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Multiparty secret sharing of quantum information via cavity QED
Optics Communications 261 (2006) 199–202 www.elsevier.com/locate/optcom
Multiparty secret sharing of quantum information via cavity QED Zhan-jun Zhang School of Physics and Material Science, Anhui University, Hefei, Anhui 230039, China Received 6 July 2005; received in revised form 23 November 2005; accepted 25 November 2005
Abstract The scheme on multiparty secret sharing of an atomic quantum state information via entanglement swapping in cavity QED [Y.Q. Zhang, X.R. Jin, S. Zhang, Phys. Lett. A 341 (2005) 380] is revisited and accordingly an improved version is proposed. The possible decoherence effect in the original version can be avoided after revision of assuming the prior distribution of entanglement. Moreover, the success probability of teleportation has been raised from 6.25% in the original version to 100% in the present version. Ó 2005 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 03.65.w; 42.50.Dv Keywords: Multiparty secret sharing; Quantum information; Entanglement swapping; Cavity QED
The principles of quantum mechanics supplied many interesting application in the field of information in the last decade. Quantum secret sharing (QSS) is an important branch of quantum information. It is a generalization of classical secret sharing [1,2] to a quantum scenario [3]. QSS is likely to play a key role in protecting secret quantum information, e.g., in secure operations of distributed quantum computation, sharing difficult-to-construct ancilla states and joint sharing of quantum money, and so on [4]. Hence, after the pioneering work presented by Hillery, Buzek and Berthiaume in 1999 by using three-particle and four-particle GHZ states [3], many attentions have been concentrated on the realization of QSS in both theoretical and experimental aspects [4–21]. All these works [3–21] can be divided into two kinds, one only deals with the QSS of classical messages (i.e., bits) [5–13], or only deals with the QSS of quantum information [4,14–20] where the secret is an arbitrary unknown state in a qubit; and the other [3,21] studies both, that is, deals with QSS of classical messages and QSS of quantum information simultaneously. In this paper, I will revisit a recently pro-
posed scheme on multiparty secret sharing of an atomic quantum state information via entanglement swapping in cavity QED. I will point out two disadvantages of the scheme. By overcoming the two disadvantages, I will propose an improved version of the scheme. Let us first consider a three-party quantum information secret sharing scheme. Suppose a three-party system consists of Alice, Bob and Charlie. Alice has a secret quantum information jui1 ¼ ajei1 þ bjgi1 ;
0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.11.060
2
where a and b are unknown coefficients and jaj + jbj = 1, jei and jgi are atomic excited and ground states, respectively. Alice can assign either Bob or Charlie to construct the unknown state at her will. The designee can certainly reconstruct the unknown state only if the other potential designee collaborates with him. To achieve the above goal, Zhang et al. [18] proposed their three-party scheme via entanglement swapping in cavity QED. In their scheme, Alice has other 5 two-level atoms besides the atom 1. The joint state of the six atoms is jwi123456 ¼ jui1 jni234 jvi56 ;
E-mail address: [email protected].
ð1Þ 2
where
ð2Þ
200
Z.-j. Zhang / Optics Communications 261 (2006) 199–202
1 jni234 ¼ pffiffiffi ðjeeei234 þ jgggi234 Þ; ð3Þ 2 1 jvi56 ¼ pffiffiffi ðjeei56 þ jggi56 Þ. ð4Þ 2 Alice introduces two identical single-mode cavities and sends simultaneously the atoms 1, 2 and the atoms 3, 5 into the two cavities, respectively. The two atoms interact with the single-mode cavity and are driven by a classical field. In the strong driving regime (i.e., X d and X n) and in the case d n, where X is the Rabi frequency, d is the deturning between the atomic transition frequency and the cavity frequency, and n is the atom-cavity coupling strength, there is no energy exchange between the atomic system and the cavity. Consequently, in the interaction picture, the effective Hamiltonian reads [29,30], " # 2 2 X f X þ þ þ He ¼ ðjgill hgj þ jeill hejÞ þ ðrl rk þ rl rk þ hcÞ ; 2 l¼1 l;k¼1;l6¼k ð5Þ n2 . 2d
