- Email: [email protected]
New deterministic quantum communication via symmetric W state
Optics Communications 283 (2010) 4397–4400
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
New deterministic quantum communication via symmetric W state Chia-Wei Tsai, Tzonelih Hwang ⁎ National Cheng-Kung University, Department of Computer Science and Information Engineering, No. 1, Ta-Hsueh Road, Tainan City 701, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 19 March 2010 Received in revised form 10 June 2010 Accepted 10 June 2010 Keywords: Quantum secure direct communication Dense coding Symmetric W state Decoy photon
a b s t r a c t In 2006, Xue et al. [Chin. Phys. 15, 1421] proposed a dense coding on the symmetric W state j W〉 = p1ffiffi3 ð j 001〉 + j 010〉 + j 100〉Þ to establish a deterministic quantum communication (DQC) protocol. In the dense coding, however, the encoded message can be recovered with a 67% probability. Therefore, within one execution of their protocol, the sender can only successfully transmit 67% of information to the receiver. The same protocol has to be repeated several times before a message is fully transmitted. The purpose of this study is to propose a novel coding function of the symmetric W state for constructing a DQC protocol which can fully transmit the message within one execution of DQC. Furthermore, the security of the protocol is analyzed to show that any eavesdropper will be detected with a very high probability if he/she attempts to steal any information. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Unlike a quantum key distribution protocol (QKDP) [1], which allows the participants to share a secret key without pre-sharing any private information, a quantum direct communication (QDC) protocol [2] or a deterministic quantum communication (DQC) protocol [3] allows two participants to directly communicate messages between each other in the same condition as the QKDP. Many related studies have been proposed: some are based on maximally entangled states [5–11]; some use single photons [4,12,13]; and the others are based on W state [14–17]. In addition, Xu et al. [18] further considered the practical network environment and proposed multi-user quantum communication protocol. Furthermore, the technology of continuous variables has been used to reduce the information revealing to an eavesdropper in [19,20]. Recently, Xue et al. proposed a dense coding on the symmetric W state jW〉 = p1ffiffi3 ð j001〉 + j010〉 + j100〉Þ to establish a DQC protocol [14]. Unfortunately, in their dense coding, only 67% of the encoded messages can be correctly recovered. That is, 33% of the messages will be lost, and have to be resent again. Thus, the sender and the receiver have to repeat the same DQC approximately 5 times ((33%)5 ≈ 0) to retransmit the lost information. Although later the four-photon W state jW〉 = 12 ð j0001〉 + j 0010〉 + j0100〉 + j1000〉Þ [15] and the 1 asymmetric W state jWs 〉 = αj 100〉 + βj010〉 + pffiffiffi j001〉 [16,17,21] 2
have been used to establish a DQC protocol without having this problem, the indeterminate situation in the symmetric W state has not yet been solved.
⁎ Corresponding author. E-mail address: [email protected] (T. Hwang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.06.039
This study attempts to propose a new coding function for the symmetric W state to avoid the indeterminate situation in Xue et al.'s DQC and also use it to construct a new QDC with better performance. In the new coding function for the symmetric W state, the sender has to use a unitary operation pair (Um1 ⊗ Um2) to encode messages, where Umi belongs to the set of four Pauli operations {I, σz, σx, iσy}. And the receiver also has to adopt two different strategies to measure the photons. Furthermore, considering the eavesdroppers may exist in the quantum channel, this study uses the technology of decoy photons [22,23] to detect the eavesdroppers and uses the reordering technique [24,25] to ensure that the transmitted messages will not be disclosed to the eavesdroppers. This paper is organized as follows. First, Section 2 briefly reviews the Xue et al.'s dense coding and DQC. Then, Section 3 describes the proposed coding function and the proposed DQC. The security analysis is also given here. Finally, Section 4 summarizes the contributions of this study and gives a conclusion. 2. Xue et al.'s dense coding and DQC Dense coding is an application of quantum entanglement, in which a sender can encode an x-bit classical message by performing unitary operations on y photons of an entangled state, where x N y. For example, a two-bit classical message can be encoded by performing a unitary operation on one photon of EPR pair. Liu et al. [26] proposed a general dense coding scheme for high-dimensional state. Moreover, the dense coding of two-photon has been experimentally established by Mattle et al. [27] and Fang et al. [28]. For three-photon state, Wei et al. [29] also proposed an experiment realization by using nuclear magnetic resonance (NMR). In this section, Xue et al.'s dense coding for the symmetric W state is first reviewed. Then, the DQC based on this dense coding is described.
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2.1. Xue et al.'s dense coding
Table 1 The corresponding unitary operations based on two-bit messages.
