Fault-Tolerant Quantum Dialogue Without Information Leakage Based on Entanglement Swapping between Two Logical Bell States

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1 Commun. Theor. Phys. 63 (015) Vol. 63, No. 4, April 1, 015 Fault-Tolerant Quantum Dialogue Without Information Leakage Based on Entanglement Swapping between Two Logical Bell States YE Tian-Yu (ãí ) College of Information & Electronic Engineering, Zhejiang Gongshang University, Hangzhou , China (Received October 10, 014; revised manuscript received January 1, 105) Abstract At present, the anti-noise property and the information leakage resistant property are two great concerns for quantum dialogue (QD). In this paper, two anti-noise QD protocols without information leakage are presented by using the entanglement swapping technology for two logical Bell states. One works well over a collective-dephasing noise channel, while the other takes effect over a collective-rotation noise channel. The negative influence of noise is erased by using logical Bell states as the traveling quantum states. The problem of information leakage is avoided by swapping entanglement between two logical Bell states. In addition, only Bell state measurements are used for decoding, rather than four-qubit joint measurements. PACS numbers: Dd, Hk, Pp Key words: quantum dialogue (QD), information leakage, entanglement swapping, logical Bell state, collective-dephasing noise, collective-rotation noise 1 Introduction In 1984, Bennett and Brassard [1] proposed the first quantum key distribution (QKD) protocol, giving birth to the novel concept named quantum cryptography. Quantum cryptography is a kind of cryptography which absorbs the principle of quantum mechanics into classical cryptography, thus it can be considered as the combination of quantum mechanics and classical cryptography. What QKD quantum cryptography can do is just to establish secret keys between two participants through the transmission of quantum signal. That is to say, secret messages cannot be transmitted directly through QKD. If one wants to transmit secret messages, another one-time-pad (OTP) process after QKD will be needed. Up to now, some good OKD protocols [ 5] have been constructed from different implemental ways. Naturally, one may think whether there exists a kind of quantum cryptography qualified for transmitting secret messages directly. This question motivates the appearance of quantum secure direct communication (QSDC). The first QSDC protocol was proposed by Long and Liu [6] in 00, where secret messages are represented by Einstein Podolsky Rosen (EPR) pairs and sent directly through block transmission of EPR pairs from the sender to the receiver. Due to the ability to transmit secret messages directly, QSDC has been actively pursued in recent years. [7 17] Similarly, one may also think whether there exists a special kind of quantum cryptography qualified for transmitting secret messages forth and back between two participants. In 004, Nguyen [18] and Zhang et al. [19 0] gave a positive answer to this question. That is to say, quantum dialogue (QD) was first put forward by them to answer this question. In Nguyen s QD protocol, [19] one EPR pair is encoded with Bob and Alice s secret messages one after another through unitary operations. When Bob publishes the final state of encoded EPR pair, they can read out each other s secret messages immediately. Since the concept of QD was put forward, a variety of QD protocols [1 30] have been constructed. Unfortunately, QD always runs the risk of information leakage, which was first pointed out by Gao et al. [31 3] in 008. Actually, the problem of information leakage is aroused by the existence of classical correlation, discovered by Tan and Cai [33] in 008. It is a common sense that one quantum cryptography protocol will be meaningless