Two-Step Efficient Deterministic Secure Quantum Communication Using Three-Qubit W State

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1 Commun. Theor. Phys. 55 (2011) Vol. 55, No. 6, June 15, 2011 Two-Step Efficient Deterministic Secure Quantum Communication Using Three-Qubit W State YUAN Hao ( ), 1, ZHOU Jun ( ), 1,2 ZHANG Gang ( ), 1 WEI Xiang-Fei (ï ì), 1 and LIU Xiang-Yuan ( å ) 1 1 Department of Material and Chemical Engineering, West Anhui University, Lu an 27012, China 2 Department of Material Science and Engineering, University of Science and Technology of China, Hefei 20026, China (Received September 9, 2010) Abstract A two-step deterministic secure quantum communication (DSQC) scheme using blocks of three-qubit W state is proposed. In this scheme, the secret messages can be encoded by employing four two-particle unitary operations and directly decoded by utilizing the corresponding measurements in Bell basis or single-particle basis. Comparing with most previous DSQC protocols, the present scheme has a high total efficiency, which comes up to 50%. Apart from this, it has still the advantages of high capacity as each W state can carry two bits of secret information, and high intrinsic efficiency because almost all the instances are useful. Furthermore, the security of this communication can be ensured by the decoy particle checking technique and the two-step transmitting idea. PACS numbers: 0.67.Dd, 0.67.Hk Key words: deterministic secret quantum communication, three-qubit W states, decoy particle checking technique, two-step transmitting idea 1 Introduction Over the past two decades, the principles of quantum mechanics have been applied in the field of information, which has produced many interesting and important developments in quantum secret communication and quantum computing. Quantum key distribution (QKD) is an important branch of quantum secret communication, in which two remote legitimate users (Alice and Bob) establish a shared secret key through the transmission of quantum signals and thereafter they can use this key to encrypt messages. Its ultimate advantage is the unconditional security, the feat in cryptography. Hence, much attention [1 7] has been focused on QKD after the pioneering work of Bennett and Brassard published in 1984 (referred to as BB84 QKD protocol). [1] In these works, various properties of quantum mechanics, such as no-cloning theorem, uncertainty principle, entanglement, indistinguishability of nonorthogonal states, non-locality, and so on, are used to accomplish QKD tasks. In recent years, a novel and quite different quantum communication, i.e., the quantum secure direct communication (QSDC) has been put proposed and actively pursued by some groups. [8 22] Different from QKD, in a QSDC protocol the secret message can be transmitted directly without first establishing a private key to encrypt it. Due to its instantaneous, QSDC is important in some applications, which has been shown by Boström et al. [8] and Deng et al. [12 1] On the other hand, another class of instantaneous quantum communication is called deterministic secure quantum communication (DSQC). [2] In the framework of DSQC, the receiver can read out the secret message only after the transmission of at least one bit of additional classical information for each qubit, different from QSDC in which the secret message can be read out directly without exchanging any classical information. Comparing with QKD, DSQC can be used to obtain deterministic information, not a randomly binary string. So far, DSQC has progressed quickly and a lot of correlative protocols have been advanced. [2 45] In these schemes, various kinds of entangled states, such as GHZ states, W states, cluster states, single-photon states, and EPR pairs, etc. are used as quantum channel. Just as requiring consuming some class information, the total efficiency of most previous DSQC protocols is usually not high. For example, in Refs. [0, 6 7, 4, 45] it is 20%, in Refs. [24, 8 9, 42] it is 25%, in Ref. [28] is 0%, in Ref. [25] is.