This subsection briefly describes Xue et al.'s dense coding and then points out the indeterminate coding situation on their dense coding. A symmetric W state can be shown by j W〉123 = p1ffiffi3 ð j 001〉 + j010〉 + j100〉Þ123 . Let wi denote the i-th photon of |W〉123. The four unitary operations {I, σx, iσy, σz} can be denoted by four twobit messages: “00”, “01”, “10”, and “11”, respectively, where I = |0〉 〈0| + |1〉〈1|; σx = |0〉〈1| + |1〉〈0|; iσy = |0〉〈1| − |1〉〈0|; and σz = |0〉 〈0| − |1〉〈1|. Thus, based on the two bits of a message, the sender first performs a corresponding unitary operation on w 1 . For example, if the two bits are “11”, the unitary operation σz will be performed on the w1. The sender keeps w3 to himself/herself whereas conveys w1 and w2 to the receiver. After the sender completes the encoding process on w1, the state of |W〉123 can be presented in the following four equations, respectively: 1 I⋅ jW〉123 = pffiffiffi ð j001〉 + j010〉 + j100〉Þ123 3
ð1Þ
1 σx ⋅ jW〉123 = pffiffiffi ð j101〉 + j110〉 + j000〉Þ123 3
ð2Þ
1 iσy ⋅ jW〉123 = pffiffiffi ð− j101〉−j110〉 + j000〉Þ123 3
ð3Þ
1 σz ⋅ jW〉123 = pffiffiffi ð j001〉 + j010〉− j100〉Þ123 : 3
ð4Þ
qffiffiffiffiffi pffiffiffi þ þ − = 2=3 jΨ 〉12 j0〉3 + 1 6 jΦ 〉 + jΦ 〉 j1〉3
σx ⋅ jW〉123
12
qffiffiffiffiffi pffiffiffi þ þ − = 2=3 jΦ 〉12 j0〉3 + 1 6 jΨ 〉 + jΨ 〉 j1〉3 12
iσy ⋅ jW〉123 = σz ⋅ jW〉123 =
qffiffiffiffiffi pffiffiffi þ − − 2 =3 jΦ 〉12 j0〉3 + 1 6 j Ψ 〉 + jΨ 〉 j1〉3 12
qffiffiffiffiffi pffiffiffi þ − − 2 =3 jΨ 〉12 j0〉3 + 1 6 jΦ 〉 + jΦ 〉 j1〉3
, where jΦF 〉12 =
12
1 pffiffiffið j00〉Fj11〉Þ12 2
and jΨF 〉12 =
Unitary operation
00
I ⊗ I or σz ⊗ σz
01
I ⊗ σx or σz ⊗ iσy
10
σx ⊗ σz or iσy ⊗ I
11
σx ⊗ iσy or iσy ⊗ σx
Quantum state of |W〉123 qffiffiffi 2 1 F j Ψþ 〉12 j 0〉3 + pffiffiffi j 00〉12 j1〉3 3 qffiffiffi 3 2 1 jΦþ 〉12 j0〉3 Fpffiffiffi j 01〉12 j1〉3 3 q3ffiffiffi 2 1 jΦ− 〉12 j 0〉3 Fpffiffiffi j10〉12 j 1〉3 3 3 qffiffiffi 2 1 F j Ψ− 〉12 j0〉3 −pffiffiffi j 11〉12 j 1〉3 3
3
performed by the sender is σx (see also Eq. (6)); thus, the message sent from Alice is “01”. However, if the measuring result of w3 is |1〉, Bob cannot recover Alice's message because the measuring result of (w1, w2) is indeterminate (i.e., either |Ψ+〉 or |Ψ−〉). Therefore, only 67% of the transmitted a message can be received by a receiver, i.e., 33% of the transmitted message will be lost. Thus, before a message can be sent completely to the receiver, the sender has to execute the same DQC protocol at least 5 times because 1 − (33%)5 ≐ 1. 3. A new coding function for symmetric W state and DQC protocol
In the decoding processes, the sender measures w3 by the computational basis {|0〉,|1〉} (called R-basis for short), and then the receiver performs the Bell measurement on w1 and w2. To better understand the measuring processes, the above equations can be rewritten as follows: I⋅ jW〉123
Two-bit message
ð5Þ ð6Þ ð7Þ ð8Þ
1 pffiffiffið j01〉Fj10〉Þ12 2
are Bell bases. According to the Eqs. (5)–(8), the measurement result of (w1, w2) is a determinate one if the measurement result of w3 is equal to |0〉. Otherwise, it will be indeterminate (i.e., it may be |Φ+〉 or |Φ−〉(|Ψ+〉 or |Ψ−〉)). 2.2. Xue et al.'s DQC protocol Based on the above dense coding, a DQC has been constructed among a sender, Alice, a receiver, Bob, and an assistant, Charlie. Assume that a symmetric W state |W〉123's photons w1, w2, and w3 have been distributed to Alice, Bob, and Charlie, respectively. Alice encodes a two-bit message by performing a corresponding unitary operation (as described in Section 2.1) on w1, and then she sends the encoded w1 to Bob. After Charlie measures w3 with R-basis, Bob can perform a Bell measurement on w1 and w2 to recover Alice's message. However, due to the indeterminate situation described earlier, Bob can only obtain the messages when the measurement result of w3 is equal to |0〉. For example, if the measuring result of (w1, w2) and w3 are |Φ+〉 and |0〉, respectively, Bob knows that the unitary operation