if it runs the risk of security. There is no exception for QD. Therefore, how to solve the issue of information leakage in QD quickly became an active research topic after it was first pointed out in 008. At present, several good methods have been put forward to solve it, such as the correlation extractability of Bell states, [34] the measurement correlation of entanglement swapping between two Bell states, [35] the direct transmission of auxiliary quantum states, [36 39] quantum encryption sharing, [40] and twice QSDC transmissions. [41] Besides the information leakage resistant property, another great concern for QD is the anti-noise property, as it is related to the practicability. It is well known that if photons travel inside a time window which is shorter than the variation of noise, they will be affected by the Supported by the National Natural Science Foundation of China under Grant Nos and Corresponding author, ñhappyyty@aliyun.com c 015 Chinese Physical Society and IOP Publishing Ltd

2 43 Communications in Theoretical Physics Vol. 63 same noise. [4 43] In this sense, the channel noise can be always modeled as collective noise. As decoherence-free (DF) states [4 53] are immune to collective noise, they are usually used to resist it. Apparently, only when one QD protocol possesses the anti-noise property and the information leakage resistant property simultaneously can it be put into practical use. Several QD protocols of this kind were proposed by Yang and Hwang [51] in 013 and the author [53] in 014, respectively. All of the QD protocols in Refs. [51,53] use DF states to combat collective noise and the direct transmission of auxiliary quantum states to resist the information leakage problem. Based on the above analysis, the author is devoted to solving the noise disturbance problem and the information leakage problem of QD simultaneously in this paper. Different from the QD protocols in Refs. [51,53], quantum entanglement swapping technology is employed here, which is an important technology of quantum information processing. Two anti-noise QD protocols without information leakage are constructed, where one works well over a collective-dephasing noise channel, and the other takes effect over a collective-rotation noise channel. The negative influence of noise is erased by using logical Bell states (one kind of four-qubit DF states) as the traveling quantum states. The problem of information leakage is avoided by swapping entanglement between two logical Bell states. Logical Bell States under Collective-Dephasing Noise and Their Transformations When encountering the transformations of collectivedephasing noise, the horizontal polarization state of photon 0 is kept unchanged, and the vertical polarization state of photon 1 is changed into e iϕ 1, where ϕ is the collective-dephasing noise parameter fluctuating with time. [44] Two logical qubits, 0 dp = 01 and 1 dp = 10, are immune to this kind of noise. [44] Moreover, their superpositions, ± dp ( 0 dp ± 1 dp ) ( 01 ± 10 ), are also invariant towards this kind of noise. [46] Two sets of logical measuring bases composed by the above four states are Z dp = { 0 dp, 1 dp } and X dp = { + dp, dp }. Furthermore, the four logical Bell states shown in Eq. (1), [48] are also resistant to this kind of noise. Here φ ± = (1/ )( 00 ± 11 ) and ψ ± = (1/ )( 01 ± 10 ) are the four original Bell states. These logical Bell states can be discriminated among each other after performed with two Bell state measurements on the 1st and the 3rd qubits, and on the nd and the 4th qubits, respectively. [48] Φ + dp 134 ( 0 dp 0 dp + 1 dp 1 dp ) 134 ( ) 134 ( ) 134 ( φ + φ + φ φ ) 134, Φ dp 134 ( 0 dp 0 dp 1 dp 1 dp ) 134 ( ) 134 ( ) 134 ( φ φ + φ + φ ) 134, Ψ + dp 134 ( 0 dp 1 dp + 1 dp 0 dp ) 134 ( ) 134 ( ) 134 ( ψ + ψ + ψ ψ ) 134, Ψ dp 134 ( 0 dp 1 dp 1 dp 0 dp ) 134 ( ) 134 ( ) 134 ( ψ ψ + ψ + ψ ) 134. (1) Table 1 Transformations among logical Bell states under collective-dephasing noise. Φ + dp Φ dp Ψ+ dp Ψ dp Ω I Φ + dp Φ dp Ψ+ dp Ψ dp Ω Z Φ dp Φ+ dp Ψ dp Ψ+ dp Ω X Ψ + dp Ψ dp Φ+ dp Φ dp Ω Y Ψ dp Ψ+ dp Φ dp Φ+ dp As shown in Table 1, the above four logical Bell states can be changed into each other with one of the four logical unitary operations {Ω I, Ω Z, Ω X, Ω Y }, [48 49] where Ω I = I 1 I, Ω Z = U 1 Z I, Ω X = U 1 X U X, Ω Y = U 1 Y U X. () Here I = , U X = , U Y = and U Z = are the four original unitary operations. Moreover, the superscript in each original unitary operation denotes the physical qubit performed with it. Each of the four logical unitary operations is assumed to represent a two-bit in a manner as Ω I 00, Ω Z 01, Ω X 10, Ω Y 11. (3)

3 No. 4 Communications in Theoretical Physics Logical Bell States under Collective-Rotation Noise and Their Transformations When encountering the transformations of collectiverotation noise, the horizontal polarization state of photon 0 is converted into cosθ 0 + sinθ 1, and the vertical polarization state of photon 1 is changed into sin θ 0 +cos θ 1, where θ is the collective-rotation noise parameter fluctuating with time. [46] Two logical qubits, 0 r = φ + and 1 r = ψ, are immune to this kind of noise. [46] Moreover, their superpositions, ± r ( 0 r ± 1 r ) ( φ + ± ψ ), are also invariant towards this kind of noise. [48] Two sets of logical measuring bases composed by the above four states are Z r = { 0 r, 1 r } and X r = { + r, r }. Furthermore, the four logical Bell states shown in Eq. (4), [48] are also resistant to this kind of noise. These logical Bell states can be discriminated among each other after performed with two Bell state measurements on the 1st and the 3rd qubits, and on the nd and the 4th qubits, respectively. [48] Φ + r 134 ( 0 r 0 r + 1 r 1 r ) 134 ( φ + φ + + ψ ψ ) 134 ( φ + φ + + ψ ψ ) 134, Φ r 134 ( 0 r 0 r 1 r 1 r ) 134 ( φ + φ + ψ ψ ) 134 ( φ φ + ψ + ψ + ) 134, Ψ + r 134 ( 0 r 1 r + 1 r 0 r ) 134 ( φ + ψ + ψ φ + ) 134 ( φ ψ + ψ + φ ) 134, Ψ r 134 ( 0 r 1 r 1 r 0 r ) 134 ( φ + ψ ψ φ + ) 134 ( φ + ψ ψ φ + ) 134. (4) Table Transformations among logical Bell states under collective-rotation noise. Φ + r Φ r Ψ + r Ψ r Θ I Φ + r Φ r Ψ + r Ψ r Θ Z Φ r Φ + r Ψ r Ψ + r Θ X Ψ + r Ψ r Φ + r Φ r Θ Y Ψ r Ψ + r Φ r Φ + r As shown in Table, the above four logical Bell states can be changed into each other with one of the four logical unitary operations {Θ I, Θ Z, Θ X, Θ Y }, [48 49] where Θ I = I 1 I, Θ Z = U 1 Z U Z, Θ X = U 1 Z U X, Θ Y = I 1 U Y. (5) Each of the four logical unitary operations is assumed to represent a two-bit in a manner as Θ I 00, Θ Z 01, Θ X 10, Θ Y 11. (6) 4 Fault Tolerant QD Protocols without Information Leakage Suppose that Alice has a secret consisting of N bits, i.e., {(i 1, j 1 )(i, j ) (i n, j n ) (i N, j N )}, and Bob has a secret consisting of N bits, i.e., {(k 1, l 1 )(k, l ) (k n, l n ) (k N, l N )}, where i n, j n, k n, l n {0, 1}, n {1,,..., N}. The fault tolerant QD protocol without information leakage against collectivedephasing (collective-rotation) noise is made up of the following steps. Step 1: Preparation Bob prepares a logical Bell state sequence, where each state is in the state of Φ + dp AB( Φ + r AB). This sequence is denoted by {(A 1, B 1 ), (A, B ),..., (A N, B N )}, where the letters A and B denote two logical qubits in one logical Bell state, and the subscripts 1,,...,N indicate the orders of logical Bell states in the sequence. Bob picks out the first logical qubit in order from each logical Bell state to form sequence S A, i.e., S A = [A 1, A,...,A N ]. The remaining logical qubits form sequences B, i.e., S B = [B 1, B,...,B N ]. Then, Bob adopts the decoy photon technology [54 55] to guarantee the transmission security of S A. That is, Bob