%. It should be noted that, the particle arrangement technique proposed in Refs. [29, 2 5] is novel and interesting. However, a large number of class bits are required to recover the positions of the transmitting particles. Hence, the total efficiency of the Refs. [29, 2 5] is also not high. In this paper, based Supported by the Key Project of the Education Department of Anhui Province under Grant No. KJ2010A2, the Talent Project of the Anhui Province for Outstanding Youth under Grant Nos. 2009SQRZ190, 2010SQRL186, 2010SQRL187 and 2011SQRL147, the Natural Science Research Programme of the Education Department of Anhui Province under Grant No. KJ2009B018Z yuanhao@wxc.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 No. 6 Communications in Theoretical Physics 985 on the two-step transmitting idea [4,12 1,15] and the decoy particle checking technique, [46 47] we will propose a new novel DSQC scheme by employing the three-qubit W states. In the presented scheme, the secret messages can be encoded by four two-particle unitary operations and decoded by utilizing some assistant classical information and the corresponding measurement bases. It will be shown that the present scheme has a high capacity as each W state can carry two bits of secret messages, and has a high intrinsic efficiency because almost all the instances are useful. The distinct advantage of the present scheme is that it s total efficiency comes up to 50%, more higher than that in Refs. [24 25, 28 0, 2 9, 42 4, 45]. Furthermore, there existing two-level security checking procedure, which ensures the present scheme be preventible to any type of attack strategy. 2 Two-Step DSQC Using Three-Qubit W State Now, let us describe the details of our DSQC scheme. Suppose there are two remote legitimate communicators, Alice and Bob. Alice wants to transmit n two-bit secret classical messages to Bob, which may be implemented by the following nine-step scheme. Step 1 Preparing a triplet sequence P. Alice prepares a sequence of n ordered triplets of entangled particles P. Each triplet is in the three-qubit W state W, W ( ) ab1b 2, (1) where 0 and 1 are the up and down eigenstates of the σ z, and the subscript a, b 1, and b 2 represent the three particles in one W state. We denote the ordered n triplet in the sequence P with [P 1 (a, b 1, b 2 ), P 2 (a, b 1, b 2 ),..., P n (a, b 1, b 2 )], here the subscript 1, 2,..., n indicate the order of each triplet in the sequence P, respectively. Step 2 Encoding secret information on the sequence P. Alice performs one of the four unitary operations {U 00, U 01, U 10, U 11 } on the particles b 1 and b 2 of each triplet in the sequence P to encode her secret messages {00, 01, 10, 11}, where and U 00 = I I, U 01 = I iσ y, U 10 = σ x I, U 11 = σ x iσ y, (2) I = , σ x = , iσ y = , σ z = () The operation U ij (i, j {0, 1}) will transform the state W into the state W ij, where W 00 ( ) ab1b 2 ( 2 0 a ψ + b1,b a 00 b1b 2 ), (4) W 01 ( ) ab1b 2 ( 2 0 a φ b1b 2 1 a 01 b1b 2 ), (5) W 10 ( ) ab1b 2 ( 2 0 a φ + b1b a 10 b1b 2 ), (6) W 11 ( ) ab1b 2 ( 2 0 a ψ b1b 2 1 a 11 b1b 2 ), (7) here ψ +, ψ, φ +, and φ are the four Bell states, which are defined as follows, ψ ± = 1 ( 0 1 ± 1 0 ), 2 φ ± 2 ( 0 0 ± 1 1 ), (8) they compose of B-basis. Step Dividing the sequence P into three correlated