No. 12 Probabilistic teleportation of an arbitrary Suppose that the sender (Ali) wants to transmit an unknown arbitrary three-particle state t

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1 Vol 12 No 12, Demr 2003 cfl 2003 Chin. Phys. Soc /2003/12(12)/ Chinese Physics and IOP Publishing Ltd Probabilistic teleportation of an arbitrary three-particle state via a partial entangled four-particle state and a partial entangled pair Dai Hong-Yi( Λ ), Li Cheng-Zu(Ξ Φ), and Chen Ping-Xing( Π±) College of Scien, National University of Defense Technology, Changsha , China (Reived 8 January 2003; revised manuscript reived 29 April 2003) We present a scheme to probabilistically teleport an arbitrary and unknown three-particle state via a two-particle non-maximally entangled state and a four-particle non-maximally entangled state as the quantum channel. With the help of Bell-state measurements, an arbitrary three-particle state can perfectly teleported if a reiver introdus a collective unitary transformation. All kinds of unitary transformations are given in greater detail. This scheme can generalized to the teleportation of an arbitrary and unknown multiparticle state. Keywords: probabilistic teleportation, arbitrary three-particle state, unitary transformation, fourparticle entangled state PACC: 0365 The quantum teleportation pross, proposed by Bennett et al [1] can transmit an unknown quantum state from a sender (Ali) to reiver (Bob) at a distant location via a quantum channel with the aid of some classical information. Teleportation of the polarized photon and a single coherent mode of the radiation field has en realized experimentally by using parametric down-conversion. [2 4] Davidovich et al [5] have presented a protocol to teleport an unknown atomic state tween two high-q cavities initially prepared in an entangled Fock state. Cirac and Parkins [6] also investigated another QED proposal for quantum teleportation of an atomic state. Rently, more and more attention has en paid to teleportation of an unknown entangled state. Gorbachev and Trubilko [7] Shi et al, [8;9] Lee and Kim [10] Ikram et al, [11] Li et al, [12] Zeng et al, [13] Marinatto and Wer [14] and Yan et al [15] have investigated teleportation of a two-particle entangled state under various conditions. The proposals for teleportation of a three-particle entangled state have en put forward by Yang and Guo [16] and Lu and Guo [17] by using three Einstein Podolski Rosen (EPR) pairs and one EPR pair as well as the Greenbareer Horne Zeillinger (GHZ) triplet. Meanwhile, Liu and Guo [18] have proposed a method for teleportation of a three-particle entangled GHZ state by using three partly entangled pairs as the quantum channel. Zheng et al [19] and Chen [20] have presented a scheme to teleport a threeparticle entangled state with the help of one EPR pair and GHZ triplet. However, how to probabilistically teleport an arbitrary three-particle state is an important and interesting problem in quantum information. In previous schemes for teleporting a three-particle entangled state, almost all schemes are restricted to the quantum channels such as the EPR pair, the GHZ state, etc. Furthermore, they only considered the teleportation of a three-particle GHZ state or three-particle entangled W state, while an arbitrary three-particle state includes all kinds of three-particle states, either the entangled three-particle GHZ state and entangled threeparticle W state, or the disentangled three-particle state. Here, we study the teleportation of an arbitrary and unknown three-particle state: a four-particle nonmaximally entangled state and a two-particle nonmaximally entangled state are chosen as the quantum channel. We show that the probabilistic teleportation of the arbitrary and unknown three-particle state can realized by performing three generalized Bell-state measurements at the sender's side and introducing an appropriate unitary transformation in the reiver's laboratory.

