Int J Theor PhysDOI 10.1007/s10773-014-2047-3

Efficient and Economic Schemes for Remotely Preparinga Four-Qubit Cluster-Type Entangled State

Shu-Yu Zhao ·Hao Fu ·Xiao-Wei Li ·Gui-Bin Chen ·Peng-Cheng Ma ·You-Bang Zhan

Received: 4 December 2013 / Accepted: 5 February 2014© Springer Science+Business Media New York 2014

Abstract We propose two novel schemes for remotely preparing a four-qubit cluster-type entangled state (FCES) with complex coefficients by using four EPR pairs and twothree-qubit GHZ states as the quantum channel, respectively. To complete the remote statepreparation (RSP) schemes, several novel sets of four-and two-qubit measuring basis wereintroduced. In these schemes, after the sender performs two different projective measure-ments, the receiver should introduce two auxiliary qubits and employ suitable C-NOT gateson his qubits, the original state can be reconstructed with unit successful probability. Com-pared with the previous schemes for the RSP of a FCES, the advantage of the presentschemes is that the entanglement resource can be reduced.

Keywords Deterministic remote state preparation · Four-qubit cluster-type entangledstate · Unit successful probability

1 Introduction

Multipartite entanglement play a fundamental role in the field of quantum information the-ory and its applications. So far multipartite entanglement has been well studied theoreticallyand experimentally (e.g. [1–10]). A few years ago, Briegel et al. [11] introduced a specialkind of multipartite entangled states, the so-called cluster states. It has been shown that one-dimensional N–qubit cluster states are generated in arrays of N qubit with an Ising-typeinteraction. It has been shown that the cluster states are more immune to decoherence than

S.-Y. Zhao (�) · H. Fu · X.-W. Li · G.-B. Chen · P.-C. Ma · Y.-B. ZhanSchool of Physics and Electronic Electrical Engineering,Huaiyin Normal University, Huaian 223300, People’s Repulic of Chinae-mail: zsy@hytc.edu.cn

S.-Y. ZhaoModern Education Technology Centre, Huaiyin Normal University,Huaian 223300, People’s Repulic of China

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Greenbrger-Horne-Zeilinger (GHZ) states. For N = 4, the cluster state can be written in thestandard form

|C〉 = 1

2(|0000〉 + |0011〉 + |1100〉 + |1111〉). (1)

The cluster state has important application in the quantum information field [12–18].In the last decade, Lo [19], Pati [20], and Bennett et al. [21] presented a new quantum

communication scheme that uses classical communication and a previously shared entan-gled resource to remotely prepare a quantum state. This communication scheme is calledremote state preparation (RSP). In RSP, Alice is assumed to know fully the transmittedstate to be prepared by Bob, so RSP is called the teleportation [22] of a known state.Compared with the teleportation, RSP requires less classical communication cost than tele-portation. Since then, RSP has attracted much attention, various theoretical schemes forgeneralization of RSP have been proposed and experimental implementation of RSP havebeen presented [23–42]. Recently, several authors [43–46] investigated various theoreticalschemes for remote preparation of a four-qubit cluster-type entangled state (FCES). Ma andZhan [43] proposed a scheme of a FCES by a set of four-qubit orthogonal basis projectivemeasurement. In this scheme [43], the success probability of RSP process with maximallyentangled states as quantum channel is 1/4. Ma et al. [44] proposed two schemes for remotepreparation a FCES. In their schemes [44], two different entanglement resources are usedas the quantum channel: one is an EPR pair and two three-particle GHZ states, and theother is two four-qubit GHZ states, and the total successful probabilities can reach 1/2 and1, respectively. More recently, Zhan et al. [46] presented a scheme for remotely prepar-ing a FCES with complex coefficients by using six EPR pairs as the quantum channel. Inthe scheme [46], the original state can be constructed by the receiver with unit successfulprobability.

