique. On the other hand, Yeo and Chua [14] have alreadyshown that the quadqubit entangled state can be used to teleportan arbitrary two-qubit state. Thus, it is not the best idea to teleportan arbitrary two-qubit state with the pentaqubit entangled state,which add the difficulty of the scheme and consumed one moreentangled qubit. Besides the genuine pentaqubit entangled stateproposed by Man et al. [5], we have shown that the Brown statecan also be used as the resource for CT. In other words, the univer-sal resource for CT of an arbitrary two-qubit state is not unique.Thus, in the future it may be very interesting to search for otherpentaqubit entangled states which can also be used to implementperfect CT and CDC.

Acknowledgement

This work was supported by the Scientific Research Fund of Hu-nan Provincial Education Department, China (Grant No. 08c343).

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