2 No. 12 Probabilistic teleportation of an arbitrary Suppose that the sender (Ali) wants to transmit an unknown arbitrary three-particle state to the reiver (Bob) who is spatially separated. The arbitrary three-particle state, which will teleported, can expressed as jψ i 123 = 1 j001i j010i j100i 123 where 8X i=1 + 4 j110i j101i j011i j000i j111i 123 ; (1) j i j 2 = 1. Without loss of generality, the quantum channel, composed of a four-particle nonmaximally entangled state and a two-particle nonmaximally entangled state, reads as jψ i 4567 =aj0000i bj1001i cj0110i dj1111i 4567 ; (2) jψ i 89 = ej00i 89 + f j11i 89 ; (3) where the coefficients a, b, c and d are real, jdj is smaller than the absolute value of other coefficients, jaj 2 + jbj 2 + jcj 2 + jdj 2 = 1; real coefficients e and f satisfy jej 2 + jf j 2 = 1, jej jf j. Particles 4, 6 and 8 long to the sender Ali's side while particles 5, 7 and 9 are at the reiver Bob's side. Therefore, the initial state of the whole system composed of particles (1, 2, 3) and quantum channel formed by entangled states (4, 5, 6, 7) and (8, 9) is given by jψ i = jψ i 123 ΩjΨ i 4567 ΩjΨ i 89 : (4) Fig.1. Sketch of the teleportation of an arbitrary three-particle state. Ali wants to teleport an arbitrary three-particle state (particles 1, 2 and 3) to Bob. A four-particle (particles 4, 5, 6 and 7) partial entangled state and a two-particle (particles 8 and 9) partial entangled state are used as a quantum channel. Particles 4, 6 and 8 long to Ali while particles 5, 7 and 9 are at the reiver Bob's side. U.T." in the box denotes the unitary transformation Bob must perform in order to retrieve the original state. C.C." represents classical communication. In order to realize teleportation, Ali must perform Bell-state measurements on particles (1, 4), particles (2, 6) and particles (3, 8), respectively. After the three Bell-state measurements, a new entanglement is established among particles 5, 7 and 9, and entanglement swapping occurs. The schematic diagram for teleporting an arbitrary three-particle state is shown in Fig.1. The possible 64 kinds of results are: 38hΦ ± j 26 hφ ± j 14 hφ ± jψ i = 1 2 p 2 [±3 af 1 j001i ± 2 2 j100i ± 1 3 j010i ± 2 (± 1 )de 4 j110i ± 3 (± 1 )bf 5 j011i ± 3 (± 2 )cf 6 j101i + 7 j000i ± 3 (± 2 )(± 1 ) 8 j111i] 579 ; (5) 38hΨ ± j 26 hφ ± j 14 hφ ± jψ i = 1 2 p 2 [±3 1 j000i ± 2 cf 2 j101i ± 1 bf 3 j011i ± 2 (± 1 ) 4 j111i ± 3 (± 1 ) 5 j010i ± 3 (± 2 ) 6 j100i + af 7 j001i ± 3 (± 2 )(± 1 )de 8 j110i] 579 ; (6) 38hΦ ± j 26 hψ ± j 14 hφ ± jψ i = 1 2 p 2 [±3 cf 1 j101i ± 2 2 j000i ± 1 de 3 j110i ± 2 (± 1 ) 4 j010i ± 3 (± 1 ) 5 j111i ± 3 (± 2 )af 6 j001i + 7 j100i ± 3 (± 2 )(± 1 )bf 8 j011i] 579 ; (7) 38hΨ ± j 26 hψ ± j 14 hφ ± jψ i = 1 2 p 2 [±3 1 j100i ± 2 af 2 j001i ± 1 3 j111i ± 2 (± 1 )bf 4 j011i ± 3 (± 1 )de 5 j110i ± 3 (± 2 ) 6 j000i + cf 7 j101i ± 3 (± 2 )(± 1 ) 8 j010i] 579 ; (8)