where f ¼ Since the effective Hamiltonian is independent of the cavity field state, the cavity field is also permitted in a thermal state. In the above case, the evolution operator of the system reads [29,30] ! 2 X iH 0 t iH e t þ U ðtÞ ¼ e e exp iXt rl rl eiH e t . ð6Þ l¼1
By selecting the interaction time satisfy ft = p/4 and making the Rabi frequency satisfy Xt = p, the state of the system evolves to 1 jw0 i123456 ¼ jeeeei1235 ðajeei46 bjggi46 Þ 4 15 X þ ci j/i i1235 j/i i46 ; ð7Þ i¼1
where 1235h/ij/ji1235 = dij and 1235h/ijeeeei1235 = 0. After the interactions, Alice sends the atoms 4 and 6 to Bob and Charlie, respectively. Here, we stress that in Zhang et al.’s scheme only after interactions the atoms 4 and 6 are sent. After confirming that Bob and Charlie each has received an atom, Alice measures the atoms 1, 2, 3 and 5 and publishes her measurement outcomes. If the outcomes are jeeeei1235, then the state of atoms 4 and 6 collapses to jwI i46 ¼ ajeei46 bjggi46 . ð8Þ The subscript I will appear clear in what follows. This equation can be rewritten as jwI i46 ¼ ajeei46 bjggi46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajei6 bjgi6 Þ 2 2 1 þ pffiffiffi ðjei4 jgi4 Þðajei6 þ bjgi6 Þ 2 1 1 ¼ pffiffiffi pffiffiffi ðjei6 þ jgi6 Þðajei4 bjgi4 Þ 2 2 1 þ pffiffiffi ðjei6 jgi6 Þðajei4 þ bjgi4 Þ . ð9Þ 2
If Alice assigns Charlie to recover the unknown state, then Bob measures the atom 4 in the basis fp1ffiffi2 ðjei þ jgiÞ; p1ffiffi ðjei jgiÞg. If Bob collaborates with Charlie, then con2 ditioned on Bob’s measurement result Charlie can recover the unknown state with 100% certainty. That is, if Bob obtains fp1ffiffi2 ðjei4 jgi4 Þg, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6; if Bob obtains fp1ffiffi2 ðjei4 þ jgi4 Þg, conditioned Bob’s result, Charlie needs to perform a rz operation to transform the state ajei6 bjgi6 into ajei6 + bjgi6. Thus far, Zhang et al.’s scheme of three-party secret sharing of quantum information [18] has been briefly reviewed. Note that the success probability of Zhang et al.’s scheme is only 1/16. Incidentally, the security of this scheme is confirmed [18]. In Zhang et al.’s scheme, there are two disadvantages: (1) As mentioned before, only after the atoms interact with the cavities, the atoms 4 and 6 are sent to Bob and Charlie, respectively. In this case, during the transmission it is quite possible the decoherence happens due to the environment noises, i.e., the entanglement between the atom 4 and other atoms might change or even lose during the transmissions. This situation might also happen for the atom 6. If so, though Alice performs measurements on the atoms 1, 2, 3 and 5, the state of the atoms 4 and 6 will not collapse to the state jwi46 = ajeei46 bjggi46 even if the state jeeeei1235 is obtained by Alice. In this case, obviously, the secret sharing of the quantum information in Zhang et al.’s scheme [18] cannot be achieved anymore. More serious is that the unknown state which needs to be teleported has already been destroyed and thus the sharing of this unknown state completely fails. (2) As stressed before, the success probability of Zhang et al.’s scheme is only 1/16. With probability of 15/16 the secret sharing is not achieved and the unknown state is destroyed. This is not a satisfactory result. To overcome the above two disadvantages, I improve Zhang et al.’s scheme [18] as follows. Instead of transmitting the atoms 4 and 6 after the atoms 1, 2, 3 and 5 interact with the two cavities, we suppose the atom distributions are achieved in advance. That is, Alice and Bob have securely shared in advance the 3-atom maximally entangled state jwi234 ¼ p1ffiffi2 ðjeeei234 þ jgggi234 Þ. Alice has the atoms 2 and 3 in her site