sequences. Alice takes the particle a, b 1, and b 2 from each triplet in the sequence P to form three correlated sequences, say, [P 1 (a), P 2 (a),..., P n (a)], [P 1 (b 1 ), P 2 (b 1 ),..., P n (b 1 )], and [P 1 (b 2 ), P 2 (b 2 ),..., P n (b 2 )], called the sequence P A, P b1, and P b2 respectively. Step 4 Adding some decoy particles into the sequence P b1. Before sending the sequence P b1 to Bob, Alice has to add some decoy particles in it. The purpose of this step is to check for eavesdropping in the transmission of the sequence P b1 subsequently. The detailed process is as follows. Alice prepares k (k n) decoy particles each is randomly in one of the four states { 0, 1, +, }, here + = ( )/ 2 and = ( 0 1 )/ 2 are the up and down eigenstates of the σ x, they compose of X-basis. Then she randomly inserts the k decoy particles into the P b1 sequence. Thus, a new sequence P b1+k is formed. Since the states and the positions of the decoy particles are only known for Alice herself, the eavesdropping done by an eavesdropper will inevitably disturb these decoy particles and will be detected. Step 5 Sending the sequence P b1+k. Alice sends the P b1+k sequence to Bob and keeps the other two sequences in her site. Step 6 Checking the transmitting security of the sequence P b1+k. After confirming Bob has received the P b1+k sequence, Alice announces publicly the positions and the states of the k decoy particles. Then Bob performs a suitable measurement on each decoy particle with

3 986 Communications in Theoretical Physics Vol. 55 the same basis as Alice chose for preparing it. By comparing his measurement results with Alice s announcement, Bob can then evaluate the error rate of the transmission of the P b1+k sequence. If the error rate exceeds the threshold, then Alice and Bob abandon the communication and repeat the procedure from the beginning. Otherwise, they continue to the next step. Step 7 Performing the same operations as that used in the steps (4) (6) on the sequence P b2. That is, Alice prepares and randomly inserts some decoy particles into the sequence P b2 and sends the new sequence to Bob. Then Alice and Bob check the transmitting security of the new sequence. If the error rate exceeds the threshold, then Alice and Bob abandon the communication and repeat the procedure from the beginning. Otherwise, they continue to the next step. Step 8 Merging the two sequences P b1 and P b2 into one sequence. Now, Bob has owned two sequences, i.e., the sequence P b1 and the sequence P b2. He merges them into a new sequence [P 1 (b 1, b 2 ), P 2 (b 1, b 2 ),..., P n (b 1, b 2 )], called sequence P B. Step 9 Deducing secret messages by utilizing the corresponding measurements. Alice performs Z-basis measurements on her particles (i.e., the particle a) in the P A sequence, where Z = { 0, 1 }. If her measurement result is 0 a, then Alice sends classical information 0 to Bob, otherwise 1 is sent. According to Alice s announcement, Bob chooses one of the two bases, i.e., Z Z-basis or B-basis to measure the partner particles b 1 and b 2 in the sequence P B, where Z Z represents a local measurement on each qubit of the particle pairs in Z-basis. That is, if Alice s announcement is 0, then Bob measures the partner particles b 1 and b 2 in B-basis, otherwise, he measures the partner particles in Z Z-basis. Once obtaining the measurement result, Bob can deduce the secret message according to Eqs. (4) to (7). Table 1 shows the joint correlations of the results for measurements made by Alice and Bob and all possible cases of quantum communication in the present DSQC scheme. Table 1 Corresponding relations among Alice s measurement results (AMR), Alice s classical information (ACI), Bob s measurement results (BMR), and the secret messages in the presented DSQC scheme. AMR ACI BMR 0 a 0 ψ + b1 b 2 φ b1 b 2 φ + b1 b 2 ψ b1 b 2 1 a 1 00 b1 b 2 01 b1 b 2 10 b1 b 2 11 b1 b 2 Secret messages To explicitly demonstrate the decoding process in our protocol, let us show an example. Suppose Alice s measurement results are { 0 a, 1 a, 0 a, 0 a, 1 a,...