Int J Theor Phys (2013) 52:2615–2622DOI 10.1007/s10773-013-1549-8

Deterministic Remote Preparation of a Four-QubitCluster-Type Entangled State

You-Bang Zhan · Hao Fu · Xiao-Wei Li ·Peng-Cheng Ma

Received: 14 December 2012 / Accepted: 22 February 2013 / Published online: 12 March 2013© Springer Science+Business Media New York 2013

Abstract We propose a scheme for remotely preparing a four-qubit cluster-type state withcomplex coefficients by using six EPR pairs as the quantum channel. To complete the re-mote state preparation scheme, a novel set of four-qubit mutually orthogonal basis vectorshas been introduced. It is shown that, after the sender performs two different four-qubitprojective measurements, the receiver can reconstruct the original state (to be prepared re-motely) with unit successful probability. Moreover, the scheme is also generalized to thecase that non-maximally two-qubit entangled states are taken as the quantum channel.

Keywords Remote state preparation · Four-qubit cluster-type state · Unit successprobability

1 Introduction

In the last decade, Lo [1], Pati [2], and Bennett et al. [3] presented a new quantum com-munication scheme that uses classical communication and a previously shared entangledresource to remotely prepare a quantum state. This communication scheme is called remotestate preparation (RSP). In RSP, Alice is assumed to know fully the transmitted state to beprepared by Bob, so RSP is called the teleportation [4] of a known state. Compared withthe teleportation, RSP requires less classical communication cost than teleportation. In re-cent years, RSP has attracted much attention, various schemes for generalization of RSPhave been presented using variant kinds of methods, including low-entanglement RSP [5],optimal RSP [6], generalized RSP [7], oblivious RSP [8], high-dimensional RSP [9, 10],continuous variable RSP [11, 12], joint RSP [13–15], etc. Meanwhile, some RSP schemes

Y.-B. Zhan (�) · H. Fu · X.-W. Li · P.-C. MaSchool of Physics and Electronic Electrical Engineering, Huaiyin Normal University, Huaian 223300,People’s Republic of Chinae-mail: ybzhan@hytc.edu.cn

P.-C. MaLaboratory of Nanophotonic Functional Materials and Devices, Laboratory of Quantum, InformationTechnology, South China Normal University, Guangzhou 510006, People’s Republic of China

2616 Int J Theor Phys (2013) 52:2615–2622

have already been implemented experimentally [16–20]. One can notice easily that all theabove RSP schemes can not be realized with unit success probability.

Recently, Briegel et al. [21] introduced a special kind of multipartite entangled states, theso-called cluster states, which can be written in the form

|ψN 〉 = 1

2N/2

N⊗

a=1

(|0〉aσ (a+1)z + |1〉a

), (1)

with the convention σ (N+1)z ≡ 1. It has been shown that one-dimensional N-qubit cluster

states are generated in arrays of N qubit with an Ising-type interaction. When N = 2 (or 3),the cluster states are equivalent to Bell states and Greenberger-Horne-Zeilinger (GHZ) statesrespectively under stochastic local operation and classical communication (LOCC), butwhen N > 3, the cluster state and the N-qubit GHZ state cannot be transformed into eachother by LOCC [21]. It has been shown that the cluster states are more immune to deco-herence than GHZ states [22]. For N = 4, the cluster state can be written in the standardform

|ψ4〉 = 1

2

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

The four-qubit cluster state has been shown to be useful for quantum information, such asquantum computation [21, 23, 24], quantum error correction [25], quantum dense coding[26], quantum teleportation [26, 27], quantum information splitting [28], etc. In a recentpaper [29], we presented a scheme for remotely preparing a four-qubit cluster-type state bya set of new four-qubit orthogonal basis projective measurement. In this scheme [29], thesuccess probability of RSP process with maximally entangled states as quantum channelis 1/4. More recently, Wang and Yan [30] presented a scheme for deterministic remotepreparation of an arbitrary quantum state with complex coefficients. In the scheme [30], theoriginal state can be reconstructed by the receiver with unit successful probability. Inspiredby Refs. [14, 30], in this paper, we propose a scheme for remote preparation of a four-qubitcluster-type state with complex coefficients by two sets of novel four-qubit orthogonal basisprojective measurement. It is shown that the receiver can reconstruct the unknown originalstate with unit success probability.

