Remote Preparation of Multipartite Equatorial Entangled States in High Dimensions with Three Parties

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1 Commun. Theor. Phys. (Beiing, China) 51 (2009) pp c Chinese Physical Society an IOP Publishing Lt Vol. 51, No. 4, April 15, 2009 Remote Preparation of Multipartite Equatorial Entangle States in High Dimensions with Three Parties HOU Kui, WANG Jing, an SHI Shou-Hua School of Physics & Material Science, Anhui University, Hefei 2009, China (Receive May 29, 2008; Receive September 26, 2008) Abstract A scheme for probabilistic remotely preparing N-particle -imensional equatorial entangle states via entangle swapping with three parties is presente. The quantum channel is compose of N 1 pairs of bipartite -imensional non-maximally entangle states an a tripartite -imension non-maximally entangle state. It is shown that the sener can help either of the two receivers to remotely prepare the original state, an the N-particle proective measurement an the generalize Haamar transformation are neee in this scheme. The total success probability an classical communication cost are calculate. PACS numbers: a, 0.67.Hk, w Key wors: remoter state preparation, -imensional, multiparticle equatorial entangle states, proective measurement, classical communication cost 1 Introuction Different from classical information, quantum information theory has opene up the possibility of novel form of information processing tasks. In recent years, there have been two important quantum information processing tasks: teleportation [1] an remote state preparation (RSP). [2 4] In teleportation, one can transmit an unknown quantum state from a sener to a spatially istant receiver using the entanglement channel by means of some classical information. Li et al. [5] propose a protocol to teleportation of a single-particle state by using a partial entangle state as the quantum channel. They showe that the original state coul be realize with a certain probability. Since then, quantum teleportation has always been a topic of interest ue to its important applications in quantum communication processing an quantum computation. [6] Another important application of the entanglement is remote state preparation which can transmit pure known quantum state using a prior share entanglement an some classical communication. Since the sener knows completely the remote state, in RSP there is a trae-off between the classical an the entanglement resources require, which are not possible in teleportation process. Up to now, remote preparation of two-level state has receive much attention both theoretically [7 16] an experimentally. [17,18] For the other han, with evelopment of the quantum information, there has been a growing interest in quantum informational processes base on multi-level systems. Teleportation of single particle in N-imensional Hilbert space was first propose by Bennett et al. [1] The quantum channel was compose of a pair of N-level particles in a maximally entangle state. Later, teleportation of an unknown multi-level quantum state attracte much attention. [19 22] Meanwhile, various RSP protocols in higher imensions have been put forwar. For example, Zeng et al. [2] prove that the remote state preparation scheme in real Hilbert space coul only be implemente when the imension of the space was 2, 4, or 8. Agrawa et al. [24] presente remote state preparation scheme for special class of states for multiparties, an they have generalize RSP scheme for qubits, qutrits an quits as well. Yu et al. [25] propose a protocol for the remote preparation of a quit using maximally entangle states. Liu et al. [26] propose two kins of schemes for remotely preparing multi-particle -imensional equatorial entangle states with unit probability. Wang et al. [27] presente a scheme for probabilistic remote preparation of a -imensional equatorial quantum state. Mikami et al. [28] reporte an experimental emonstration of remote preparation of an arbitrary pure qutrit state. In this paper, we propose a scheme for preparing remotely the multiparticle -imensional equatorial entangle states from a sener to either of two receivers by using N 1 pairs of bipartite partial entangle states an a partial tripartite entangle states as the share resource. Since the quantum channel is compose of non-maximally