In this section, a new coding function for the symmetric W state is introduced. Furthermore, a DQC protocol is proposed using this coding function and decoy photons. In the proposed DQC protocol, the sender can transmit a secret message to the receiver, and the receiver can immediately recover the message without communicating with the sender again. Finally, the security analysis of this protocol is discussed. 3.1. The new coding function In our proposed coding function, a unitary operation pair Um1 ⊗ Um2 is used to encode a two-bit message, and then two measuring bases, R-basis
{|0〉,|1〉} and Bell-basis j ΦF 〉 = p1ffiffiffið j 00〉F j 11〉Þ, j ΨF 〉 = p1ffiffiffi ð j01〉F j01〉Þg, 2
2
are employed to recover the message. Assume that |Ψ+〉, |Φ+〉, |Φ−〉, and |Ψ−〉 (|00〉,|01〉,|10〉, and |11〉) denote the two-bit messages “00”, “01”, “10”, and “11”, respectively. The encoding and the decoding functions are explained in the following. 3.1.1. Encoding function Alice, the sender, prepares a symmetric W state jW〉123 = p1ffiffi ð j001〉 + j010〉 + j100〉Þ 123 , and then encodes her two-bit mes3 sage by performing the corresponding unitary operations on w1 and w2 based on the Table 1.1 Then, she keeps w3 to herself whereas sends the others to Bob, the receiver. For example, if Alice wants to transmit the “01” to Bob, she will perform the unitary operations I ⊗ σx on w1 and w2. The encoded state of |W〉123 can be shown as follows: 1 I⊗σx ⊗I jW〉123 = pffiffiffi ð j11〉 + j00〉Þ12 j0〉3 + j01〉12 j1〉3 3 rffiffiffi 2 þ 1 jΦ 〉12 j0〉3 + pffiffiffi j01〉12 j1〉3 : = 3 3
ð9Þ
3.1.2. Decoding function After Bob receives the photons, Alice uses the R-basis to measure w3, and then she sends the measuring result r_mA to Bob. If r_mA = |0〉, Bob performs the Bell-basis measurement on w1 and w2; otherwise, he uses the R-basis to measure. Finally, he can extract the two-bit message sent from Alice based on the measuring result r_mB. 1 Table 1 shows the corresponding relationship between the unitary operations and the quantum states based on the two-bit messages.
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Based on the newly proposed coding function on the symmetric W state and the block transmission method which is similar to [2], a DQC can be constructed as follows (see also Fig. 1).
Step 6 Alice measures H with R_basis, and then announces the sequence of measuring results R _ MA = {r _ mA1, r _ mA2, …, r _ mAm} and the permutation of TM to Bob. Step 7 According to the permutation, Bob recovers TM from TM' . Based on R_MA, Bob performs the corresponding measurement on TM. That is, if the r _mAI =|0〉, Bell-basis is used to measure (t1i , ti2); otherwise, the R-basis is used. Finally, Bob can correctly extract the messages sent from Alice based on the sequence of measuring results TM = {r _ mB1,.r _ mB2, …, r _ mBm}.
Step 1 Alice prepares m symmetric W states. Each of them can be denoted as
Obviously, Alice can transmit the messages to Bob within one execution of the protocol.
