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Procedia Engineering
Procedia Engineering 00 (2011)29000–000 Procedia Engineering (2012) 568 – 572 www.elsevier.com/locate/procedia
2012 International Workshop on Information and Electronics Engineering (IWIEE)
High-Capacity and High-Efficient Deterministic Secure Quantum Communication Based on Four-Qubit W State Guo ZHAOa∗, Dong WANGa,b, Zhe-Ran ZHUa, Xiao-Ming MAOa a
School of Communication and Information Engineering, Xi’an University of Posts and Telecommunications, Xi’an 710061, China b School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710061, China
Abstract A novel high-capacity and high-efficient deterministic secure quantum communication protocol with four-qubit W states is proposed. By utilizing four kinds of unitary operations, the two legitimate users can directly transmit the secret messages based on the Bell measurements and some additional classical information. Analysis shows that our protocol has a high capacity as each W state can carry two bits of secret information, and has a high intrinsic efficiency because almost all the instances are useful. Moreover, this protocol is unconditionally secure and feasible with present-day technique.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology Open access under CC BY-NC-ND license. Keywords: deterministic secret quantum communication; four-qubit W state; Bell measurements;
1. Introduction The quantum key distribution (QKD) is an ingenious application of quantum mechanics, in which two remote legitimate users, say Alice and Bob, can establish a shared secret key through the transmission of quantum signals. Its ultimate advantage is the unconditional security, the feat in cryptography. Hence, after Bennett and Brassard’s pioneering work published in 1984, [1] much attention has been focused on QKD. [2-10] Another class of quantum communication used to transmit secret messages is called as deterministic secure quantum communication (DSQC). It is different from QSDC, and in DSQC the receiver can read out the secret message only after exchanging at least one bit of classical information from the sender for ∗ Corresponding author. Tel.: +86-18202907635. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2012.01.005
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each qubit. Since Beige et al. [11] put forward the pioneering work in 2002, DSQC has been actively pursued by some groups.[12-16] As a matter fact, these works can be divided into two categories: one utilizes the single photons [11, 15], another adopts the entangled state, such as the EPR pairs [12, 13], the GHZ states [14], W states [16], etc. Similar to QSDC, another class of quantum cryptography is called deterministic secure quantum communication (DSQC).[17-21] 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 been actively pursued by some groups. 2. DSQC protocol with four-qubit W state Now, let us describe the details of our DSQC protocol. Here, we only consider the ideal condition, i.e., there is no noise and losses in the quantum channel. Suppose there are two legitimate anticipators, Alice and Bob. Alice is the sender, and Alice wants to transmit N two-bit secret classical message to Bob, which can be achieved with the following seven steps. • Step 1 Preparing a quartet sequence P Alice produces a sequence of N ordered quartets of entangled particles P. Each quartet is in the fourqubit W state, which is defined as 1 ( 0001 + 0010 + 0100 + 1000 )A A B B 2
W = 4
1 2 1 2
(1)
Where the subscript A1 , A2 , B1 and B2 represent the four particles of the W 4 state. We denote the ordered n quartets in the sequence P with { p1 ( A1 , A2 ,B1 ,B2 ), p2 ( A1 , A2 ,B1 ,B2 ) ...pn ( A1 , A2 ,B1 ,B2 )} where the subscripts 1,2,...,n indicate the order of each particle quartet 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 A2 and B1 of each quartet in the sequence P to encode her secret messages {00,01,10,11}, where U 00 =⊗ I A I B ,U 01 = σ x A ⊗ I B ,U 10 = iσ y A ⊗ σ z B ,U 11 = σ z A ⊗ σ zB 2
1
2
2
2
1
2
1
(2)
and ⎛1 0⎞ ⎛0 1⎞ ⎛ 0 1⎞ ⎛1 0 ⎞ I =⎜ ⎟ σx = ⎜ ⎟ −iσ y = ⎜ ⎟ δz = ⎜ ⎟ ⎝0 1⎠ ; ⎝1 0⎠ , ⎝ −1 0 ⎠ , ⎝ 0 −1 ⎠
(3)
The operation U ij ( i, j ∈ { 0,1} ) will transform the state W 4 into the state Wmn , Where W00 =
1 ( 0001 + 0010 + 0100 + 1000 )A A B B 2
1 2 1 2
1 = [( φ + + φ − )A A ψ + 2 1 2
W01 =
B1B2
+ ψ+
A1 A2
( φ + + φ − )B B ] 1 2
(4)
1 ( 0101 + 0110 + 0000 + 1100 )A A B B 2
1 2 1 2
1 =[( ψ + + ψ − )A A ψ + 2 1 2
B1B2
+ φ+
A1 A2
( φ + + φ − )B B ] 1 2
(5)
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W10 =
1 ( 0101 − 0110 − 0000 + 1100 )A A B B 2
1 2 1 2
1 =[( ψ + + ψ − )A A ψ − 2 1 2
W11=
B1B2
− φ−
A1 A2
( φ + + φ − )B B ] 1 2
(6)
1 ( 0001 − 0010 − 0100 + 1000 )A A B B 2
1 2 1 2
1 = [( φ + + φ − )A A ψ − 2 1 2
Here φ , ψ ±
3
±
B1B2
− ψ−
A1 A2
( φ + + φ − )B B ] 1 2
(7)
are the four Bell states, which are defined as the follows, φ± =
1 1 ( 00 ± 11 ), ψ ± = ( 01 ± 10 ) 2 2
(8)