3 1356 Dai Hong-Yi et al Vol hΦ ± j 26 hφ ± j 14 hψ ± jψ i = 1 2 p 2 [±3 bf 1 j011i ± 2 de 2 j110i ± 1 3 j000i ± 2 (± 1 ) 4 j100i ± 3 (± 1 )af 5 j001i ± 3 (± 2 ) 6 j111i + 7 j010i ± 3 (± 2 )(± 1 )cf 8 j101i] 579 ; (9) 38hΨ ± j 26 hφ ± j 14 hψ ± jψ i = 1 2 p 2 [±3 1 j010i ± 2 2 j111i ± 1 af 3 j001i ± 2 (± 1 )cf 4 j101i ± 3 (± 1 ) 5 j000i ± 3 (± 2 )de 6 j110i + bf 7 j011i ± 3 (± 2 )(± 1 ) 8 j100i] 579 ; (10) 38hΦ ± j 26 hψ ± j 14 hψ ± jψ i = 1 2 p 2 [±3 1 j111i ± 2 2 j010i ± 1 3 j100i ± 2 (± 1 ) 4 j000i ± 3 (± 1 )cf 5 j101i ± 3 (± 2 )bf 6 j011i + de 7 j110i ± 3 (± 2 )(± 1 )af 8 j001i] 579 ; (11) 38hΨ ± j 26 hψ ± j 14 hψ ± jψ i = 1 2 p 2 [±3 de 1 j110i ± 2 bf 2 j011i ± 1 cf 3 j101i ± 2 (± 1 )af 4 j001i ± 3 (± 1 ) 5 j100i ± 3 (± 2 ) 6 j010i + 7 j111i ± 3 (± 2 )(± 1 ) 8 j000i] 579 ; (12) where the Bell-states, in which the particles (i, j) are measured, are defined as jφ ± i ij = 1 p 2 (j00i ij ±j11i ij ); (13) jψ ± i ij = 1 p 2 (j01i ij ±j10i ij ); (14) and where ± 1, ± 2, and ± 3 correspond to the superscripts for the Bell-state composed of particles (1, 4), (2, 6) and (3, 8), respectively. For instan, if Ali's measurement results are jφ i 14, jψ + i 26 and jφ i 38, i.e. the corresponding superscripts are, + and the state of particles 5, 7, 9, as shown by Eq.(7), will collapse into jψ i 579 = 38 hφ j 26 hψ + j 14 hφ jψ i = 1 2 p 2 ( cf 1j101i + 2 j000i de 3 j110i 4 j010i + 5 j111i af 6 j001i + 7 j100i bf 8 j011i) 579 : (15) After operations, Ali informs Bob of her measurements results via a classical communication. First, Bob needs to establish a corresponden so that the coefficients i (i = 1; ; 8) can correspond to j001i 579, j010i 579, j100i 579, j110i 579, j101i 579, j011i 579, j000i 579 and j111i 579, respectively. This can realized by performing a unitary transformation U 1 on particles 5, 7 and 9. All possible unitary transformations U 1 on the states of particles 5, 7 and 9 are given in Table 1. For instan, the corresponding unitary transformation U 1 [20] U 1 =(j0ih1j 1 + j1ih0j 4 + j0ih1j 7 + j1ih0j 8) 5 Ω (j1ih0j 2 + j0ih1j 3 + j0ih1j 5 + j1ih0j 6) 7 (16) (here, subscript i denotes that the operater is applied to the state of coefficient i ) can transform the state expressed by Eq.(15) into jψ i 579 = 1 2 p 2 ( cf 1j001i + 2 j010i de 3 j100i 4 j110i + 5 j101i af 6 j011i + 7 j000i bf 8 j111i) 579 : (17) Secondly, Bob introdus an auxiliary particle A with an initial state j0i A and performs another unitary transformation U 2 on particles 5, 7, 9 and A. In order for Bob to reincarnate the original state under the basis fj0010i 579A, j0100i 579A, j1000i 579A, j1100i 579A, j1010i 579A, j0110i 579A, j0000i 579A, j1110i 579A, j0011i 579A, j0101i 579A, j1001i 579A, j1101i 579A, j1011i 579A, j0111i 579A, j0001i 579A, j1111i 579A g, the unitary transformation (a matrix) may take the form U 2 = A 1 A 2 A 2 A 1 1 A ; (18)