In this paper, we propose two schemes for remote preparation of a FCES with complexcoefficients. To present our schemes more clearly, here we consider only maximally entan-gled channel. We first propose a scheme for remote preparation of a FCES via two sets offour- and two-qubit measuring basis with four EPR pairs as the quantum channel. In the sec-ond scheme, we present the remote preparation of a FCES by using two sets of two-qubitsmeasure basis with two three-qubit GHZ states as the quantum channel. Different fromprevious schemes (e.g. [43–46]), in the present schemes, after the sender performs the pro-jective measurement on her qubits, according to the result of the measurement, the receivershould introduce two auxiliary qubits and employs appropriate C-NOT gates on his qubits,the sender’s original state can be reconstructed with unit success probability. Comparedwith the previous schemes in Ref. [46], the advantage of our schemes is that the entangle-ment resource can be reduced. Hence, we can believe that our schemes are more efficientand economical.

2 RSP of a FCES with Four EPR Pairs as the Quantum Channel

Suppose that a sender Alice wishes to help the receiver Bob remotely prepare a FCES

|φ〉 = x0|0000〉 + x1eiη1 |0011〉 + x2e

iη2 |1100〉 + x3eiη3 |1111〉, (2)

where xj (j = 0, 1, 2, 3) and ηl(l = 1, 2, 3) are real, and∑

j x2j = 1(j = 0, 1, 2, 3).

Suppose that the sender Alice wishes to help the receiver Bob remotely prepare the state (2).Assume that Alice knows the original state (3), i.e., Alice knows xj and ηl completely, but

Int J Theor Phys

Bob does not know them at all. We also suppose that the quantum channel shared by Aliceand Bob are four EPR Pairs

|ψ1〉 = 1√2(|00〉 + |11〉)A1A2 ,

|ψ2〉 = 1√2(|00〉 + |11〉)A3B1 ,

|ψ3〉 = 1√2(|00〉 + |11〉)A4A5 ,

|ψ4〉 = 1√2(|00〉 + |11〉)A6B2 ,

(3)

where the qubits A1, A2, · · · , A6 belong to Alice, and the qubits B1, B2 belong to Bob,respectively. In order to help Bob remotely prepare the original state |φ〉 , what Aliceneeds to do is to perform four-qubit and two-qubit projective measurement on her qubitsA1, A3, A4, A6and A2, A5, respectively. The first measurement basis chosen by Alice isa set of mutually orthogonal basis vectors (MOBVs) { |τp〉}(p = 1, 2, · · · , 16), which isgiven by

⎛

⎜⎜⎝

|τ1〉|τ2〉|τ3〉|τ4〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ1〉|ζ2〉|ζ3〉|ζ4〉

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

|τ5〉|τ6〉|τ7〉|τ8〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ5〉|ζ6〉|ζ7〉|ζ8〉

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

|τ9 〉|τ10〉|τ11〉|τ12〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ9 〉|ζ10〉|ζ11〉|ζ12〉

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

|τ13〉|τ14〉|τ15〉|τ16〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ13〉|ζ14〉|ζ15〉|ζ16〉

⎞

⎟⎟⎠ ,

(4)

where

F =

⎛

⎜⎜⎝

x0 x1 x2 x3x1 −x0 x3 −x2

x2 −x3 −x0 x1x3 x2 −x1 −x0

⎞

⎟⎟⎠ , (5)

and|ζ1〉 = |0000〉, |ζ2〉 = |0011〉, |ζ3〉 = |1100〉, |ζ4〉 = |1111〉,|ζ5〉 = |0001〉, |ζ6〉 = |0010〉, |ζ7〉 = |1101〉, |ζ8〉 = |1110〉,|ζ9〉 = |0100〉, |ζ10〉 = |0111〉, |ζ11〉 = |1000〉, |ζ12〉 = |1011〉,|ζ13〉 = |0101〉, |ζ14〉 = |0110〉, |ζ15〉 = |1001〉, |ζ16〉 = |1010〉.