and Bob has the atom 4 in his site. Similarly, Alice and Charlie have securely shared in advance the 2atom maximal entangled state jwi56 ¼ p1ffiffi2 ðjeei56 þ jggi56 Þ. Alice has the atom 5 in her site and Charlie has the atom 6 in his site. In fact, so far a highly efficient procedure involving a sequence of entanglement purification and teleportation has been devised that allows the reliable sharing of such maximal entangled states between two arbitrarily distant locations [23–28]. Moreover, the generally used twomeasuring-basis method [22] can be employed to detect whether the atoms are attacked by eavesdropper during the atom distributions. Hence, it is reasonable to assume the secure sharing of the maximally entangled atom states
Z.-j. Zhang / Optics Communications 261 (2006) 199–202
between arbitrarily distant parties. If assumed, then the decoherence arising from the transmissions in Zhang et al.’ scheme [18] can be avoided. This ensures the success of the improved scheme. After the atom–cavity interactions with the given times, the state of the whole atom system evolves to 1 jwi ¼ ½jeeeei1235 ðajeei46 bjggi46 Þ 4 þ jeeegi1235 ðajegi46 bjgei46 Þ þ jeegei1235 ðajegi46 þ bjgei46 Þ þ jeeggi1235 ðajeei46 þ bjggi46 Þ þ jegeei1235 ðajggi46 þ bjgei46 Þ þ jegegi1235 ðajgei46 þ bjegi46 Þ þ jeggei1235 ðajgei46 bjegi46 Þ þ jegggi1235 ðajggi46 bjeei46 Þ þ jegggi1235 ðajggi46 bjeei46 Þ þ jgeegi1235 ðajgei46 bjegi46 Þ þ jgegei1235 ðajgei46 þ bjegi46 Þ þ jgeggi1235 ðajggi46 þ bjeei46 Þ þ jggeei1235 ðajeei46 þ bjggi46 Þ þ jggegi1235 ðajegi46 þ bjgei46 Þ þ jgggei1235 ðajegi46 bjgei46 Þ þ jggggi1235 ðajeei46 bjggi46 Þ.
ð10Þ
Alice measures the atoms 1, 2, 3 and 5 and publishes her measurement outcomes. In Zhang et al.’s scheme [18], the authors considered only the case that the measurement outcome is the state jeeeei1235 and therefore the success probability of their scheme is 1/16. Now in this paper I will further consider other measurement outcome cases. Incidentally, without loss of generality, in the following I will assume Charlie is assigned to recover the unknown state with help nof Bob and Bob always measures the atom 4 in o the basis p1ffiffi2 ðjei þ jgiÞ; p1ffiffi2 ðjei jgiÞ . Case 1: If Alice’s measurement outcome is the state jeeeei1235, the state of atoms 4 and 6 also collapses to jwIi46 = ajeei46 bjggi46. As shown by Zhang et al. [18], it is natural that conditioned on Alice’s public announcements Charlie can recover the unknown state with the help of Bob. However, one can easily find that if the measurement outcome is the state jggggi1235, then the state of atoms 4 and 6 also collapses to jwIi46. In this case, obviously, conditioned on Alice’s public announcements Charlie can also recover the unknown state with the help of Bob. Case 2: Alice’s measurement outcome is the state jeeggi1235 or the state jggeei1235. In this case, the state of atoms 4 and 6 collapses to jwII i46 ¼ ajeei46 þ bjggi46 . It can be rewritten as
ð11Þ
201
1 1 jwII i46 ¼ pffiffiffi pffiffiffi ðjei4 jgi4 Þðajei6 bjgi6 Þ 2 2 1 þ pffiffiffi ðjei4 þ jgi4 Þðajei6 þ bjgi6 Þ . ð12Þ 2 Bob measures the atom 4. If Bob collaborates with Charlie, then conditioned on Bob’s measurement result Charlie can recover the unknown state with 100% certainty. That is, if Bob obtains fp1ffiffi2:ðjei4 þ jgi4 Þg, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6; if Bob obtains p1ffiffi2 ðjei4 jgi4 Þ, Charlie needs to perform a rz operation to transform the state ajei6 bjgi6 into ajei6 + bjgi6. Case 3: Alice’s measurement outcome is the state jeggei1235 or the state jgeegi1235. In this case, the state of atoms 4 and 6 collapses to jwIII i46 ¼ ajgei46 bjegi46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajei6 bjgi6 Þ 2 2 1 ð13Þ pffiffiffi ðjei4 jgi4 Þðajei6 þ bjgi6 Þ . 