For example, if r_mA = |1〉, then Bob uses R_basis to measure w1 and w2. If the measuring result is r_mB = |01〉, then Bob can determine that the message sent from Alice is “01” according to the Table 1. 3.2. DQC based on the new coding function
1 jW〉t 1 t 2 hi = pffiffiffi ð j001〉 + j010〉 + j100〉Þt 1 t 2 hi i i i i 3
ð10Þ
, where i = 1 to m. She divides these m W states into two sequences: TM ={(t11, t12), (t12, t22), … , (t1m, t2m)} and H={h1, h2,…, hm}. In addition, she prepares n decoy single photons from {|0〉, |1〉, |+〉, |−〉} as the checking sequence TC ={d1, d2, …, dn} for checking eavesdrop1 1 pings, where j+〉 = pffiffiffið j0〉 + j1〉Þ and j−〉 = pffiffiffið j0〉− j1〉Þ. 2
2
Next, she employs the coding function described in Section 3.1 to encode the 2m-bit messages on (t1i , t2i ), where i=1, 2, …m. Then, ′ (i.e., the order of each she permutes the encoded TM into TM particle of TM is reordered), and mixes TM with TC to get T. Alice keeps H to herself, and sends T to Bob. Step 2 Bob sends an acknowledgment to Alice upon receiving T. Step 3 Alice announces the positions and the corresponding bases of TC to Bob. Step 4 Bob extracts TC from T, and then he measures TC with these corresponding bases. Finally, Bob returns the measuring results R _ MC = {r _ mC1,.r _ mC2, …, r _ mCn} to Alice. Step 5 Alice checks the existence of eavesdropping based on the R_MC. If the detected error rate exceeds a predetermined threshold τ (τ ≐ 2% ∼ 8.9% depending on the channel situation (e.g., distance, etc.)[32–36]), she aborts this communication and restarts the protocol.
3.3. Analysis of eavesdropping This section analyzes two popular attacks, the Intercept-andResend attack and the Entangle-and-Measure attack, against the proposed protocol. (i) Intercept-and-Resend. Attack We first assume that the quantum channel is ideal and that an attacker (called Eve) intercepts s photons sent from Alice, stores them and resends a sequence of forged photons to Bob in hope to pass the eavesdropping checking. Because the decoy photons are randomly inserted in TM by Alice, without knowing their positions and states, each of Eve's fake photons in the checking positions could be detected by the public discussion between Alice and Bob with the 1 probability of (i.e., the eavesdropper can pass that check with 4 3 the probability of ). Assume 2 m = n (i.e., half of Eve's 4 intercepting photons are decoy photons). Eve can be detected s2 3 s in the public discussion with the probability of 1− ≐1, if 4 2 is large. However, even if Eve can pass the eavesdropping check, she may not be able to obtain Alice's message. Because the order of TM is permuted, Eve can obtain useful information only if two intercepted photons (e.g., t1i , t2i ) belong to the corresponding W 1 state (e.g., |W〉t1i ti2hi). The probability of that is 2m + n which is C2
negligible if 2 m+n is large enough. The probability of intercepting
Fig. 1. The proposed DQC.
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s 2
valid photon pairs is
s
3 2 4
×
s = 4−1 1 ∏ 2m + n−2j j = 0 C2
s =4−1 1 ∏ 2m + n−2j j = 0 C2
. Thus, Eve has the probability
s 2
to obtain -bit message. It is obvious that this
probability is very small. If, however, the quantum channel is not ideal, Eve may hide her attack by employing the noise of quantum channel [31]. Suppose that the quantum bit error rate (QBER) caused by the channel noise is τ (≐2%∼8.9%). If the QBER cased by the Eve's attack is smaller than τ, then Eve's attacking behavior may not be detected by a legitimate participant. However, in the proposed protocol, the QBER caused by the attack is about 25% which is larger than τ. Thus, this kind of attack eventually will be detected in the eavesdropper checking process. (ii) Entangle-and-Measure Attack. In this attack, Eve entangles an ancillary photon on each of the traveling photons by using a unitary operation Û (i.e., Uˆ † Uˆ = Uˆ Uˆ † = I), and then measure the ancillary photons to steal Alice's messages. However, similar to the analysis proposed in [30], if she wants to steal the message, the Û operation will disturb the states of decoy photons. Therefore, Eve's attack can be detected by the eavesdropping check between Alice and Bob. 4. Conclusion This paper proposes a new coding function for the symmetric W state. Based on the new encoding function, a DQC protocol is constructed. The message can be transmitted within one execution of DQC, which is rather efficient as compared to Xue et al.'s scheme (also using the symmetric W state) which requires about five-time executions of DQC protocol to complete a message transmission. Moreover, the eavesdroppers would be detected under both the ideal quantum channels and the noisy quantum channels. The security analysis shows that the proposed protocol is a quasi-secure DQC. In particular, the proposed DQC protocol is convenient for experimental implementation because only the symmetric W state, unitary operations, and the projected measurements are used. Acknowledgments The authors would like to thank the anonymous reviewers for their very valuable comments that enhance the clarity of this paper a lot.
We also like to thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC 98-2221-E-006-097-MY3.