(6)

The second measuring basis chosen by Alice is a set of MOBVs {|εj 〉} (j = 1, 2, 3, 4)which given by ⎛

⎜⎜⎝

|ε1〉|ε2〉|ε3〉|ε4〉

⎞

⎟⎟⎠ = G

⎛

⎜⎜⎝

|00〉|01〉|10〉|11〉

⎞

⎟⎟⎠ , (7)

where

G =

⎛

⎜⎜⎝

1 δ1 δ2 δ3

1 −δ1 δ2 −δ31 −δ1 −δ2 δ31 δ1 −δ2 −δ3

⎞

⎟⎟⎠ , (8)

where δl = e−iηl (l = 1, 2, 3).

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Now let Alice first perform the four-qubit projective measurement on her qubitsA1, A3, A4, A6 under the basis {τp〉}(p = 1, 2, · · · , 16). Next, according to her measure-ment result, Alice should employ suitable unitary operations on qubits A2 and A5, andthen measures the qubits A2 and A5. After these measurements, Alice informs Bob of herresults of measurement by the classical channel. For instance, without loss of generality,assume Alice’s first measurement outcome is |τ6〉A1A3A4A6 ,the qubits A2, A5, B1, B2 willbe collapsed into the state

|u〉 = 1

4(x1(|0001〉 − x0|0010〉 + x3|1101〉 − x2|1110〉)A2B1A5B2 , (9)

she can carry out an unitary operation (−iσy) on the qubit A5, the state (9) will betransformed to the state

|u′〉 = 1

4(x0(|0000〉 + x1|0011〉 + x2|1100〉 + x3|1111〉)A2B1A5B2 . (10)

Then Alice measures the qubits A2 and A5 in the basis {|εj 〉}(j = 1, 2, 3, 4). AssumeAlice’s second measurement outcome is |ε2〉A2A5 , the qubits B1 and B2 will collapse intothe state

|u′′〉 = 1

4(x0|00〉 − x1e

iη1 |01〉 + x2eiη2 |10〉 − x3e

iη3 |11〉)B1B2 . (11)

According to Alice’s public announcements, Bob can perform the local unitary operationUB = (I )B1 ⊗ (σz)B2 on his B1 and B2, then the state (11) can be transformed into

|q〉 = 1

2(x0|00〉 + x1e

iη1 |01〉 + x2eiη2 |10〉 + x3e

iη3 |11〉)B1B2 , (12)

the state (12) contains full information of the original state |φ〉. In order to complete theRSP, Bob introduces two auxiliary qubits B3 and B4 with the initial states |0〉B3 and |0〉B4 ,and the state (12) will be described as

|q ′〉 = 1

2(x0|00〉 + x1e

iη1 |01〉 + x2eiη2 |10〉 + x3e

iη3 |11〉)B1B2 ⊗ |0〉B3 ⊗ |0〉B4 . (13)

Then Bob employs in turn four C-NOT gates CB2−B3 , CB2−B4 , CB1−B2 and CB3−B2 onthe qubits B1, B2, B3 and B4, where Ci−j denotes that i as control qubit and j as targetone. After that, the state (13) can be transformed into the original state |φ〉 and the RSPsucceeds in this situation. If Alice’s first measurement results are the other 15 cases, sheshould choose suitable unitary operations on the qubits A2 and A5, and then measures themunder the basis {|εj 〉}. After that, in accord with Alice’s public announcements, Bob canemploy the appropriate unitary operations on his B1 and B2, the state |q〉 (see (12)) canbe obtained. Next, by above the same approach, i.e., Bob introduces two auxiliary qubitsB3 and B4 with the initial states |0〉B3 and |0〉B4 , and then perform four C-NOT gatesCB2−B3 , CB2−B4 , CB1−B2 and CB3−B2 on his qubits, the desired state |φ〉 can be recon-structed. It is easily found that, for all the 64 measurement outcomes of Alice, the receiverBob can reconstruct the original state |φ〉 with unit successful probability, and the requiredclassical communication cost (CCC) is six bits in this scheme.