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6. Otherwise, Charlie needs to perform a rz operation on the atom 6. Case 4: Alice’s measurement outcome is the state jgegei1235 or the state jegegi1235. In this case, the state of atoms 4 and 6 collapses to jwIV i46 ¼ ajgei46 þ bjegi46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajei6 þ bjgi6 Þ 2 2 1 pffiffiffi ðjei4 jgi4 Þðajei6 bjgi6 Þ . ð14Þ 2 If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ via his measurement, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6. Otherwise, Charlie needs to perform a rz operation on the atom 6. Case 5: Alice’s measurement outcome is the state jgggei1235 or the state jeeegi1235. In this case, the state of atoms 4 and 6 collapses to jwV i46 ¼ ajegi46 bjgei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 jgi4 Þðajgi6 þ bjei6 Þ 2 2 1 þ pffiffiffi ðjei4 þ jgi4 Þðajgi6 bjei6 Þ . ð15Þ 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. Case 6: Alice’s measurement outcome is the state jggegi1235 or the state jeegei1235. In this case, the state of atoms 4 and 6 collapses to
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jwVI i46 ¼ ajegi46 þ bjgei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 jgi4 Þðajgi6 þ bjei6 Þ 2 2 1 ð16Þ þ pffiffiffi ðjei4 þ jgi4 Þðajgi6 bjei6 Þ . 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. Case 7: Alice’s measurement outcome is the state jegggi1235 or the state jgeeei1235. In this case, the state of atoms 4 and 6 collapses to jwVII i46 ¼ ajggi46 bjeei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajgi6 bjei6 Þ 2 2 1 pffiffiffi ðjei4 jgi4 Þðajgi6 þ bjei6 Þ . ð17Þ 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. Case 8: Alice’s measurement outcome is the state jegeei1235 or the state jgeggi1235. In this case, the state of atoms 4 and 6 collapses to jwVIII i46 ¼ ajggi46 þ bjeei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajgi6 þ bjei6 Þ 2 2 1 pffiffiffi ðjei4 jgi4 Þðajgi6 bjei6 Þ . ð18Þ 2 If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. So far I have explicitly presented the improved version of Zhang et al.’s scheme for secret sharing of quantum information via entangled swapping in cavity QED. In contrast to the original scheme, the distinct advantages of the improved version are: (1) the decoherence problem arising from the atom transmissions in the original scheme [18] is avoided by introducing the prior distributions of maximally entangled atoms; (2) the original probabilistic
(6.25%) scheme has been improved to be a deterministic (100%) one by considering Alice’s other measurement outcomes. Moreover, in Zhang et al.’s paper [18] a generalization to multiparty quantum secret sharing scheme is presented in Section 3. Hence, the present method can be directly applied to the multi-party case also. Here, I will not state it anymore. Acknowledgements I am grateful to the anonymous referees for their detailed suggestions. This work is supported by the National Natural Science Foundation of China under Grant No. 10304022. References [1] B. Schneier, Applied Cryptography, Wiley, New York, 1996, p. 70. [2] J. Gruska, Foundations of Computing, Thomson Computer Press, London, 1997, p. 504. [3] M. Hillery, V. Buzk, A. Berthiaume, Phys. Rev. A 59 (1999) 1829. [4] Y.M. Li, K.S. Zhang, K.C. Peng, Phys. Lett. A 324 (2004) 420. [5] D. Gottesman, Phys. Rev. A 61 (1999) 042311. [6] W. Tittel, H. Zbinden, N. Gisin, Phys. Rev. A 63 (2001) 042301. [7] V. Karimipour, A. Bahraminasab, Phys. Rev. A 65 (2000) 042320. [8] H.F. Chau, Phys. Rev. A 66 (2002) 060302. [9] S. Bagherinezhad, V. Karimipour, Phys. Rev. A 67 (2003) 044302. [10] G.P. Guo, G.C. Guo, Phys. Lett. A 310 (2003) 247. [11] L. Xiao, G.L. Long, F.G. Deng, J.W. Pan, Phys. Rev. A 69 (2004) 052307. [12] Z.J. Zhang, Phys. Lett. A 342 (2005) 60. [13] Z.J. Zhang, Z.X. Man, Phys. Rev. A72 (2005) 022303. [14] R. Cleve, D. Gottesman, H.K. Lo, Phys. Rev. Lett. 83 (1999) 648. [15] S. Bandyopadhyay, Phys. Rev. A 62 (2000) 012308. [16] L.Y. Hsu, Phys. Rev. A 68 (2003) 022306. [17] A.M. Lance, T. Symul, W.P. Bowen, B.C. Sanders, P.K. Lam, Phys. Rev. Lett. 92 (2004) 177903. [18] Y.Q. Zhang, X.R. Jin, S. Zhang, Phys. Lett. A 341 (2005) 380. [19] Z.J. Zhang, J. Yang, Z.X. Man, Y. Li, Eur. Phys. J. D 33 (2005) 133. [20] Z.J. Zhang, Z.X. Man, Phys. Lett. A 341 (2005) 50. [21] Z.J. Zhang, Y. Li, Z.X. Man, Phys. Rev. A 71 (2005) 044301. [22] F.G. Deng, G.L. Long, X.S. Liu, Phys. Rev. A 68 (2003) 042317. [23] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. [24] S.J. van Enk, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 78 (1997) 4293. [25] H.