2 RSP with Six EPR Pairs as the Quantum Channel

A four-qubit cluster-type state with complex coefficients can be always expressed as

|φ〉 = x0|0000〉 + x1eiη1 |0011〉 + x2e

iη2 |1100〉 + x3eiη3 |1111〉, (3)

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

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

pose that the sender Alice wish to help the receiver Bob remotely prepare the state (3).Assume that Alice knows the original state (3), i.e., Alice knows xj and ηl completely, butBob does not know them at all. We also suppose that the quantum channel shared by Aliceand Bob are six EPR Pairs

|ψ1(2,3,4,5,6)〉 = 1√2

(|00〉 + |11〉)A1B1(C1A2,B2C2,A3B3,C3A4,B4C4)

, (4)

where qubits A1,A2,A3,A4,B1,B2,B3,B4 belong to Alice, and qubits C1,C2,C3,C4 be-long to Bob, respectively. The joint system can be written as |�〉 = �6

n=1|ψn〉. In order

Int J Theor Phys (2013) 52:2615–2622 2617

to help Bob remotely prepare the original state |φ〉, what Alice needs to do is to performfour-qubit projective measurement on her qubits Am and Bm (m = 1,2,3,4) respectively.The first measurement basis chosen by Alice is a set of mutually orthogonal basis vectors(MOBVs) {|τp〉}(p = 1,2, . . . ,16), which is given by

⎛

⎜⎜⎝

|τ1〉|τ2〉|τ3〉|τ4〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ1〉|ζ2〉|ζ3〉|ζ4〉

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

|τ5〉|τ6〉|τ7〉|τ8〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ5〉|ζ6〉|ζ7〉|ζ8〉

⎞

⎟⎟⎠ ,

(5)⎛

⎜⎜⎝

|τ9〉|τ10〉|τ11〉|τ12〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ9〉|ζ10〉|ζ11〉|ζ12〉

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

|τ13〉|τ14〉|τ15〉|τ16〉

⎞

⎟⎟⎠ = F

⎛

⎜⎜⎝

|ζ13〉|ζ14〉|ζ15〉|ζ16〉

⎞

⎟⎟⎠ ,

where

F =

⎛

⎜⎜⎝

x0 x1 x2 x3

x1 −x0 x3 −x2

x2 −x3 −x0 x1

x3 x2 −x1 −x0

⎞

⎟⎟⎠ , (6)

and

|ζ1〉 = |0000〉, |ζ2〉 = |0011〉, |ζ3〉 = |1100〉, |ζ4〉 = |1111〉,|ζ5〉 = |0001〉, |ζ6〉 = |0010〉, |ζ7〉 = |1101〉, |ζ8〉 = |1110〉,

(7)|ζ9〉 = |0100〉, |ζ10〉 = |0111〉, |ζ11〉 = |1000〉, |ζ12〉 = |1011〉,|ζ13〉 = |0101〉, |ζ14〉 = |0110〉, |ζ15〉 = |1001〉, |ζ16〉 = |1010〉.

The second measurement bases chosen by Alice are four sets of MOBVs {|μ(k)r 〉} (r =

1,2, . . . ,16; k = 1,2,3,4), which are given by

⎛

⎜⎜⎜⎜⎜⎝

|μ(k)

1 〉|μ(k)

2 〉|μ(k)

3 〉|μ(k)

4 〉

⎞

⎟⎟⎟⎟⎟⎠= G(k)

⎛

⎜⎜⎜⎜⎝

|ζ1〉|ζ2〉|ζ3〉|ζ4〉

⎞

⎟⎟⎟⎟⎠,

⎛

⎜⎜⎜⎜⎜⎝

|μ(k)

5 〉|μ(k)

6 〉|μ(k)

7 〉|μ(k)