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Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
New deterministic quantum communication via symmetric W state Chia-Wei Tsai, Tzonelih Hwang ⁎ National Cheng-Kung University, Department of Computer Science and Information Engineering, No. 1, Ta-Hsueh Road, Tainan City 701, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 19 March 2010 Received in revised form 10 June 2010 Accepted 10 June 2010 Keywords: Quantum secure direct communication Dense coding Symmetric W state Decoy photon
a b s t r a c t In 2006, Xue et al. [Chin. Phys. 15, 1421] proposed a dense coding on the symmetric W state j W〉 = p1ffiffi3 ð j 001〉 + j 010〉 + j 100〉Þ to establish a deterministic quantum communication (DQC) protocol. In the dense coding, however, the encoded message can be recovered with a 67% probability. Therefore, within one execution of their protocol, the sender can only successfully transmit 67% of information to the receiver. The same protocol has to be repeated several times before a message is fully transmitted. The purpose of this study is to propose a novel coding function of the symmetric W state for constructing a DQC protocol which can fully transmit the message within one execution of DQC. Furthermore, the security of the protocol is analyzed to show that any eavesdropper will be detected with a very high probability if he/she attempts to steal any information. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Unlike a quantum key distribution protocol (QKDP) [1], which allows the participants to share a secret key without pre-sharing any private information, a quantum direct communication (QDC) protocol [2] or a deterministic quantum communication (DQC) protocol [3] allows two participants to directly communicate messages between each other in the same condition as the QKDP. Many related studies have been proposed: some are based on maximally entangled states [5–11]; some use single photons [4,12,13]; and the others are based on W state [14–17]. In addition, Xu et al. [18] further considered the practical network environment and proposed multi-user quantum communication protocol. Furthermore, the technology of continuous variables has been used to reduce the information revealing to an eavesdropper in [19,20]. Recently, Xue et al. proposed a dense coding on the symmetric W state jW〉 = p1ffiffi3 ð j001〉 + j010〉 + j100〉Þ to establish a DQC protocol [14]. Unfortunately, in their dense coding, only 67% of the encoded messages can be correctly recovered. That is, 33% of the messages will be lost, and have to be resent again. Thus, the sender and the receiver have to repeat the same DQC approximately 5 times ((33%)5 ≈ 0) to retransmit the lost information. Although later the four-photon W state jW〉 = 12 ð j0001〉 + j 0010〉 + j0100〉 + j1000〉Þ [15] and the 1 asymmetric W state jWs 〉 = αj 100〉 + βj010〉 + pffiffiffi j001〉 [16,17,21] 2
have been used to establish a DQC protocol without having this problem, the indeterminate situation in the symmetric W state has not yet been solved.
⁎ Corresponding author. E-mail address: [email protected] (T. Hwang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.06.039
This study attempts to propose a new coding function for the symmetric W state to avoid the indeterminate situation in Xue et al.'s DQC and also use it to construct a new QDC with better performance. In the new coding function for the symmetric W state, the sender has to use a unitary operation pair (Um1 ⊗ Um2) to encode messages, where Umi belongs to the set of four Pauli operations {I, σz, σx, iσy}. And the receiver also has to adopt two different strategies to measure the photons. Furthermore, considering the eavesdroppers may exist in the quantum channel, this study uses the technology of decoy photons [22,23] to detect the eavesdroppers and uses the reordering technique [24,25] to ensure that the transmitted messages will not be disclosed to the eavesdroppers. This paper is organized as follows. First, Section 2 briefly reviews the Xue et al.'s dense coding and DQC. Then, Section 3 describes the proposed coding function and the proposed DQC. The security analysis is also given here. Finally, Section 4 summarizes the contributions of this study and gives a conclusion. 2. Xue et al.'s dense coding and DQC Dense coding is an application of quantum entanglement, in which a sender can encode an x-bit classical message by performing unitary operations on y photons of an entangled state, where x N y. For example, a two-bit classical message can be encoded by performing a unitary operation on one photon of EPR pair. Liu et al. [26] proposed a general dense coding scheme for high-dimensional state. Moreover, the dense coding of two-photon has been experimentally established by Mattle et al. [27] and Fang et al. [28]. For three-photon state, Wei et al. [29] also proposed an experiment realization by using nuclear magnetic resonance (NMR). In this section, Xue et al.'s dense coding for the symmetric W state is first reviewed. Then, the DQC based on this dense coding is described.
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2.1. Xue et al.'s dense coding
Table 1 The corresponding unitary operations based on two-bit messages.