-J. Briegel, W. Dur, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 81 (1998) 5932. [26] W. Dur, H.-J. Briegel, J.I. Cirac, P. Zoller, Phys. Rev. A 59 (1999) 169. [27] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wotters, Phys. Rev. Lett. 70 (1993) 1895. [28] H.K. Lo, H.F. Chau, Science 283 (1999) 2050. [29] S.B. Zheng, G.C. Guo, Phys. Rev. Lett. 85 (2000) 2392. [30] S.B. Zheng, Phys. Rev. A 68 (2003) 035801.
Multiparty secret sharing of quantum information via cavity QED Zhan-jun Zhang School of Physics and Material Science, Anhui University, Hefei, Anhui 230039, China Received 6 July 2005; received in revised form 23 November 2005; accepted 25 November 2005
Abstract The scheme on multiparty secret sharing of an atomic quantum state information via entanglement swapping in cavity QED [Y.Q. Zhang, X.R. Jin, S. Zhang, Phys. Lett. A 341 (2005) 380] is revisited and accordingly an improved version is proposed. The possible decoherence effect in the original version can be avoided after revision of assuming the prior distribution of entanglement. Moreover, the success probability of teleportation has been raised from 6.25% in the original version to 100% in the present version. Ó 2005 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 03.65.w; 42.50.Dv Keywords: Multiparty secret sharing; Quantum information; Entanglement swapping; Cavity QED
The principles of quantum mechanics supplied many interesting application in the field of information in the last decade. Quantum secret sharing (QSS) is an important branch of quantum information. It is a generalization of classical secret sharing [1,2] to a quantum scenario [3]. QSS is likely to play a key role in protecting secret quantum information, e.g., in secure operations of distributed quantum computation, sharing difficult-to-construct ancilla states and joint sharing of quantum money, and so on [4]. Hence, after the pioneering work presented by Hillery, Buzek and Berthiaume in 1999 by using three-particle and four-particle GHZ states [3], many attentions have been concentrated on the realization of QSS in both theoretical and experimental aspects [4–21]. All these works [3–21] can be divided into two kinds, one only deals with the QSS of classical messages (i.e., bits) [5–13], or only deals with the QSS of quantum information [4,14–20] where the secret is an arbitrary unknown state in a qubit; and the other [3,21] studies both, that is, deals with QSS of classical messages and QSS of quantum information simultaneously. In this paper, I will revisit a recently pro-
posed scheme on multiparty secret sharing of an atomic quantum state information via entanglement swapping in cavity QED. I will point out two disadvantages of the scheme. By overcoming the two disadvantages, I will propose an improved version of the scheme. Let us first consider a three-party quantum information secret sharing scheme. Suppose a three-party system consists of Alice, Bob and Charlie. Alice has a secret quantum information jui1 ¼ ajei1 þ bjgi1 ;
0030-4018/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.11.060
2
where a and b are unknown coefficients and jaj + jbj = 1, jei and jgi are atomic excited and ground states, respectively. Alice can assign either Bob or Charlie to construct the unknown state at her will. The designee can certainly reconstruct the unknown state only if the other potential designee collaborates with him. To achieve the above goal, Zhang et al. [18] proposed their three-party scheme via entanglement swapping in cavity QED. In their scheme, Alice has other 5 two-level atoms besides the atom 1. The joint state of the six atoms is jwi123456 ¼ jui1 jni234 jvi56 ;
E-mail address: [email protected].