8 〉

⎞

⎟⎟⎟⎟⎟⎠= G(k)

⎛

⎜⎜⎜⎜⎝

|ζ5〉|ζ6〉|ζ7〉|ζ8〉

⎞

⎟⎟⎟⎟⎠,

⎛

⎜⎜⎜⎜⎜⎝

|μ(k)

9 〉|μ(k)

10 〉|μ(k)

11 〉|μ(k)

12 〉

⎞

⎟⎟⎟⎟⎟⎠= G(k)

⎛

⎜⎜⎜⎜⎝

|ζ9〉|ζ10〉|ζ11〉|ζ12〉

⎞

⎟⎟⎟⎟⎠,

⎛

⎜⎜⎜⎜⎜⎝

|μ(k)

13 〉|μ(k)

14 〉|μ(k)

15 〉|μ(k)

16 〉

⎞

⎟⎟⎟⎟⎟⎠= G(k)

⎛

⎜⎜⎜⎜⎝

|ζ13〉|ζ14〉|ζ15〉|ζ16〉

⎞

⎟⎟⎟⎟⎠,

(8)

2618 Int J Theor Phys (2013) 52:2615–2622

where

G(1) =

⎛

⎜⎜⎝

1 δ1 δ2 δ3

1 −δ1 δ2 −δ3

1 −δ1 −δ2 δ3

1 δ1 −δ2 −δ3

⎞

⎟⎟⎠ , G(2) =

⎛

⎜⎜⎝

δ1 1 δ3 δ2

δ1 −1 δ3 −δ2

δ1 −1 −δ3 δ2

δ1 1 −δ3 −δ2

⎞

⎟⎟⎠ ,

G(3) =

⎛

⎜⎜⎝

δ2 δ3 1 δ1

δ2 −δ3 1 −δ1

δ2 −δ3 −1 δ1

δ2 δ3 −1 −δ1

⎞

⎟⎟⎠ , G(4) =

⎛

⎜⎜⎝

δ3 δ2 δ1 1δ3 −δ2 δ1 −1δ3 −δ2 −δ1 1δ3 δ2 −δ1 −1

⎞

⎟⎟⎠ ,

(9)

where δl = e−iηl (l = 1,2,3), and |ζr〉 (r = 1,2, . . . ,16) is still in Eq. (7).Now let Alice first performs four-qubit projective measurement on the qubits A1, A2,

A3, A4 by using the basis {|τp〉} (p = 1,2, . . . ,16). Next, according to the result of mea-surement, Alice chooses one of measuring bases {|μ(k)

r 〉} (r = 1,2, . . . ,16; k = 1,2,3,4)

to measure qubits B1,B2,B3,B4. After these measurements, Alice informs Bob of her out-comes of measurement by classical channel. In accord with Alice’s results, Bob can re-construct the original state |φ〉. For instance, without loss of generality, assume Alice’sfirst measurement result is |τ1〉A1A2A3A4 , she should choose measuring basis {|μ(1)

r 〉} (r =1,2, . . . ,16) to measure qubits B1,B2,B3,B4, and then publicly announces her measure-ment outcomes. If Alice’s measurement result is |μ(1)

10 〉B1,B2,B3,B4 , the qubits C1,C2,C3 andC4 will collapse into the state 1

8 (x0|ζ9〉 − x1eiη1 |ζ10〉 + x2e

iη2 |ζ11〉 − x3eiη3 |ζ12〉)C1C2C3C4 .

According to Alice’s public announcement, Bob should perform the local unitary transfor-mation U

(1)

10 = (I )C1 ⊗ (σx)C2 ⊗ (σz)C3 ⊗ (I )C4 on his qubits C1,C2,C3 and C4, then Bobcan reconstruct the original state |φ〉 at his side.If Alice’s second measurement results arethe other 15 cases in the basis {|μ(1)

r 〉} (r = 1,2, . . . ,9,11, . . . ,16), Bob can perform suit-able local unitary transformation U(1)

r on qubits C1,C2,C3 and C4, the original state canbe recovered. The relation between the measurement results obtained by Alice and the ap-propriate unitary transformations performed by Bob is shown in Table 1. If Alice’s firstmeasurement outcomes are the other 15 cases in the basis {|τp〉A1A2A3A4} (p = 2, . . . ,16),