entangle states, the receiver can obtain the original state with a certain probability if an only if another party agrees to collaborate. We also calculat the total success probability an the total classical communication cost in this RSP scheme. The organization of this paper is as follows. In Sec. 2, we propose a scheme for remotely preparing a threeparticle three-imensional equatorial entangle state with three parties. In Sec., we sketch the generation of the three-imensional three-particle scheme to -imensional mutiparticle case. Conclusions are given in Sec. 4. The proect supporte by the Specialize Research Fun for the Doctoral Program of Higher Eucation uner Grant No hkahx@yahoo.com.cn ssh@ahu.eu.cn

2 642 HOU Kui, WANG Jing, an SHI Shou-Hua Vol Probabilistic Remotely Preparing a Three- Particle Three-Dimensional Equatorial Entangle State with Three Parties For simplicity, we firstly present a scheme for remote preparation of three-imensional equatorial entangle state with three parties. Our scheme consists of one sener (Alice) an two remote receivers (Bob an Charlie). Suppose that Alice washes to help the either one of the receivers Bob or Charlie to remotely prepare a triparticle three-imensional entangle state, φ = 1 ( e iη e iη ), (1) the parameters η 1 an η 2 are real numbers. Assume that Alice knows the original state φ completely, but neither Bob nor Charlie knows it at all. The quantum channel share by Alice, Bob, an Charlie is compose of two non-maximally two-particle three-imensional entangle states an a three-particle three-imensional nonmaximally entangle state which as follows, ϕ 11 = a a a , ϕ 22 = b b b , ϕ 4 = z z z , (2) a 2 = 1, b 2 = 1, z 2 = 1, a 2 < a ( = 0, 1), b 2 < b ( = 0, 1), z 2 < z ( = 0, 1). The initial state of the combine system is Ξ = ϕ 11 ϕ 22 ϕ 4. () Assume particles 1, 2, an belong to Alice, particles an 4 belong to Bob an Charlie an particles 1 an 2 belong to either one of the receivers (Bob or Charlie), respectively. In orer to help the receiver to reconstruct the original state, Alice shoul perform a three-particle proective measurement on the particles 1, 2, an in the basis vectors of Hilbert space of three particles with three imensions, [26,29] ψ p0 p 1 p 2 12 = 1 e 2πip0/ e iη 1 + p p 2, (4) + p 1 = [+p 1 mo ] an + p 2 = [+p 2 mo ], an η 0 = 0;, p 0, p 1, p 2 = 0, 1, 2, respectively. The entangle basis states { ψ p0 p 1 p 2 } form a complete orthonormal basis in a quantum system of the 27-imensional Hilbert space. Then, the resulting seven-particle state can be written as ψ p0 p 1 p 2 12 ψ p0 p 1 p 2 Ξ = ψ p0 p 1 p 2 12 ξ p0 p 1 p , (5) ξ p0 p 1 p = 1 2 a b +p1 z +p2 e 2iπp0/ e iη 1 + p p 2 + p 2 4. (6) Obviously, after Alice s three-particle proective measurement, the initial entanglement hel by Alice an the two receives between particles 1 an 1, 2 an 2, an, 4 isappears, but a new entangle state ξ p0 p 1 p 2, which oes not share any common past, is establishe between particles 1, 2,, an 4. Thus, a process of entanglement swapping appears (see Fig. 1). As mentione before, Alice can help Bob or Charlie remotely prepare the original state. Here, we only consier the case that Alice intens to restore the original state in Bob s place. Particles 1 an 2 belongs to him an the classical message of the three-particle proective measurement result must be sent to him. In orer to obtain the original state φ, Bob can ask Charlie to perform the Haamar transformation on particle 4 uner the basis { λ k }, which is given by Fig. 1 A scheme for remote preparation of the three-particle three-imensional equatorial entangle states via entangle swapping with three parties is presente. Two pairs of bipartite partial entangle state an a partial tripartite entangle state are use as the quantum channel. The scheme consists of one sener (Alice) an two remote receivers (Bob an Charlie). The sketch only shows that Bob to reconstruct the original state. A soli ot enotes a particle. The soli lines connecting particle represent their entanglement. (a) Alice performs three-particle proective measurement (TPM) on her particles 1, 2, an. Charlie performs the generalize Haamar transformation (GHT) on her particle 4. Particles 1, 2 an belongs to Bob; (b) Alice an Charlie inform Bob of their measurement results, an S 1 an S 2 stans for classical information.