ð1Þ 2
where
ð2Þ
200
Z.-j. Zhang / Optics Communications 261 (2006) 199–202
1 jni234 ¼ pffiffiffi ðjeeei234 þ jgggi234 Þ; ð3Þ 2 1 jvi56 ¼ pffiffiffi ðjeei56 þ jggi56 Þ. ð4Þ 2 Alice introduces two identical single-mode cavities and sends simultaneously the atoms 1, 2 and the atoms 3, 5 into the two cavities, respectively. The two atoms interact with the single-mode cavity and are driven by a classical field. In the strong driving regime (i.e., X d and X n) and in the case d n, where X is the Rabi frequency, d is the deturning between the atomic transition frequency and the cavity frequency, and n is the atom-cavity coupling strength, there is no energy exchange between the atomic system and the cavity. Consequently, in the interaction picture, the effective Hamiltonian reads [29,30], " # 2 2 X f X þ þ þ He ¼ ðjgill hgj þ jeill hejÞ þ ðrl rk þ rl rk þ hcÞ ; 2 l¼1 l;k¼1;l6¼k ð5Þ n2 . 2d
where f ¼ Since the effective Hamiltonian is independent of the cavity field state, the cavity field is also permitted in a thermal state. In the above case, the evolution operator of the system reads [29,30] ! 2 X iH 0 t iH e t þ U ðtÞ ¼ e e exp iXt rl rl eiH e t . ð6Þ l¼1
By selecting the interaction time satisfy ft = p/4 and making the Rabi frequency satisfy Xt = p, the state of the system evolves to 1 jw0 i123456 ¼ jeeeei1235 ðajeei46 bjggi46 Þ 4 15 X þ ci j/i i1235 j/i i46 ; ð7Þ i¼1
where 1235h/ij/ji1235 = dij and 1235h/ijeeeei1235 = 0. After the interactions, Alice sends the atoms 4 and 6 to Bob and Charlie, respectively. Here, we stress that in Zhang et al.’s scheme only after interactions the atoms 4 and 6 are sent. After confirming that Bob and Charlie each has received an atom, Alice measures the atoms 1, 2, 3 and 5 and publishes her measurement outcomes. If the outcomes are jeeeei1235, then the state of atoms 4 and 6 collapses to jwI i46 ¼ ajeei46 bjggi46 . ð8Þ The subscript I will appear clear in what follows. This equation can be rewritten as jwI i46 ¼ ajeei46 bjggi46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajei6 bjgi6 Þ 2 2 1 þ pffiffiffi ðjei4 jgi4 Þðajei6 þ bjgi6 Þ 2 1 1 ¼ pffiffiffi pffiffiffi ðjei6 þ jgi6 Þðajei4 bjgi4 Þ 2 2 1 þ pffiffiffi ðjei6 jgi6 Þðajei4 þ bjgi4 Þ . ð9Þ 2
If Alice assigns Charlie to recover the unknown state, then Bob measures the atom 4 in the basis fp1ffiffi2 ðjei þ jgiÞ; p1ffiffi ðjei jgiÞg. If Bob collaborates with Charlie, then con2 ditioned on Bob’s measurement result Charlie can recover the unknown state with 100% certainty. That is, if Bob obtains fp1ffiffi2 ðjei4 jgi4 Þg, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6; if Bob obtains fp1ffiffi2 ðjei4 þ jgi4 Þg, conditioned Bob’s result, Charlie needs to perform a rz operation to transform the state ajei6 bjgi6 into ajei6 + bjgi6. Thus far, Zhang et al.’s scheme of three-party secret sharing of quantum information [18] has been briefly reviewed. Note that the success probability of Zhang et al.’s scheme is only 1/16. Incidentally, the security of this scheme is confirmed [18]. In Zhang et al.’s scheme, there are two disadvantages: (1) As mentioned before, only after the atoms interact