• Step 3 Dividing the particles in the sequence. Alice takes the particles A1 and A2 from each quartet in the sequence P to form an ordered particle pair sequence { p1 ( A1 , A2 ), p2 ( A1 , A2 ),..., pn ( A1 , A2 )} , named the p A sequence. The remaining partner particle pair { p1 ( B1 ,B2 ), p2 ( B1 ,B2 ),..., pn ( B1 ,B2 )} is called the pB sequence. • Step 4 Preparing the checking single photons. Before sending the pB sequence to Bob, Alice has to add some decoy particles in it. The purpose of this step is to guard for eavesdropping in the transmission of the pB sequence. Alice prepares the non= ± orthogonal decoy particles each randomly in one of the four states { 0 , 1 , + , − } , here
1 ( 0 ± 1 ). 2
Then she randomly inserts the decoy particles into the pB sequence. Thus, a new sequence p'B 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 Transmitting the p'B sequence. After Alice added decoy particles into the pB sequence, she sends the p'B sequence to Bob, and keeps the p A sequence in her hand. • Step 6 Checking eavesdropping of the quantum channel. After confirming Bob has received the p'B sequence, Alice announces publicly the positions and the states of the decoy photons. Then Bob performs a suitable measurement on each decoy photon with 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'B sequence. If the error rate exceeds the threshold, they abort this communication and repeat the procedures from the beginning. Otherwise, they continue to the next step. Alice exposes the secret transmitted order of the p'B sequence. According to this information Bob can adjust the disturbed sequence p'B to the original sequence pB . • Step 7 Decoding secret information Alice performs Bell measurements on her particle pairs (i.e., the particles A1 and A2 ) in the p A sequence. Bob performs Bell-basis measurements on the partner particle pairs (i.e., the particles B1 and B2 ) in the pB sequence, respectively. Subsequently, Alice publishes her measurement results of the particles A1 and A2 in the p A sequence. Thus Bob can obtain Alice’s secret message by comparing his measurement result with Alice’s measurement result according to Eqs. (4)~(7). Table 1 shows the joint correlations of the results for measurements made by Alice for particles A1 and A2 , Bob for particles B1 and B2 in all the possible cases of quantum communication in our DSQC protocol.
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Table 1. Corresponding relations between Alice’s measurement results and Bob’s measurement results and the secret messages in the presented DSQC protocol.
Alice’s measurements φ or φ +
Bob’s measurement ψ
−
+
ψ + or ψ −
ψ+
ψ + or ψ −
ψ−
φ + or φ − ) ψ
ψ−
+
φ or φ −
φ+
φ + or φ −
φ
−
φ + or φ −
ψ−
φ + or φ −
+
the secret messages 00 01 10 11 00 01 10 11
3. Security and Efficiency analysis Security is an important issue of quantum secure communication, especially to DSQC or QSDC protocol. To gain some useful information about the secret message, Eve must attack the quantum channel during the transmitting process. She may use some types of man-in-the-middle attack strategy, such as (1) Measure-resend attack: Eve measures the qubits emerging from Alice and then resends them to Bob. (2) Entangle-measure attack: Eve entangles her ancilla with the particle pair pi ( B1 ,B2 ) ( i ∈ { 1,2,...,n } ) before pi ( B1 ,B2 ) reaches Bob. After Bob measures his particle pair pi ( B1 ,B2 ) , Eve does so with her ancilla and deduce Bob’s measurement result. Unfortunately, all of the above types of attack can be forbidden by the decoy-particle checking procedure explained in the preceding section, The step 6, in our protocol, each decoy photon is prepared randomly in one of the four states { 0 , 1 , + , − } , and is distributed in the sequence pB randomly. That is to say, the states and the positions of the decoy photons are unknown for Eve. Hence, Once Eve manipulates the particle pairs in the sequence pB , she will inevitably disturb these decoy photons and be easily detected by the authorized users. 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 legitimate users Alice and Bob. That is to say, the states and the positions of the single-photon states are unknown for Eve. Hence, any eavesdropping done by Eve will inevitably disturb these particles and be detected by the legitimate users. Following, we investigate the intrinsic efficiency and the total efficiency of our DSQC protocol, respectively. The definition of intrinsic efficiency of a quantum communication scheme is [22] qu qt
(9)
bs qt + bt
(10)
ηq =
Where qu is the number of useful qubits in the quantum communication and qt is the number of total qubits used (not the ones transmitted). Next, we calculate the total efficiency of our DSQC protocol. Let’s employ the definition in Ref. [34] to the total efficiency of a quantum communication: ηt =
Where bs and bt are the numbers of message transmitted and the classical bits exchanged, respectively. In the present DSQC protocol, the legitimate communicators need two bits of classical information and four bits of quantum information to communicate two bits of secret message, that is, bs = 2, qt = 4 and bt = 2. Thus, it’s total efficiency is our ηt = 2/(4+2) = 1/3=33.3% in theory.