3 RSP of a FCES with Two Three-Qubit GHZ States as the Quantum Channel

Now let us further propose a more economic scheme for remote preparation of a FCES withtwo three-qubit GHZ states as the quantum channel. Suppose that the state Alice wishes to

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help Bob remotely prepare is still in state |φ〉 (see (2)) and the states shared by Alice andBob as the quantum channel are two three-qubit GHZ states, which are given by

|ϕ1〉 = 1√2(|000〉 + |111〉)A1A2B1 ,

|ϕ2〉 = 1√2(|000〉 + |111〉)A3A4B2 ,

(14)

where the qubits A1, A2, A3, A4 belong to Alice, and the qubits B1, B2 belong to Bob,respectively. The first measurement basis chosen by Alice is a set of MOBVs {|νk〉}(k =1, 2, 3, 4), which is given by

⎛

⎜⎜⎝

|ν1〉|ν2〉|ν3〉|ν4〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|00〉|01〉|10〉|11〉

⎞

⎟⎟⎠ , (15)

where F is still in (5). The second measuring basis chosen by Alice is still in (7) and (8).Now let Alice first perform the two-qubit projective measurement on her qubits A1

and A3 under the basis {|νk〉}(k = 1, 2, 3, 4). Without loss of generality, assume Alice’smeasurement result is |ν3〉A1A3 , the qubits A2, A4, B1, B2 will be collapsed into the state

|r〉 = 1

2(x2(|0000〉 − x3|0011〉 − x0|1100〉 + x1|1111〉)A2B1A4B2 , (16)

she should employ an unitary operations (−iσy) on her qubit A2, the state (16) will betransformed into the state

|r ′〉 = 1

2(x2(|1000〉 − x3|1011〉 + x0|0100〉 − x1|0111〉)A2B1A4B2 . (17)

Then, Alice measures her qubits A2 and A4 under the basis {|εi〉}(j = 1, 2, 3, 4) andinforms Bob of her measurement outcomes. Assume Alice’s second result of measurementis |ε2〉A2A4 , the qubits B1 and B2 will be collapsed into the state

|r ′′〉 = 1

2(x0|10〉 + x1e

iη1 |11〉 + x2eiη2 |00〉 + x3e

iη3 |01〉)B1B2 . (18)

According to Alice’s public announcements, Bob can carry out an unitary operation (σx)

on his qubit B1, the state (18) can be transformed into the state |q〉 (see (12)). By the sameapproach in Section 2, the receiver Bob can recover the original state |φ〉. If Alice’s firstmeasurement results are the other 3 cases, she should choose suitable unitary operationson the qubits A2 and A4, and then measures them under the basis {|εj 〉}. After that, inaccord with Alice’s public announcements, Bob can employ the appropriate unitary opera-tions on his B1 and B2, the state |q〉 (see (12)) can be always obtained. Next, by above thesame approach,the desired state |φ〉 can be reconstructed. It is easily found that, for all 16measurement results of Alice in this scheme, the total success probability is still 1 and therequired CCC is four bits.

4 Conclusion

In conclusion, we have proposed two novel schemes for remote preparation of a FCES withcomplex coefficients. In these schemes, four EPR pairs and two three-qubit GHZ stateshave been used as the quantum channel, respectively, and several sets of four- and two-qubitmeasuring basis have been employed. In the present schemes, after the sender performed

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first projective measurement on her qubits, in accord with the result of measurement, thesender should employ suitable unitary operations on her other qubits and then measuredthese qubits, and informed the receiver of her outcomes of measurements. According tothe sender’s public announcements, the receiver should introduce two auxiliary qubits andemploy appropriate C-NOT gates on his qubits, the sender’s original state can be recon-structed with unit successful probability. Compared with the previous scheme [46], theadvantage of our schemes is that the entanglement resource can be reduced. Hence, we canbelieve that our schemes are more efficient and economical.

Acknowledgments This work is supported by the National Natural Science Foundation of China underGrants No.11074088.

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