4 No. 12 Probabilistic teleportation of an arbitrary Table 1. Unitary transformations U1 on the states of particles 5, 7 and 9. States of particles 5, 7 and 9 38hΦ ± j 26 hφ ± j 14 hφ ± jψi 38hΨ ± j 26 hφ ± j 14 hφ ± jψi Unitary transformations U1 38hΦ ± j 26 hψ ± j 14 hφ ± jψi 38hΨ ± j 26 hψ ± j 14 hφ ± jψi (j0ih1j 1 + j1ih0j 4 + j0ih1j 7 + j1ih0j 8 ) 5 (j0ih1j 1 + j1ih0j 4 + j1ih0j 7 + j1ih0j 8 ) 5 38hΦ ± j 26 hφ ± j 14 hψ ± jψi 38hΨ ± j 26 hφ ± j 14 hψ ± jψi 38hΦ ± j 26 hψ ± j 14 hψ ± jψi 38hΨ ± j 26 hψ ± j 14 hψ ± jψi (j0ih1j 1 + j1ih0j 4 + j0ih1j 7 + j1ih0j 8 ) 5 (j0ih1j 1 + j1ih0j 4 + j0ih1j 7 + j1ih0j 8 ) 5 where A i (i=1, 2 ) is an 8 8 matrix, and may written, respectively, as A 1 = diag(a 1 ;a 2 ;a 3 ;a 4 ;a 5 ;a 6 ;a 7 ;a 8 ); (19) q1 a 21 ; q1 a 22 ; q1 a 23 ; q A 2 =diag 1 a 2; 4 q q1 a q1 25 ; a q1 26 ; a 27 ; 1 a 2 8 ; (20) where a i (i = 1; 2; ; 8, and ja i j» 1) depends on the state of particles 5, 7 and 9. For example, Bob introdus an auxiliary particle A with an initial state j0i A, the state descrid in Eq.(17) comes jψ i 579 Ωj0i A = 1 2 p 2 ( cf 1j001i + 2 j010i If we choose de 3 j100i 4 j110i + 5 j101i af 6 j011i + 7 j000i bf 8 j111i) 579 j0i A : (21) (a 1 ;a 2 ;a 3 ;a 4 ;a 5 ;a 6 ;a 7 ;a 8 ) = d c ; ; f e ; ; 1; d a ; ; d ; (22) b the unitary transformation U 2 will transform the state in Eq.(21) into 2 p 2 ( 1j001i + 2 j010i + 3 j100i + 4 j110i + 5 j101i + 6 j011i + 7 j000i + 8 j111i) 579 j0i A p 2 [ fp c 2 d 2 1 j001i p () 2 () 2 2 j010i 579 d p e 2 f 2 3 j100ii 579 p () 2 () 2 4 j110i 579 f p a 2 d 2 6 j011ii p () 2 () 2 7 j000i 579 f p b 2 d 2 8 j111ii 579 ]j1i A : (23) Of course, U 1 and U 2 may incorporated into a single transformation U = U 2 (U 1 Ω I A ). Such a unitary transformation can decomposed into universal quantum logical operations [21] that have en demonstrated experimentally in some physical systems. [22]

5 1358 Dai Hong-Yi et al Vol. 12 Table 2 shows the values of the coefficients a i (i = 1; ; 8) corresponding to every state, where ± 1, ± 2, and ± 3 of values a i (i = 1; ; 8) correspond to the superscripts of the Bell-state composed of particles (1, 4), particles (2, 6) and particles (3, 8), respectively. For example, if Ali makes Bell measurements jψ i 14, jφ i 26 and jψ i 38, values a i (i = 1; ; 8) are chosen as (a 1 ;a 2 ;a 3 ;a 4 ;a 5 ;a 6 ;a 7 ;a 8 ) = ; 1; d a ; d c ; ; f e ; d b ; : (24) Table 2. Values a i (i = 1; ; 8) of unitary transformation U2 corresponding to the states of particles 5, 7 and 9. States of particles 5, 7 and 9 a1 a2 a3 a4 a5 a6 a7 a8 38hΦ ± j 26 hφ ± j 14 hφ ± jψi ± 3 d a ± 2 ± 1 ± 2 (± 1 ) f e ± 3 (± 