with the cavities, the atoms 4 and 6 are sent to Bob and Charlie, respectively. In this case, during the transmission it is quite possible the decoherence happens due to the environment noises, i.e., the entanglement between the atom 4 and other atoms might change or even lose during the transmissions. This situation might also happen for the atom 6. If so, though Alice performs measurements on the atoms 1, 2, 3 and 5, the state of the atoms 4 and 6 will not collapse to the state jwi46 = ajeei46 bjggi46 even if the state jeeeei1235 is obtained by Alice. In this case, obviously, the secret sharing of the quantum information in Zhang et al.’s scheme [18] cannot be achieved anymore. More serious is that the unknown state which needs to be teleported has already been destroyed and thus the sharing of this unknown state completely fails. (2) As stressed before, the success probability of Zhang et al.’s scheme is only 1/16. With probability of 15/16 the secret sharing is not achieved and the unknown state is destroyed. This is not a satisfactory result. To overcome the above two disadvantages, I improve Zhang et al.’s scheme [18] as follows. Instead of transmitting the atoms 4 and 6 after the atoms 1, 2, 3 and 5 interact with the two cavities, we suppose the atom distributions are achieved in advance. That is, Alice and Bob have securely shared in advance the 3-atom maximally entangled state jwi234 ¼ p1ffiffi2 ðjeeei234 þ jgggi234 Þ. Alice has the atoms 2 and 3 in her site and Bob has the atom 4 in his site. Similarly, Alice and Charlie have securely shared in advance the 2atom maximal entangled state jwi56 ¼ p1ffiffi2 ðjeei56 þ jggi56 Þ. Alice has the atom 5 in her site and Charlie has the atom 6 in his site. In fact, so far a highly efficient procedure involving a sequence of entanglement purification and teleportation has been devised that allows the reliable sharing of such maximal entangled states between two arbitrarily distant locations [23–28]. Moreover, the generally used twomeasuring-basis method [22] can be employed to detect whether the atoms are attacked by eavesdropper during the atom distributions. Hence, it is reasonable to assume the secure sharing of the maximally entangled atom states
Z.-j. Zhang / Optics Communications 261 (2006) 199–202
between arbitrarily distant parties. If assumed, then the decoherence arising from the transmissions in Zhang et al.’ scheme [18] can be avoided. This ensures the success of the improved scheme. After the atom–cavity interactions with the given times, the state of the whole atom system evolves to 1 jwi ¼ ½jeeeei1235 ðajeei46 bjggi46 Þ 4 þ jeeegi1235 ðajegi46 bjgei46 Þ þ jeegei1235 ðajegi46 þ bjgei46 Þ þ jeeggi1235 ðajeei46 þ bjggi46 Þ þ jegeei1235 ðajggi46 þ bjgei46 Þ þ jegegi1235 ðajgei46 þ bjegi46 Þ þ jeggei1235 ðajgei46 bjegi46 Þ þ jegggi1235 ðajggi46 bjeei46 Þ þ jegggi1235 ðajggi46 bjeei46 Þ þ jgeegi1235 ðajgei46 bjegi46 Þ þ jgegei1235 ðajgei46 þ bjegi46 Þ þ jgeggi1235 ðajggi46 þ bjeei46 Þ þ jggeei1235 ðajeei46 þ bjggi46 Þ þ jggegi1235 ðajegi46 þ bjgei46 Þ þ jgggei1235 ðajegi46 bjgei46 Þ þ jggggi1235 ðajeei46 bjggi46 Þ.