Table 1 The relation between the measurement results (MR) of Alice and the local unitary transforma-

tions U(1)r (r = 1,2, . . . ,16) performed on the qubits C1,C2,C3,C4 by Bob. (λ1 → |τ1〉A1A2A3A4 , ξ

(1)r →

|μ(1)r 〉B1B2B3B4 ; r = 1,2, . . . ,16)

MR U(1)r MR U

(1)r

λ1ξ(1)1 (I )C1 ⊗ (I )C2 ⊗ (I )C3 ⊗ (I )C4 λ1ξ

(1)9 (I )C1 ⊗ (σx)C2 ⊗ (I )C3 ⊗ (I )C4

λ1ξ(1)2 (I )C1 ⊗ (I )C2 ⊗ (σz)C3 ⊗ (I )C4 λ1ξ

(1)10 (I )C1 ⊗ (σx)C2 ⊗ (σz)C3 ⊗ (I )C4

λ1ξ(1)3 (I )C1 ⊗ (σz)C2 ⊗ (σz)C3 ⊗ (I )C4 λ1ξ

(1)11 (σz)C1 ⊗ (σx)C2 ⊗ (σz)C3 ⊗ (I )C4

λ1ξ(1)4 (I )C1 ⊗ (σz)C2 ⊗ (I )C3 ⊗ (I )C4 λ1ξ

(1)12 (σz)C1 ⊗ (σx)C2 ⊗ (I )C3 ⊗ (I )C4

λ1ξ(1)5 (I )C1 ⊗ (I )C2 ⊗ (I )C3 ⊗ (σx)C4 λ1ξ

(1)13 (I )C1 ⊗ (σx)C2 ⊗ (I )C3 ⊗ (σx)C4

λ1ξ(1)6 (I )C1 ⊗ (I )C2 ⊗ (σz)C3 ⊗ (σx)C4 λ1ξ

(1)14 (I )C1 ⊗ (σx)C2 ⊗ (σz)C3 ⊗ (σx)C4

λ1ξ(1)7 (I )C1 ⊗ (σz)C2 ⊗ (σz)C3 ⊗ (σx)C4 λ1ξ

(1)15 (σz)C1 ⊗ (σx)C2 ⊗ (σz)C3 ⊗ (σx)C4

λ1ξ(1)8 (I )C1 ⊗ (σz)C2 ⊗ (I )C3 ⊗ (σx)C4 λ1ξ

(1)16 (σz)C1 ⊗ (σx)C2 ⊗ (I )C3 ⊗ (σx)C4

Int J Theor Phys (2013) 52:2615–2622 2619

she should choose appropriate measuring bases {|μ(k)r 〉} to measure her qubits B1,B2,B3

and B4. The corresponding relation of the first measurement outcome |τp〉A1A2A3A4 and thesecond measuring bases |μ(k)

r 〉 can be described as

|τ1〉, |τ5〉, |τ9〉, |τ13〉 → ∣∣μ(1)r

⟩, |τ2〉, |τ6〉, |τ10〉, |τ14〉 → ∣∣μ(2)

r

⟩,

|τ3〉, |τ7〉, |τ11〉, |τ15〉 → ∣∣μ(3)r

⟩, |τ4〉, |τ8〉, |τ12〉, |τ16〉 → ∣∣μ(4)

r

⟩,

(10)

where r = 1,2, . . . ,16. It is easily found that, for all the 256 measurement results of Alice,the receiver Bob can reconstruct the original state |φ〉 by performing appropriate unitary op-erations {|μ(k)

r 〉} (r = 1,2, . . . ,16; k = 1,2,3,4), the success probability of the RSP processbeing 1. The required classical communication cost is 8 bits in the scheme.