}, then she sends the classical bit sequence {0, 1, 0, 0, 1,...} to Bob. After receiving Alice s announcement, Bob chooses the bases {B, Z Z, B, B, Z Z,...} to measure his partner particles in the sequence P B, respectively. Assumed the measurement results are { ψ b1b 2, 00 b1b 2, φ b1b 2, ψ + b1b 2, 10 b1b 2,...}, then Bob can deduce the secret messages are {11, 00, 01, 00, 10,...} according to Table 1. So far we have expatiated a two-step DSQC scheme with the three-qubit W state. Security Analysis Security is the vital issue of quantum secure communication, especially to DSQC or QSDC protocol. To ensure the security of the communication, the present scheme exploits the decoy particle checking technique and the twostep transmitting idea. The decoy particle checking technique can decide whether a potential eavesdropper exists or not, and the two-step transmitting idea is used to prevent Eve from obtaining secret messages. As mentioned ahead, the particles b 1 and b 2 in each W state are sent from Alice to Bob in two sequences P b1 and P b2 with two steps. The first step is transmitting all particles b 1 in the sequence P b1. After confirming the quantum channel is secure in this stage, Alice sends all particles b 2 in the sequence P b2 to Bob in the second step, and implements the security checking procedure on them. Here, the security checking of the sequences P b1 and P b2 is based on the decoy particle checking technique. That is, each decoy particle is produced randomly in one of the four states { 0, 1, +, }, and distributed in the sequences P b1 and P b2 randomly. No one knows the positions and the states of the decoy particles except for Alice herself. Therefore, any eavesdropping done by Eve will inevitably disturb the states of the decoy particles and ultimately be detected by the two authorized users. As a matter of fact, from the view of an eavesdropper, the procedure of checking eavesdropping done by the two authorized users in the present protocol is essentially the same as that in BB84 QKD scheme. [1] As far as the security checking method are considered, the difference between this protocol and BB84 QKD scheme is that, Bob measures each decoy particle using the same basis as Alice chooses for preparing it and all the decoy particles are transmitted in a quantum data block in our protocol, while in the BB84 QKD protocol, Bob measurements the checking particles in Z- basis or X-basis at random and all checking photons are transmitted in a stream. Therefore, strictly speaking, the checking process in our protocol is more reasonable and efficient than that in BB84 QKD protocol. The BB84 QKD protocol has been proven unconditionally secure by several groups. [48 50] Therefore, the transmission of sequences P b1 and P b2 in this scheme can also be ensured secure by the decoy particle checking technique. That is to say, once eavesdropping the quantum channel, then Eve