3 No. 4 Remote Preparation of Multipartite Equatorial Entangle States in High Dimensions with Three Parties 64 λ k = 1 2 e 2πik/, (7), k = 0, 1, 2. In this case, the equation which escribe in Eq. (5) can be rewritten as λ k 4 λ k ψ p0 p 1 p 2 12 ψ p0 p 1 p 2 Ξ = ψ p0 p 1 p 2 12 λ k 4 ζ p0 p 1 p 2 1 2, (8) ζ p0 p 1 p is given by ζ p0 p 1 p = 1 a b +p1 z +p2 e 2iπ[p 0+(+p 2 )k]/ e iη 1 + p p 2. (9) As the same as Eq. (5), After Charlie s measurement, another entanglement swapping among particles 1, 2,, an 4 appears. Then, Charlie informs Bob of the measurement result through a classical channel. After having receive Alice s an Charlie s classical message, Bob knows that the particles 1, 2, an have been foun in the state { ζ p0 p 1 p }. Now, let us see how the original state can be prepare from the state in Eq. (9). Firstly, Bob nees to establish a corresponence that the coefficients a 0 b 0+p1 z 0+p2 e 2iπp2k/, a 1 b 1+p1 z 1+p2 e 2iπ[p 0+(1+p 2 )k]/ e iη 1, a 2 b 2+p1 z 2+p2 e iη 2 e 2iπ[2p 0+(2+p 2 )k]/ can correspon to , eiη , e iη Accoring to Alice s an Charlie s measurement outcomes, it may be realize by performing the following unitary transformation U 1 = (U p0 p 1 ) 2 (U p2 ), (10) with its matrix elements U p0 p 1 = e 2iπ[p 0+(+p 2 )k]/ + p 1, U p2 = + p 2. (11) Seconly, Bob introuces an auxiliary particle M in the state 0 M an performs another unitary transformation [5] U 2 on particles 1, 2,, an M uner the basis { 0000, 1110, 2220, 0001, 1111, 2221, 0002, 1112, 2222 }, the unitary transformation U 2 may take the form of the following 9 9 matrix M 1 M 2 0 U 2 = M 2 M 1 0, (12) 0 0 I M i (i = 1, 2) are iagonal matrix an I is a ientity matrix, respectively. M 1 an M 2 take the forms as follows, M 1 = iag (τ 1, τ 2, τ ), M 2 = iag ( 1 τ 2 1, 1 τ 2 2, 1 τ 2 ), (1), τ 1, τ 2, an τ only epen on the state escribe in Eq. (9). One may take a 2 b 2 z 2 a 2 b 2 z 2 τ 1 =, τ 2 =, a 0 b 0+p1 z 0+p2 a 1 b 1+p1 z 1+p2 τ = a 2 b 2 z 2 a 2 b 2+p1 z 2+p2. (14) The unitary transformation U 1 an U 2 will transform the state ζ p0 p 1 p M into U 2 U 1 ζ p0 p 1 p M = a 2b 2 z 2 ( eiη eiη ) 0 M + 1 (a b +p1 b +p2 ) 2 (a 2 b 2 z 2 ) M. (15) Finally, Bob measures the state of auxiliary particle M in the basis { 0, 1, 2 }. If the measurement result is 1 M, the RSP fails. However, if he fins 0 M, Bob knows the original state φ has been reconstructe on his particles 1, 2, an with the probability of a 2 b 2 z 2 2 /. After Alice performs three-particle proective measurement on her particles 1, 2, an, an Charlie carries out the generalize Haamar transformation on her particle 4, there are possible = 81 results. Obviously, we can easily fin that 81 kins of states can be realize successfully with the equal probability of a 2 b 2 z 2 2 /. Thus, the total probability of successful RSP is 81 a 2 b 2 z 2 2 / = 27 a 2 b 2 z 2 2. Specially, if the quantum channel is compose of two maximally entangle three-imension two-particle state an a maximally entangle three-imension three-particle state ( a 0 = a 1 = a 2 = 1/, b 0 = b 1 = b 2 = 1/, an z 0 = z 1 = z 2 = 1/ ), the total probability equals 1. By the similar metho, if Charlie possesses the particle 1 or 2, Alice can help her to remotely prepare the original state with help of Bob s classical information an the total probability of successful RSP is also 27 a 2 b 2 z 2 2. Since classical communication cost is one of the important investigation in RSP, more RSP schemes were iscusse etailely [11 1] for non-maximally entangle quantum channel. Now, let us iscuss the classical communication cost in our remote state preparation scheme. Here, we only consier the case that the sener Alice intens to restore the original state in Bob s location. Similar to the scheme of Ref. [12], the classical message in our