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4. Conclusion In summary, we have presented an efficient DSQC protocol based on the four-qubit W states and single-photon eavesdropping check. In this protocol, the two legitimate users exploit the four local unitary operations I ,δ x ,iδ y ,δ z , to encode their secret messages, and then decode them by using Bell-basis measurements with help of classical information exchanging. Since the quantum carriers are transmitted only one time, the opportunity of Eve intercepts them is greatly reduced. The security of the present DSQC scheme is assured by utilizing the single-photon to check eavesdropping. Analysis shows it is secure and feasibility. It’s intrinsic efficiency for qubits and source capacity are both high as almost all of the instances are useful and each W state can carry two bits of information. The total efficiency of our protocol is 33.3%, which is higher than that of some existing DSQC protocols. Hence, our protocol is more efficient. Furthermore, since W state has been applied in one-way quantum computer, the communication of our protocol is easily implemented by this kind of quantum computer. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10902083) and the Natural Science Foundation of Shannxi Province (Grant No. 2009JM1007). References [1] C.H. Bennett and G. Brassard, in Proceedings of IEEE International Conference on Computer, Systems and Signal Processing, Bangalore, India, IEEE, New York, 1984, 175–179 [2] A.K. Ekert, Phys.Rev.Lett.67, 661 (1991) [3] C.H. Bennett, G. Brassard, and N.D. Mermin, Phys. Rev.Lett 68:557 (1992) [4] Long, G.L., Liu, X.S.: Phys. Rev. A 65, 032302 (2002) [5] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys.74 145 (2002) [6] F.G. Deng, G.L. Long, Phys. Rev. A 68 042315 (2003) [7] Deng, F.G., Long, G.L.: Phys. Rev. A 68, 042315 (2003) [8] Deng, F.G., Long, G.L., Wang, Y., Xiao, L.: Chin. Phys. Lett. 21 2097 (2004) [9] Z.J. Zhang, Z.X. Man , S.H. Shi, Int. J. Quantum Information 3 555 (2005) [10] W.H. Kye, C.M. Kim, M.S. Kim, Y.J. Park, Phys.Rev. Lett. 95 040501(2005) [11] A. Beige, B. G. Englert, C. Kurtsiefer and H. Weinfurter, Acta Phys. Pol. A 101,357(2002) [12] Z. X. Man, Z. J. Zhang and Y. Li, Chin. Phys. Lett. 22 18 (2005) [13] F. L. Yan and X. Zhang, Euro. Phys. J. B vol. 41 75 (2004) [14] T. Gao, F. L. Yan, and Z. X. Wang, J. Phys. A:Math. Gen., vol. 38, 5761 (2005) [15] X. H. Li, F. G. Deng, C. Y. Li, Y. J. Liang, P. Zhou and H. Y. Zhou, J. Kerean Phys. Soc., vol. 49, 1354 (2006) [16] H. J. Cao and H. S. Song, Chin. Phys. Lett. vol. 23, 290 (2006) [17] Li, X.H., Deng, F.G., Zhou, H.Y.: Phys. Rev. A 74, 054302 (2006) [18] Wang, J., Zhang, Q., Tang, C.J.: Phys. Lett. A 358, 256 (2006) [19] Lee, H., Lim, J., Yang, H.: Phys. Rev. A 73, 042305 (2006) [20] Cao, H.J., Song, H.S.: Chin. Phys. Lett. 23, 290 (2006) [21] Xiu, X.M., Dong, L., Gao, Y.J., Chi, F.: Opt. Commun. 282, 333 (2009) [22] Deng, X.H. Li, C.Y. Li, P. Zhou, and H.Y. Zhou, Physica Scripta 76 25. (2007)
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