4 No. 6 Communications in Theoretical Physics 987 will be detected certainly. Of course, Eve can access the quantum channel but she can only capture one transmitting qubit (b 1 or b 2 ) in each W state. However, as the secret messages are encoded in the whole entangled state, Eve cannot get any useful information if she just gets one qubit of the three-qubit W state. In this way, no secret messages will be leaked to Eve. From the above analysis, one can see that the present scheme is unconditionally safe against any types of attack strategy. 4 Advantages To see the advantages of the present scheme, let us make a comparison with some previous DSQC schemes. Firstly, it has a high capacity as each W state can carry two bits of secret messages. Secondly, it has a high intrinsic efficiency, and total efficiency, which can be seen from the following calculations. As almost all the quantum source (except for the decoy photons used for eavesdropping check) can be used to carry the secret message, the intrinsic efficiency η [2] for qubits in the present scheme approaches the maximal value 100%. Here η = q u /q t, where q u is the number of useful qubits in the quantum communication and q t is the number of total qubits used. The definition of total efficiency of a quantum communication scheme is [2] ζ = b s /(q t +b t ), where b s and b t are the numbers of secret message transmitted and the classical bits exchanged, respectively. In the present DSQC scheme, the legitimate users need only one bit of classical information and three bits of quantum information (a three-qubit W state) to communicate two bits of secret messages, i.e., b s = 2, q t = 4, and b t = 1. Therefore, the total efficiency of the present scheme is ζ = 2/( + 1) = 50%. In the following Table 2, we list the b s, b t, q t, and ζ of some previous DSQC protocols. It will be helpful to clarify the advantages of the present scheme. Table 2 The different factors (i.e., b s, b t, and q t) and total efficiency ζ of some previous DSQC protocols. the Ref. b s q t b t ζ [24] 1 (an EPR pair and a single particle) 1 25% [25] 2 4 (two EPR pairs) 2.% [28] 6 (two GHZ states) 4 0% [29] (100 EPR pairs) % [0] 1 4 (a four-qubit cluster state) 1 20% [2,] (50 EPR pairs) % [4,5] (100 single particles) % [6,7] 1 4 (a four-qubit W state) 1 20% [8,9] 1 (a three-qubit GHZ state) 1 25% [42] 1 (a three-qubit W state) 1 25% [4] 1 4 (a three-qubit W state and a single particle) 1 20% [45] 1 (an EPR pair and a single particle) 2 20% Note: Due to requiring tracking the positions of transmitting qubits, the ζ of Refs. [29, 2 5] is relative with the number of secret messages transmitted. The more secret messages transmitted, the lower total efficiency of the communication. In this table, the total efficiency is obtained when the number of secret messages transmitted is only 100 bits. Once the b s exceeds 100 bits, then the total efficiency will become even lower. From the above analysis, one can see that the total efficiency of the present scheme is relatively higher than the ones in Refs. [24 25, 28 0, 2 9, 42 4, 45]. As for the controlled DSQC schemes, such as the Refs. [26 27, 40 41], the controller requires costing some classical bits to assist the accomplishment of the communication, and the more controllers, the lower total efficiency of the communication. So we do not discuss them here. 5 Summary To summarize, in this paper, by using the three-qubit W states as quantum channel, we have proposed a new novel DSQC scheme. In such scheme, the secret messages can be encoded by employing four two-particle unitary operations and directly decoded by utilizing the corresponding measurements in Bell basis or single-particle basis. It has a high source capacity as each W state can carry two bits of classical secret messages, and has a high intrinsic efficiency because almost all the instances are useful. Expressly, the total efficiency of the present scheme comes up to 50%, which is more higher than that of some previous DSQC protocols. Furthermore, the decoy particle checking technique and the two-step transmitting idea can ensure the present scheme is unconditionally safe against any types of attack strategy.