4 644 HOU Kui, WANG Jing, an SHI Shou-Hua Vol. 51 scheme also can be ivie into two transmitte processes (see Fig. 1(b)). (i) Alice informs of Bob her three-particle proective measurement result. (ii) Since Charlie nees to carry out the Haamar transformation on her particle, another classical communication cost process is from Alice to Charlie an then from Charlie to the receiver Bob. Process 1 After three-particle proective measurement on particles 1, 2, an, Alice can obtain 27 outcomes ψ p0 p 1 p 2 12 (p 0, p 1, p 2 = 0, 1, 2) with the probability of 2 (a b +p1 z +p2 ) 2 ( e 2iπp0/ e iη ) 2. Thus, the total classical communication cost in this process requires S 1 = p 0,p 1,p 2 { 2 ( ) 2 a b +p1 z +p2 e 2iπp 0 / e iη 2 [ 2 (a b +p1 z +p2 ) 2 e 2iπp0/ e iη 2 ] } 2. (16) Process 2 After knowing Alice s measurement result, Charlie shoul carry out the generalize Haamar transformation on her particle 4 in the basis vectors escribe by Eq. (7) to help Bob reconstruct the original state. As above analysis, after Charlie s separate measurement, there are 81 kins of states in all an all kins of states of particles 1, 2, an can be realize successfully with the equal probability of a 2 b 2 z 2 2 /. Therefore, the classical communication cost process is Alice to Charlie an then from Bob requires S 2 = 81 a 2b 2 z 2 2 [ a2 b 2 z 2 2 ] 2. (17) Therefore, the total classical communication for this reliable probabilistic remote preparation of a three-particle threeimensional equatorial entangle state with three parties require S = S 1 + S 2 = p 0,p 1,p 2 { 2 ( ) 2 a b +p1 z +p2 e 2iπp 0 / e iη 2 [ 2 (a b +p1 z +p2 ) 2 e 2iπp0/ e iη 2 ] } 2 [ 27 a 2 b 2 z 2 2 a2 b 2 z 2 2 ] 2. (18) Notice that, if the quantum channel is compose of maximally entangle states, the total classical information will consume 7 2 bits an Bob can obtain the original state with probability of 1. Probabilistic Remotely Preparing an N- Particle -Dimensional Equatorial Entangle States with Three Parties Our scheme also can be extene to the case of multiparticle in higher imensions. Suppose the original state that the sener Alice wants to help the two receivers (Bob or Charlie) remotely prepare is an N-particle - imensional equatorial entangle state, Φ = 1 e iη, (19) }{{} N η 0 = 0 an = 0, 1, 2,..., 1, an N is an arbitrary positive integer, respectively. The quantum channel share by Alice, Bob, an Charlie are N 1 pairs of bipartite partial entangle states an a partial tripartite entangle state, which are given as 1 Ψ 11 = a 11, 1 Ψ 22 = b 22, 1 Ψ (N 1)(N 1) = c (N 1)(N 1), 1 Ψ NN (N+1) = z NN (N+1), (20) 1 a 2 = 1, 1 c 2 = 1, 1 b 2 = 1,..., 1 z 2 = 1, a 1 < a, b 1 < b,..., c 1 < c, z 1 < z. The initial state of the combine system is Γ NN (N+1) = Ψ 11 Ψ 22 Ψ (N 1)(N 1) Ψ NN (N+1). (21) Here, particles 1, 2,...,N belong to the sener Alice, particles 1, 2,..., (N 1) belong to either of the two receivers an particle N belongs to Bob while particle N + 1 is at the receiver Charlie s sie, respectively. Since Alice knows exactly the parameters η of the state escribe in Eq. (19), she can choose to measure her N particles in any basis