3 RSP with Six Non-maximally Two-Qubit Entangled States as the QuantumChannel

This scheme can be generalized to the case that non-maximally two-qubit entangled statesare taken as the quantum channel. Without loss of generality, the quantum channel sharedAlice and Bob can be expressed as

|ϕj 〉 = aj |00〉Rj+ bj |11〉Rj

, (11)

where j = 1,2,3,4,5,6; aj and bj are real, |aj | ≥ |bj |, a2j + b2

j = 1 and R1 = A1B1, R2 =C1A1, R3 = B2C2, R4 = A3B3, R5 = C3A4, R6 = B4C4. Assume the state that Alice wantsto help Bob to remotely prepare is still in state (3). As in the previous scheme, Alice can firstperform the four-qubit projective measurement on her qubits A1, A2, A3, A4 by using thebasis {|τp〉} (p = 1,2, . . . ,16), then chooses the appropriate measuring basis {|μ(k)

r 〉} (r =1,2, . . . ,16; k = 1,2,3,4) to measure qubits B1,B2,B3,B4 in accord with the measurementoutcome |τp〉A1A2A3A4 and informs Bob of her measurement results. According to Alice’smeasurement results, Bob can first make the appropriate unitary transformation on his qubitsC1,C2,C3,C4, then introduces an auxiliary qubit A with the initial state |0〉A and performsa collective unitary transformation on his qubits. Subsequent method of operation of Bob issimilar to the scheme of Ref. [29], after that, Bob can reconstruct the original state |φ〉 in hisposition. For example,without loss of generality, assume Alice’s first measurement outcomeis |τ3〉A1A2A3A4 and her second result is |μ(3)

8 〉B1B2B3B4 , the qubits C1,C2,C3,C4 will collapseinto the state

|ε1〉 = (b1b2b3a4a5b6x|1101〉 − b1b2b3b4b5a6yeiη1 |1110〉+ a1a2a3a4a5b6ze

iη2 |0001〉 − a1a2a3b4b5a6weiη3 |0010〉)C1C2C3C4

. (12)

According to Alice’s announcement, Bob first operates an unitary transformation

U1 = (σx)C1 ⊗ (σx)C2 ⊗ (σz)C3 ⊗ (σx)C4 (13)

on his qubits C1,C2,C3 and C4, the state (12) will be transformed into the state

∣∣ε′1

⟩ = (b1b2b3a4a5b6x|0000〉 + b1b2b3b4b5a6yeiη1 |0011〉+ a1a2a3a4a5b6ze

iη2 |1100〉 + a1a2a3b4b5a6weiη3 |1111〉)C1C2C3C4

. (14)

2620 Int J Theor Phys (2013) 52:2615–2622

Then Bob introduces an auxiliary qubit A with the initial state |0〉A and make an-other unitary transformation U2 under the basis {|00000〉C1C2C3C4A, |00110〉C1C2C3C4A,

|11000〉C1C2C3C4A,|11110〉C1C2C3C4A,|00001〉C1C2C3C4A, |00111〉C1C2C3C4A,|11001〉C1C2C3C4A,

|11111〉C1C2C3C4A}, namely

U2 =(

D1 D2

D2 −D1

), (15)

where Di(i = 1,2) is a 4 × 4 matrix and can expressed as

D1 = diag(d0, d1, d2, d3),

D2 = diag(√

1 − d20 ,

√1 − d2

1 ,

√1 − d2

2 ,

√1 − d2

3

), (16)

where dj (j = 0,1,2,3 and |dj | ≤ 1) depends on the state of the qubits C1,C2,C3 and C4.In Eq. (14), one may take

d0 = b4b5

a4a5, d1 = b6

a6, d2 = b1b2b3b4b5

a1a2a3a4a5, d3 = b1b2b3

a1a2a3. (17)

The unitary transformation U2 will transform |ε′〉 ⊗ |0〉A into

b1b2b3b4b5b6(x|0000〉 + yeiη1 |0011〉 + zeiη2 |1100〉 + weiη3 |1111〉)