This subsection briefly describes Xue et al.'s dense coding and then points out the indeterminate coding situation on their dense coding. A symmetric W state can be shown by j W〉123 = p1ffiffi3 ð j 001〉 + j010〉 + j100〉Þ123 . Let wi denote the i-th photon of |W〉123. The four unitary operations {I, σx, iσy, σz} can be denoted by four twobit messages: “00”, “01”, “10”, and “11”, respectively, where I = |0〉 〈0| + |1〉〈1|; σx = |0〉〈1| + |1〉〈0|; iσy = |0〉〈1| − |1〉〈0|; and σz = |0〉 〈0| − |1〉〈1|. Thus, based on the two bits of a message, the sender first performs a corresponding unitary operation on w 1 . For example, if the two bits are “11”, the unitary operation σz will be performed on the w1. The sender keeps w3 to himself/herself whereas conveys w1 and w2 to the receiver. After the sender completes the encoding process on w1, the state of |W〉123 can be presented in the following four equations, respectively: 1 I⋅ jW〉123 = pffiffiffi ð j001〉 + j010〉 + j100〉Þ123 3
ð1Þ
1 σx ⋅ jW〉123 = pffiffiffi ð j101〉 + j110〉 + j000〉Þ123 3
ð2Þ
1 iσy ⋅ jW〉123 = pffiffiffi ð− j101〉−j110〉 + j000〉Þ123 3
ð3Þ
1 σz ⋅ jW〉123 = pffiffiffi ð j001〉 + j010〉− j100〉Þ123 : 3
ð4Þ
qffiffiffiffiffi pffiffiffi þ þ − = 2=3 jΨ 〉12 j0〉3 + 1 6 jΦ 〉 + jΦ 〉 j1〉3
σx ⋅ jW〉123
12
qffiffiffiffiffi pffiffiffi þ þ − = 2=3 jΦ 〉12 j0〉3 + 1 6 jΨ 〉 + jΨ 〉 j1〉3 12
iσy ⋅ jW〉123 = σz ⋅ jW〉123 =
qffiffiffiffiffi pffiffiffi þ − − 2 =3 jΦ 〉12 j0〉3 + 1 6 j Ψ 〉 + jΨ 〉 j1〉3 12
qffiffiffiffiffi pffiffiffi þ − − 2 =3 jΨ 〉12 j0〉3 + 1 6 jΦ 〉 + jΦ 〉 j1〉3
, where jΦF 〉12 =
12
1 pffiffiffið j00〉Fj11〉Þ12 2
and jΨF 〉12 =
Unitary operation
00
I ⊗ I or σz ⊗ σz
01
I ⊗ σx or σz ⊗ iσy
10
σx ⊗ σz or iσy ⊗ I
11
σx ⊗ iσy or iσy ⊗ σx
Quantum state of |W〉123 qffiffiffi 2 1 F j Ψþ 〉12 j 0〉3 + pffiffiffi j 00〉12 j1〉3 3 qffiffiffi 3 2 1 jΦþ 〉12 j0〉3 Fpffiffiffi j 01〉12 j1〉3 3 q3ffiffiffi 2 1 jΦ− 〉12 j 0〉3 Fpffiffiffi j10〉12 j 1〉3 3 3 qffiffiffi 2 1 F j Ψ− 〉12 j0〉3 −pffiffiffi j 11〉12 j 1〉3 3
3
performed by the sender is σx (see also Eq. (6)); thus, the message sent from Alice is “01”. However, if the measuring result of w3 is |1〉, Bob cannot recover Alice's message because the measuring result of (w1, w2) is indeterminate (i.e., either |Ψ+〉 or |Ψ−〉). Therefore, only 67% of the transmitted a message can be received by a receiver, i.e., 33% of the transmitted message will be lost. Thus, before a message can be sent completely to the receiver, the sender has to execute the same DQC protocol at least 5 times because 1 − (33%)5 ≐ 1. 3. A new coding function for symmetric W state and DQC protocol
In the decoding processes, the sender measures w3 by the computational basis {|0〉,|1〉} (called R-basis for short), and then the receiver performs the Bell measurement on w1 and w2. To better understand the measuring processes, the above equations can be rewritten as follows: I⋅ jW〉123
Two-bit message
ð5Þ ð6Þ ð7Þ ð8Þ
1 pffiffiffið j01〉Fj10〉Þ12 2
are Bell bases. According to the Eqs. (5)–(8), the measurement result of (w1, w2) is a determinate one if the measurement result of w3 is equal to |0〉. Otherwise, it will be indeterminate (i.e., it may be |Φ+〉 or |Φ−〉(|Ψ+〉 or |Ψ−〉)). 2.2. Xue et al.'s DQC protocol Based on the above dense coding, a DQC has been constructed among a sender, Alice, a receiver, Bob, and an assistant, Charlie. Assume that a symmetric W state |W〉123's photons w1, w2, and w3 have been distributed to Alice, Bob, and Charlie, respectively. Alice encodes a two-bit message by performing a corresponding unitary operation (as described in Section 2.1) on w1, and then she sends the encoded w1 to Bob. After Charlie measures w3 with R-basis, Bob can perform a Bell measurement on w1 and w2 to recover Alice's message. However, due to the indeterminate situation described earlier, Bob can only obtain the messages when the measurement result of w3 is equal to |0〉. For example, if the measuring result of (w1, w2) and w3 are |Φ+〉 and |0〉, respectively, Bob knows that the unitary operation