5 No. 4 Remote Preparation of Multipartite Equatorial Entangle States in High Dimensions with Three Parties 645 she wants. Here, Alice carries out an N-particle proective measurement on her particles in the basis vectors efine by Υ p0 p 1 p N 1 12 N = 1 e 2πip0/ e iη 1 + p p N 1 N, (22) + p N 1 = [ + p N 1 mo ], an, p 0, p 1,...,p N 1 = 0, 1, 2,..., 1, respectively. After that, we can get Υ p0 p 1...p N N Υ p0 p 1...p N 1 Γ = Υ p0 p 1...p N N Λ N (N+1), (2) Λ = 1 a b +p1...z +p(n 1) e 2πip0/ e iη 1 + p p (N 1) N + p N 1 N+1. (24) Suppose that Alice intens to restore the original state in Bob s location. In this case, particles 1, 2,...,(N 1) will belong to Bob. In orer to reconstruct the original state, Bob also nees Charlie s help an asks he to make a measurement on particle N + 1 uner the basis { Ω () 0, Ω() 1,..., Ω() 1 }, which are given by Ω () k = 1 1 e 2πik/, (25), k = 0, 1, 2,..., 1. Then Eq. (2) can be rewritten as Ω () k N+1 Ω () k Υ p 0 p 1...p N N Υ p0 p 1...p N 1 Γ = Υ p0 p 1...p N N Ω () k N+1 Θ N, (26) Θ N = 1 a b +p1...z +pn 1 e 2πi[p 0+(+p N 1 )k]/ e iη 1 + p p N 1 N. (27) Accoring to Eqs. (2) an (26), after Alice s N-particle proective measurement an Charlie s generalize Haamar transformation in high imensions, Bob will obtain the quantum state escribe in Eq. (27) via entangle swapping. Now let us see how Bob reconstructs the original state on his particles. Firstly, Bob will carry out the following unitary transformation U 1 on his particles, U 1 = (U p0 p 1 ) 2 (U p2 )... (U pn 1 ) (N ), (28) U p0 p 1 = e 2πi(p 0+(p N 1 +)k]/ + p 1, U p2 = + p 2,, U p(n 1) = + p N 1. (29) That is, unitary transformation U 1 will transform the state escribe in Eq. (27) into U 1 Θ N = 1 a b +p1... z +p(n 1) e iη 1 2 N (0) Seconly, Bob introuces an auxiliary particle A in the state 0 A an performs another unitary U 2 on his particles 1, 2,...,N an A in the basis N A, N A,, ( 1)( 1) ( 1)0 1 2 N A, N A, N A,, ( 1)( 1) ( 1)1 1 2 N A, N A, N A,, ( 1)( 1) ( 1)2 1 2 N A, the unitary transformation U 2 may take the form of the following matrix A 1 A 2 0 U 2 = A 2 A 1 0, (1) 0 0 I U 2 is a matrix, an A 1 an A 2 are iagonal matrix an I is a ientity matrix, respectively. A 1 an A 2 take the forms as follows M 1 = iag (ρ 1, ρ 2,..., ρ N ), M 2 = iag ( 1 ρ 2 1, 1 ρ 2 2,..., 1 ρ 2 N), (2) ρ 1, ρ 2,...,ρ N only epen on the state escribe in Eq. (27). One may take ρ 1 = a ( 1)b ( 1) z ( 1) a 0 b 0+p1 z 0+p(N 1), ρ 2 = a ( 1)b ( 1) z ( 1) a ( 1) b ( 1) z ( 1),..., ρ N =. () a 1 b 1+p1 z 1+p(N 1) a ( 1) b ( 1)+p1 z ( 1)+p(N 1) After these measurement, Bob can get U 2U 1 Θ 1 2 N = a ( 1 ( 1)b ( 1) z ( 1) ) e iη 1 2 N 0 A + 1 ( 1 (a b +p1...z +p(n 1) ) 2 (a ( 1) b ( 1) z ( 1) ) 2 e iη 1 2 N ) 1 A. (4)

6 646 HOU Kui, WANG Jing, an SHI Shou-Hua Vol. 51 Finally, Bob measures the state of auxiliary particle A. If the measurement result is 1 A, the RSP fails. However, if he fins 0 A, the remote preparation of N-particle -imensional equatorial entangle states is successfully realize an the successful probability is (a ( 1) b ( 1) z ( 1) ) 2 /. After Alice s N-particle proective measurement an Charlie s generalize Haamar transformation in high imensions, there are possible N = N+1 kins of states in all. Nevertheless, we can easily fin that N+1 kins of states can be realize successfully with the equal probability of (a ( 1) b ( 1) z ( 1) ) 2 /. Thus, the total probability of successful RSP is N+1 (a ( 1)b ( 1) z ( 1) ) 2 = N (a ( 1) b ( 1) z ( 1) ) 2. Specially, if the quantum channel is compose of a maximally entangle states ( a 2 = b 2 = = z 2 = 1/ ), the total probability equals 1. Eviently, Alice also can help Charlie to remotely prepare the original state with help of Bob s classical message using the similar way to analyze. The classical information plays an essential role in remote state preparation. Without loss of generality, we only consier the case that the sener Alice wants to restore the original state in Bob s sie. The