C1C2C3C4⊗ |0〉A

+(xb1b2b3b6

√(a4a5)2 − (b4b5)2|0000〉 + yeiη1b1b2b3b4b5

√a2

6 − b26|0011〉

+ zeiη2b6

√(a1a2a3a4a5)2 − (b1b2b3b4b5)2|1100〉

+ weiη3b4b5b6

√(a1a2a3)2 − (b1b2b3)2|1111〉

)

C1C2C3C4

⊗ |1〉A. (18)

Then Bob measures the state of qubit A. If the result is |1〉A, the RSP fails. If the result is|0〉A, Bob can recover the original state |φ〉 in his position. By calculation, one can find that,if Alice’s measurement results are |τ3〉A1A2A3A4 |μ(3)

5 〉B1B2B3B4 , |τ3〉A1A2A3A4 |μ(3)

6 〉B1B2B3B4 ,and |τ3〉A1A2A3A4 |μ(3)

7 〉B1B2B3B4 , the qubits C1,C2,C3,C4 will collapse into the states respec-tively

|ε2〉 = (−b1b2b3a4a5b6x|1101〉 + b1b2b3b4b5a6yeiη1 |1110〉+ a1a2a3a4a5b6ze

iη2 |0001〉 − a1a2a3b4b5a6weiη3 |0010〉)C1C2C3C4

, (19)

|ε3〉 = (−b1b2b3a4a5b6x|1101〉 − b1b2b3b4b5a6yeiη1 |1110〉+ a1a2a3a4a5b6ze

iη2 |0001〉 + a1a2a3b4b5a6weiη3 |0010〉)C1C2C3C4

, (20)

and

|ε4〉 = (b1b2b3a4a5b6x|1101〉 + b1b2b3b4b5a6yeiη1 |1110〉+ a1a2a3a4a5b6ze

iη2 |0001〉 + a1a2a3b4b5a6weiη3 |0010〉)C1C2C3C4

. (21)

From Eqs. (12), (19), (20) and (21), it is easily found that, if Alice’s measurement resulton the qubits A1,A2,A3,A4 is |τ3〉A1A2A3A4 and the results on the qubits B1,B2,B3,B4

Int J Theor Phys (2013) 52:2615–2622 2621

are |μ(3)

5 〉B1B2B3B4 , |μ(3)

6 〉B1B2B3B4 , |μ(3)7 〉B1B2B3B4 and |μ(3)

8 〉B1B2B3B4 , respectively, the uni-tary transformation U2 performed by Bob are all same as Eqs. (15)–(17). By the abovementioned method, it can easily be proven that, in all the 256 measurement results(|τp〉A1A2A3A4 |μ(k)

r 〉B1B2B3B4 ,p, r = 1,2, . . . ,16; k = 1,2,3,4) of Alice, only the 64 kinds ofthe unitary transformation U2 should be preformed by the receiver Bob. Here we no longerdepict them one by one. From the above analysis and by Eq. (18), one can find that the totalprobability of successful RSP is 256 × 1

4 × |b1b2b3b4b5b6|2. For the maximally entangledpairs, |aj | = |bj | = 1√

2(j = 1,2, . . . ,6), the total successful probability reaches 1.

4 Conclusion

In conclusion, we have presented a new scheme for the remote preparation of a four-qubitcluster-type state with complex coefficients via six EPR pairs as quantum channel. In thisscheme, the sender should first perform a four-qubit projective measurement on her fourqubits, then she can choose, according to the measurement result, another appropriate four-qubit measuring basis to measure her other four qubits. To complete the RSP scheme, anovel set of four-qubit mutually orthogonal basis vectors has been introduced. After theseprojective measurements, the receiver can recover the original state by means of appropriateunitary transformation. Furthermore, we have also considered that the quantum channel iscomposed of six non-maximally entangled states. It is shown that for such non-maximallyentangled quantum channel, after the sender’s two measurements, the receiver can recon-struct the original state by suitable unitary operations with certain probability. Comparedwith the previous scheme of RSP in Ref. [29], the advantage of the present scheme is thatthe receiver can reconstruct the unknown original state with unit successful probability. Thusour present scheme is useful in expanding RSP field of quantum information science.

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

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