In this section, a new coding function for the symmetric W state is introduced. Furthermore, a DQC protocol is proposed using this coding function and decoy photons. In the proposed DQC protocol, the sender can transmit a secret message to the receiver, and the receiver can immediately recover the message without communicating with the sender again. Finally, the security analysis of this protocol is discussed. 3.1. The new coding function In our proposed coding function, a unitary operation pair Um1 ⊗ Um2 is used to encode a two-bit message, and then two measuring bases, R-basis
{|0〉,|1〉} and Bell-basis j ΦF 〉 = p1ffiffiffið j 00〉F j 11〉Þ, j ΨF 〉 = p1ffiffiffi ð j01〉F j01〉Þg, 2
2
are employed to recover the message. Assume that |Ψ+〉, |Φ+〉, |Φ−〉, and |Ψ−〉 (|00〉,|01〉,|10〉, and |11〉) denote the two-bit messages “00”, “01”, “10”, and “11”, respectively. The encoding and the decoding functions are explained in the following. 3.1.1. Encoding function Alice, the sender, prepares a symmetric W state jW〉123 = p1ffiffi ð j001〉 + j010〉 + j100〉Þ 123 , and then encodes her two-bit mes3 sage by performing the corresponding unitary operations on w1 and w2 based on the Table 1.1 Then, she keeps w3 to herself whereas sends the others to Bob, the receiver. For example, if Alice wants to transmit the “01” to Bob, she will perform the unitary operations I ⊗ σx on w1 and w2. The encoded state of |W〉123 can be shown as follows: 1 I⊗σx ⊗I jW〉123 = pffiffiffi ð j11〉 + j00〉Þ12 j0〉3 + j01〉12 j1〉3 3 rffiffiffi 2 þ 1 jΦ 〉12 j0〉3 + pffiffiffi j01〉12 j1〉3 : = 3 3
ð9Þ
3.1.2. Decoding function After Bob receives the photons, Alice uses the R-basis to measure w3, and then she sends the measuring result r_mA to Bob. If r_mA = |0〉, Bob performs the Bell-basis measurement on w1 and w2; otherwise, he uses the R-basis to measure. Finally, he can extract the two-bit message sent from Alice based on the measuring result r_mB. 1 Table 1 shows the corresponding relationship between the unitary operations and the quantum states based on the two-bit messages.
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Based on the newly proposed coding function on the symmetric W state and the block transmission method which is similar to [2], a DQC can be constructed as follows (see also Fig. 1).
Step 6 Alice measures H with R_basis, and then announces the sequence of measuring results R _ MA = {r _ mA1, r _ mA2, …, r _ mAm} and the permutation of TM to Bob. Step 7 According to the permutation, Bob recovers TM from TM' . Based on R_MA, Bob performs the corresponding measurement on TM. That is, if the r _mAI =|0〉, Bell-basis is used to measure (t1i , ti2); otherwise, the R-basis is used. Finally, Bob can correctly extract the messages sent from Alice based on the sequence of measuring results TM = {r _ mB1,.r _ mB2, …, r _ mBm}.
Step 1 Alice prepares m symmetric W states. Each of them can be denoted as
Obviously, Alice can transmit the messages to Bob within one execution of the protocol.