classical communication cost in this N-particle -imension RSP process is similar to the three-particle case an still is ivie into two transmitte processes. (i) After Alice performs the N-particle proective measurement, she nees to inform Bob of the measurement outcome, the classical information in this process from the sener Alice to receiver Bob requires S 1 = p 0,...,p N 1 { 1 (a b +p1 z +pn 1 ) 2 e 2πip 0/ e iη 2 [ 1 2 (a b +p1 z +pn 1 ) 2 e 2πip0/ e iη 2 ] }. (5) (ii) In orer help Bob to complete the task, Charlie shoul carry out the generalize Haamar transformation in high imensions on her particle N + 1 an informs Bob of her result via a classical channel. The classical information in this process can be see that from the sener Alice to Charlie an then from Charlie to Bob nees S 2 = N+1 (a ( 1)b ( 1) z ( 1) ) 2 [ (a( 1) b ( 1) z ( 1) ) 2 ] 2. (6) Therefore, amount of the classical communication cost require in the remote preparation of N-particle -imension equatorial entangle states with three parties is S = S 1 + S 2 = p 0,...,p N 1 { 1 (a b +p1 z +pn 1 ) 2 e 2πip 0/ e iη 2 N (a ( 1) b ( 1) z ( 1) ) 2 2 [ (a( 1) b ( 1) z ( 1) ) 2 [ 1 2 (a b +p1 z +pn 1 ) 2 e 2πip0/ e iη 2 ] } ]. (7) If the quantum channel is consiste of maximally entangle states, an auxiliary particle A an unitary U 2 are not neee. In this special case, the classical information will consume (2N + 1) 2 bits in total. 4 Conclusion In summary, by virtue of the N 1 pairs of bipartite partial entangle states an a partial tripartite entangle state as the quantum channel, we have propose a scheme for probabilistic remotely preparing N-particle -imensional equatorial entangle states with three parties. The sener s N-particle proective measurement an one receivers generalize Haamar transformation are neee in our scheme. We have calculate the total successful probability an the classical communication cost in this RSP scheme. It is shown that the successful probability of RSP can improve to 1 if the quantum channel is compose by maximally entangle states, an the classical communication cost is only etermine the particle number N an the imension. Unlike other RSP schemes [2 27] in high imensions, which only consiere the RSP with two parties from a sener to a receiver, the sener (Alice) can help either one of the two receivers (Bob or Charlie) remotely prepare the original state with certainty probability in our scheme. Furthermore, if the quantum consists of N 1 pairs of two-particle -imensional non-maximally entangle states an a k+1- particle -imension non-maximally entangle state, this scheme can be irectly generalize to the remote preparation of mutiparticle equatorial entangle state from a sener to any of k receivers. Since multi-level quantum cryptographic schemes show more secure against eavesropping than their qubit-base counterparts [0] an the present experimental context makes it reasonable to consier the manipulation of more-than-two-level quantum information carriers, [1] our scheme may be helpful to realizing the potential characteristic of multi-level state. Expressly, we have calculate the total classical communication of probabilistic RSP scheme with three parties, while the earlier RSP schemes in high imensions only consiere the classical communication cost for a maximally quantum channel. Thus, our RSP scheme also can be helpful to stuying quantum communication complexity. Nowaays, a metho for preparing the entangle state of three-level atoms has been prepare with a non-resonant cavity [2] an a scheme for preparing multi-level atoms in

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