1 ) d b ± 3 (± 2 ) d c ± 3 (± 2 )(± 1 )1 38hΨ ± j 26 hφ ± j 14 hφ ± jψi ± 3 ± 2 d c ± 1 d b ± 2 (± 1 )1 ± 3 (± 1 ) ± 3 (± 2 ) d a ± 3 (± 2 )(± 1 ) f e 38hΦ ± j 26 hψ ± j 14 hφ ± jψi ± 3 d c ± 2 ± 1 f e ± 2 (± 1 ) ± 3 (± 1 )1 ± 3 (± 2 ) d a ± 3 (± 2 )(± 1 ) d b 38hΨ ± j 26 hψ ± j 14 hφ ± jψi ± 3 ± 2 d a ± 1 1 ± 2 (± 1 ) d b ± 3 (± 1 ) f e ± 3 (± 2 ) d c ± 3 (± 2 )(± 1 ) 38hΦ ± j 26 hφ ± j 14 hψ ± jψi ± 3 d b ± 2 f e ± 1 ± 2 (± 1 ) d a ± 3 (± 1 ) d a ± 3 (± 2 )1 ± 3 (± 2 )(± 1 ) d c 38hΨ ± j 26 hφ ± j 14 hψ ± jψi ± 3 ± 2 1 ± 1 d a ± 2 (± 1 ) d c ± 3 (± 1 ) ± 3 (± 2 ) f e d b ± 3 (± 2 )(± 1 ) 38hΦ ± j 26 hψ ± j 14 hψ ± jψi ± 3 1 ± 2 ± 1 ± 2 (± 1 ) ± 3 (± 1 ) d c ± 3 (± 2 ) d b f e ± 3 (± 2 )(± 1 ) d a 38hΨ ± j 26 hψ ± j 14 hψ ± jψi ± 3 f e ± 2 d b ± 1 d c ± 2 (± 1 ) d a ± 3 (± 1 ) ± 3 (± 2 ) 1 ± 3 (± 2 )(± 1 ) Finally, Bob measures the state of the auxiliary particle A. If the result j0i A is measured, then quantum teleportation is sucssfully realized with the probability of () 2 =8. Otherwise, teleportation fails. It can easily proven that 64 kinds of state have the same probability () 2 =8, therefore the total probability of sucssful teleportation is 8() 2. It is obvious that if the quantum channel is composed of a twoparticle maximally entangled state and a four-particle maximally entangled state (j0000i + j1001i + j0110i + j1111i) 4567 =2, namely jaj = jbj = jcj = jdj = 1=2, the total sucssful probability is equal to one. In conclusion, we have proposed a protocol for teleporting an arbitrary and unknown three-particle state by using a two-particle non-maximally entangled state and a four-particle partly entangled state as the quantum channel. The results show that, for such a non-maximally entangled quantum channel, there is still a rtain probability of sucssful teleportation if both the sender (Ali) performs generalized Bell-state measurements and the reiver (Bob) adopts some appropriate unitary transformations. All kinds of unitary transformations are given in greater detail. The probability of sucss is determined by the smaller coefficients of non-maximally entangled states used as the quantum channel. We must point out that this scheme can also generalized to the teleportation of an arbitrary and unknown multiparticle state. That is to say, the arbitrary 2N +1 particle entangled state can teleported by using a two-particle partial entangled state and N four-particle partial entangled states as the quantum channel; while the N fourparticle partial entangled states are used as a quantum channel to teleport an arbitrary and unknown 2N particle entangled state. Referens [1] Bennett C H, Brassard G, Grépeau C et al 1993 Phys. Rev. Lett [2] Bouwmeester D, Pan J W, Matter K et al 1997 Nature

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