ð10Þ
Alice measures the atoms 1, 2, 3 and 5 and publishes her measurement outcomes. In Zhang et al.’s scheme [18], the authors considered only the case that the measurement outcome is the state jeeeei1235 and therefore the success probability of their scheme is 1/16. Now in this paper I will further consider other measurement outcome cases. Incidentally, without loss of generality, in the following I will assume Charlie is assigned to recover the unknown state with help nof Bob and Bob always measures the atom 4 in o the basis p1ffiffi2 ðjei þ jgiÞ; p1ffiffi2 ðjei jgiÞ . Case 1: If Alice’s measurement outcome is the state jeeeei1235, the state of atoms 4 and 6 also collapses to jwIi46 = ajeei46 bjggi46. As shown by Zhang et al. [18], it is natural that conditioned on Alice’s public announcements Charlie can recover the unknown state with the help of Bob. However, one can easily find that if the measurement outcome is the state jggggi1235, then the state of atoms 4 and 6 also collapses to jwIi46. In this case, obviously, conditioned on Alice’s public announcements Charlie can also recover the unknown state with the help of Bob. Case 2: Alice’s measurement outcome is the state jeeggi1235 or the state jggeei1235. In this case, the state of atoms 4 and 6 collapses to jwII i46 ¼ ajeei46 þ bjggi46 . It can be rewritten as
ð11Þ
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1 1 jwII i46 ¼ pffiffiffi pffiffiffi ðjei4 jgi4 Þðajei6 bjgi6 Þ 2 2 1 þ pffiffiffi ðjei4 þ jgi4 Þðajei6 þ bjgi6 Þ . ð12Þ 2 Bob measures the atom 4. If Bob collaborates with Charlie, then conditioned on Bob’s measurement result Charlie can recover the unknown state with 100% certainty. That is, if Bob obtains fp1ffiffi2:ðjei4 þ jgi4 Þg, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6; if Bob obtains p1ffiffi2 ðjei4 jgi4 Þ, Charlie needs to perform a rz operation to transform the state ajei6 bjgi6 into ajei6 + bjgi6. Case 3: Alice’s measurement outcome is the state jeggei1235 or the state jgeegi1235. In this case, the state of atoms 4 and 6 collapses to jwIII i46 ¼ ajgei46 bjegi46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajei6 bjgi6 Þ 2 2 1 ð13Þ pffiffiffi ðjei4 jgi4 Þðajei6 þ bjgi6 Þ . 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6. Otherwise, Charlie needs to perform a rz operation on the atom 6. Case 4: Alice’s measurement outcome is the state jgegei1235 or the state jegegi1235. In this case, the state of atoms 4 and 6 collapses to jwIV i46 ¼ ajgei46 þ bjegi46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajei6 þ bjgi6 Þ 2 2 1 pffiffiffi ðjei4 jgi4 Þðajei6 bjgi6 Þ . ð14Þ 2 If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ via his measurement, conditioned Bob’s result Charlie knows he has directly obtained the unknown state in the atom 6. Otherwise, Charlie needs to perform a rz operation on the atom 6. Case 5: Alice’s measurement outcome is the state jgggei1235 or the state jeeegi1235. In this case, the state of atoms 4 and 6 collapses to jwV i46 ¼ ajegi46 bjgei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 jgi4 Þðajgi6 þ bjei6 Þ 2 2 1 þ pffiffiffi ðjei4 þ jgi4 Þðajgi6 bjei6 Þ . ð15Þ 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. Case 6: Alice’s measurement outcome is the state jggegi1235 or the state jeegei1235. In this case, the state of atoms 4 and 6 collapses to
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jwVI i46 ¼ ajegi46 þ bjgei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 jgi4 Þðajgi6 þ bjei6 Þ 2 2 1 ð16Þ þ pffiffiffi ðjei4 þ jgi4 Þðajgi6 bjei6 Þ . 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. Case 7: Alice’s measurement outcome is the state jegggi1235 or the state jgeeei1235. In this case, the state of atoms 4 and 6 collapses to jwVII i46 ¼ ajggi46 bjeei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajgi6 bjei6 Þ 2 2 1 pffiffiffi ðjei4 jgi4 Þðajgi6 þ bjei6 Þ . ð17Þ 2 If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. Case 8: Alice’s measurement outcome is the state jegeei1235 or the state jgeggi1235. In this case, the state of atoms 4 and 6 collapses to jwVIII i46 ¼ ajggi46 þ bjeei46 1 1 ¼ pffiffiffi pffiffiffi ðjei4 þ jgi4 Þðajgi6 þ bjei6 Þ 2 2 1 pffiffiffi ðjei4 jgi4 Þðajgi6 bjei6 Þ . ð18Þ 2 If Bob obtains p1ffiffi2 ðjei4 þ jgi4 Þ via his measurement, conditioned Bob’s result Charlie Charlie needs to perform a rx operation on the atom 6. If Bob obtains p1ffiffi2 ðjei4 jgi4 Þ, conditioned Bob’s result Charlie needs to perform the operations rxrz on the atom 6. So far I have explicitly presented the improved version of Zhang et al.’s scheme for secret sharing of quantum information via entangled swapping in cavity QED. In contrast to the original scheme, the distinct advantages of the improved version are: (1) the decoherence problem arising from the atom transmissions in the original scheme [18] is avoided by introducing the prior distributions of maximally entangled atoms; (2) the original probabilistic
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