5 988 Communications in Theoretical Physics Vol. 55 References [1] C.H. Bennett and G. Brassard, in: Proceedings of the IEEE International Conference on Computers Systems and Signal Processings, Bangalore, India, IEEE, New York (1984) p [2] C.H. Bennett, G. Brassard, and N.D. Mermin, Phys. Rev. Lett. 68 (1992) 557. [] A. Cabello, Phys. Rev. Lett. 85 (2000) 565. [4] G.L. Long and X.S. Liu, Phys. Rev. A 65 (2002) [5] F.G. Deng and G.L. Long, Phys. Rev. A 68 (200) [6] F.G. Deng and G.L. Long, Phys. Rev. A 70 (2004) [7] W.H. Kye, C.M. Kim, M.S. Kim, and Y.J. Park, Phys. Rev. Lett. 95 (2005) [8] K. Boström and T. Felbinger, Phys. Rev. Lett. 89 (2002) [9] A. Wójcik, Phys. Rev. Lett. 90 (200) [10] Q.Y. Cai, Phys. Rev. Lett. 91 (200) [11] Q.Y. Cai and B.W. Li, Phys. Lett. A 69 (2004) [12] F.G. Deng, G.L. Long, and X.S. Liu, Phys. Rev. A 68 (200) [1] F.G. Deng and G.L. Long, Phys. Rev. A 69 (2004) [14] C. Wang, F.G. Deng, and G.L. Long, Opt. Commun. 25 (2005) 15. [15] C. Wang, F.G. Deng, Y.S. Li, X.S. Liu, and G.L. Long, Phys. Rev. A 71 (2005) [16] Q.Y. Cai, Phys. Lett. A 51 (2006) 2. [17] X.H. Li, C.Y. Li, F.G. Deng, P. Zhou, Y.J. Liang, and H.Y. Zhou, Chin. Phys. 16 (2007) [18] G.L. Long, F.G. Deng, C. Wang, X.H. Li, K. Wen, and W.Y. Wang, Front. Phys. China. 2 (2007) 251. [19] W.J. Liu, H.W. Chen, Z.Q. Li, and Z.H. Liu, Chin. Phys. Lett. 25 (2008) 254. [20] X.M. Xiu, L. Dong, Y.J. Gao, and F. Chi, Commun. Theor. Phys. 52 (2009) 60. [21] D. Liu, C.X. Pei, D.X. Quan, and N. Zhao, Chin. Phys. Lett. 27 (2010) [22] F. Gao, S.J. Qin, Q.Y. Wen, and F.C. Zhu, Opt. Commun. 28 (2010) 192. [2] X.H. Li, F.G. Deng, C.Y. Li, Y.J. Liang, P. Zhou, and H.Y. Zhou, J. Korean Phys. Soc. 49 (2006) 154. [24] F.L. Yan and X.Q. Zhang, Eur. Phys. J. B 41 (2004) 75. [25] Z.J. Zhang, Z.X. Man, and Y. Li, Int. J. Quant. Inform. 2 (2004) 521. [26] T. Gao, F.L. Yan, and Z.X. Wang, Int. J. Mod. Phys. C 16 (2005) 129. [27] T. Gao, F.L. Yan, and Z.X. Wang, Chin. Phys. 14 (2005) 089. [28] T. Gao, F.L. Yan, and Z.X. Wang, J. Phys. A 8 (2005) [29] Z.X. Man, Z.J. Zhang, and Y. Li, Chin. Phys. Lett. 22 (2005) 18. [0] G.Y. Wang, X.M. Fang, and X.H. Tan, Chin. Phys. Lett. 2 (2006) [1] J.S. Shaari, M. Lucamarini, and M.R.B. Wahiddin, Phys. Lett. A 58 (2006) 85. [2] A.D. Zhu, Y. Xia, Q.B. Fan, and S. Zhang, Phys. Rev. A 7 (2006) [] X.H. Li, F.G. Deng, and H.Y. Zhou, Phys. Rev. A 74 (2006) [4] J. Wang, Q. Zhang, and C.J. Tang, Phys. Lett. A 58 (2006) 256. [5] J. Wang, Q. Zhang, and C.J. Tang, Phys. Lett. A 68 (2007) 504. [6] H.J. Cao and H.S. Song, Chin. Phys. Lett. 2 (2006) 290. [7] J. Liu, Y.M. Liu, H.J. Cao, S.H. Shi, and Z.J. Zhang, Chin. Phys. Lett. 2 (2006) [8] H. Lee, J. Lim, and H. Yang, Phys. Rev. A 7 (2006) [9] Z.J. Zhang, J. Liu, D. Wang, and S.H. Shi, Phys. Rev. A 75 (2007) [40] X.M. Xiu, L. Dong, Y.J. Gao, and F. Chi, J. Exp. Theor. Phys. 105 (2007) 112. [41] X.M. Xiu, L. Dong, Y.J. Gao, and F. Chi, Chin. Phys. B 17 (2008) 991. [42] L. Dong, X.M. Xiu, Y. Gao, and F. Chi, Commun. Theor. Phys. 50 (2008) 59. [4] L. Dong, X.M. Xiu, Y.J. Gao, and F. Chi, Commun. Theor. Phys. 49 (2008) [44] X.M. Xiu, H.K. Dong, L. Dong, Y.J. Gao, and F. Chi, Opt. Commun. 282 (2009) [45] L. Dong, X.M. Xiu, Y.J. Gao, and F. Chi, Opt. Commun. 282 (2009) [46] C.Y. Li, H.Y. Zhou, Y. Wang, and F.G. Deng, Chin. Phys. Lett. 22 (2005) [47] C.Y. Li, X.H. Li, F.G. Deng, P. Zhou, Y.J. Liang, and H.Y. Zhou, Chin. Phys. Lett. 2 (2006) [48] H.K. Lo and H.F. Chau, Science 28 (1999) [49] P.W. Shor and J. Preskill, Phys. Rev. Lett. 85 (2000) 411. [50] N. Lütkenhaus, Phys. Rev. A 61 (2000)