For example, if r_mA = |1〉, then Bob uses R_basis to measure w1 and w2. If the measuring result is r_mB = |01〉, then Bob can determine that the message sent from Alice is “01” according to the Table 1. 3.2. DQC based on the new coding function
1 jW〉t 1 t 2 hi = pffiffiffi ð j001〉 + j010〉 + j100〉Þt 1 t 2 hi i i i i 3
ð10Þ
, where i = 1 to m. She divides these m W states into two sequences: TM ={(t11, t12), (t12, t22), … , (t1m, t2m)} and H={h1, h2,…, hm}. In addition, she prepares n decoy single photons from {|0〉, |1〉, |+〉, |−〉} as the checking sequence TC ={d1, d2, …, dn} for checking eavesdrop1 1 pings, where j+〉 = pffiffiffið j0〉 + j1〉Þ and j−〉 = pffiffiffið j0〉− j1〉Þ. 2
2
Next, she employs the coding function described in Section 3.1 to encode the 2m-bit messages on (t1i , t2i ), where i=1, 2, …m. Then, ′ (i.e., the order of each she permutes the encoded TM into TM particle of TM is reordered), and mixes TM with TC to get T. Alice keeps H to herself, and sends T to Bob. Step 2 Bob sends an acknowledgment to Alice upon receiving T. Step 3 Alice announces the positions and the corresponding bases of TC to Bob. Step 4 Bob extracts TC from T, and then he measures TC with these corresponding bases. Finally, Bob returns the measuring results R _ MC = {r _ mC1,.r _ mC2, …, r _ mCn} to Alice. Step 5 Alice checks the existence of eavesdropping based on the R_MC. If the detected error rate exceeds a predetermined threshold τ (τ ≐ 2% ∼ 8.9% depending on the channel situation (e.g., distance, etc.)[32–36]), she aborts this communication and restarts the protocol.
3.3. Analysis of eavesdropping This section analyzes two popular attacks, the Intercept-andResend attack and the Entangle-and-Measure attack, against the proposed protocol. (i) Intercept-and-Resend. Attack We first assume that the quantum channel is ideal and that an attacker (called Eve) intercepts s photons sent from Alice, stores them and resends a sequence of forged photons to Bob in hope to pass the eavesdropping checking. Because the decoy photons are randomly inserted in TM by Alice, without knowing their positions and states, each of Eve's fake photons in the checking positions could be detected by the public discussion between Alice and Bob with the 1 probability of (i.e., the eavesdropper can pass that check with 4 3 the probability of ). Assume 2 m = n (i.e., half of Eve's 4 intercepting photons are decoy photons). Eve can be detected s2 3 s in the public discussion with the probability of 1− ≐1, if 4 2 is large. However, even if Eve can pass the eavesdropping check, she may not be able to obtain Alice's message. Because the order of TM is permuted, Eve can obtain useful information only if two intercepted photons (e.g., t1i , t2i ) belong to the corresponding W 1 state (e.g., |W〉t1i ti2hi). The probability of that is 2m + n which is C2
negligible if 2 m+n is large enough. The probability of intercepting
Fig. 1. The proposed DQC.
4400
C.-W. Tsai, T. Hwang / Optics Communications 283 (2010) 4397–4400
s 2
valid photon pairs is
s
3 2 4
×
s = 4−1 1 ∏ 2m + n−2j j = 0 C2
s =4−1 1 ∏ 2m + n−2j j = 0 C2
. Thus, Eve has the probability
s 2
to obtain -bit message. It is obvious that this
probability is very small. If, however, the quantum channel is not ideal, Eve may hide her attack by employing the noise of quantum channel [31]. Suppose that the quantum bit error rate (QBER) caused by the channel noise is τ (≐2%∼8.9%). If the QBER cased by the Eve's attack is smaller than τ, then Eve's attacking behavior may not be detected by a legitimate participant. However, in the proposed protocol, the QBER caused by the attack is about 25% which is larger than τ. Thus, this kind of attack eventually will be detected in the eavesdropper checking process. (ii) Entangle-and-Measure Attack. In this attack, Eve entangles an ancillary photon on each of the traveling photons by using a unitary operation Û (i.e., Uˆ † Uˆ = Uˆ Uˆ † = I), and then measure the ancillary photons to steal Alice's messages. However, similar to the analysis proposed in [30], if she wants to steal the message, the Û operation will disturb the states of decoy photons. Therefore, Eve's attack can be detected by the eavesdropping check between Alice and Bob. 4. Conclusion This paper proposes a new coding function for the symmetric W state. Based on the new encoding function, a DQC protocol is constructed. The message can be transmitted within one execution of DQC, which is rather efficient as compared to Xue et al.'s scheme (also using the symmetric W state) which requires about five-time executions of DQC protocol to complete a message transmission. Moreover, the eavesdroppers would be detected under both the ideal quantum channels and the noisy quantum channels. The security analysis shows that the proposed protocol is a quasi-secure DQC. In particular, the proposed DQC protocol is convenient for experimental implementation because only the symmetric W state, unitary operations, and the projected measurements are used. Acknowledgments The authors would like to thank the anonymous reviewers for their very valuable comments that enhance the clarity of this paper a lot.
We also like to thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC 98-2221-E-006-097-MY3.
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