prepares enough decoy logical qubits randomly in one of the four states { 0 dp, 1 dp, + dp, dp }({ 0 r, 1 r, + r, r }), and inserts them randomly into S A. Finally, Bob exploits the block transmission technique [6] to send S A to Alice, and retains S B at his site. Step : The first security check After Alice announces the receipt of S A, they enter into the security check process. Bob tells Alice the positions and preparation bases of decoy logical qubits. Then, Alice uses the bases Bob told to measure them, and publishes her measurement results to Bob. Bob judges whether the quantum channel is secure or not by comparing Alice s measurement results with the initial state of decoy logical qubits. If the quantum channel is secure, they implement the next step; otherwise, they discard the communication. Step 3: Bob s encoding After getting rid of decoy logical qubits, both Alice and Bob divide their own sequence into groups, where a group contains two adjacent

4 434 Communications in Theoretical Physics Vol. 63 logical qubits. That is, (A n 1, A n ) forms the group n in S A, while (B n 1, B n ) forms the group n in S B. Then, Alice and Bob detect the states of (A n 1, A n ) and (B n 1, B n ) in respective hand by imposing two Bell state measurements on the 1st and the 3rd qubits and on the nd and the 4th qubits, respectively. As a result, the entanglement swapping between (A n 1, B n 1 ) and (A n, B n ) happens. That is, the quantum system evolves as follows: [5] Φ + dp A n 1B n 1 Φ + dp A nb n = 1 ( Φ+ dp A n 1A n Φ + dp B n 1B n + Φ dp A n 1A n Φ dp B n 1B n + Ψ + dp A n 1A n Ψ + dp B n 1B n + Ψ dp A n 1A n Ψ dp B n 1B n ), Φ + r An 1B n 1 Φ + r AnB n = 1 ( Φ+ r An 1A n Φ + r Bn 1B n + Φ r An 1A n Φ r Bn 1B n + Ψ + r A n 1A n Ψ + r B n 1B n + Ψ r A n 1A n Ψ r B n 1B n ). (7) Fig. 1 Physical picture of the proposed protocol in the case of collective-dephasing noise (a) Solid line with arrow denotes quantum state transmission; (b) Dotted line means to be kept stationary; (c) LBM denotes logical Bell state measurement; (d) A 1 1 denotes the first physical qubit of A 1. From Eq. (7), it is obvious that after entanglement swapping, the two initial logical Bell states Φ + dp A n 1B n 1 ( Φ + r A n 1B n 1 ) and Φ + dp A nb n ( Φ + r AnB n ) are collapsed into four combinations of two logical Bell states each with a probability of 5%. Moreover, Alice and Bob can deduce each other s measurement result of the collapsed logical Bell state according to their own one. Then, according

5 No. 4 Communications in Theoretical Physics 435 to his own measurement result, Bob reproduces a new (B n 1, B n ) with no state measurement performed. Afterward, Bob encodes his two-bit (k n, l n ) by performing Ω knl n (Θ knl n ) on the new logical qubit B n. As a result, the group n in S B is converted into (B n 1, Ω knl n B n ) ((B n 1, Θ knl n B n )). That is to say, S B is changed into S B = [B 1, Ω k1l 1 B,..., B n 1, Ω knl n B n,..., B N 1, Ω knl N B N ] (S B = [B 1, Θ k1l 1 B,..., B n 1, Θ knl n B n,..., B N 1, Θ knl N B N ]). For the transmission security of S B, Bob also adopts the decoy photon technology.[54 55] That is, Bob prepares enough decoy logical qubits randomly in one of the four states { 0 dp, 1 dp, + dp, dp } ({ 0 r, 1 r, + r, r }), and inserts them randomly into S B. Finally, Bob exploits the block transmission technique [6] to send S B to Alice. Step 4: The second security check After Alice announces the receipt of S B, they implement the same security check process as Step. If the quantum channel is secure, they implement the next step; otherwise, they discard the communication. Step 5: Alice s encoding and bidirectional communication After getting rid of decoy logical qubits, Alice encodes her two-bit (i n, j n ) by performing Ω inj n (Θ inj n ) on the logical qubit B n 1. As a result, the group n in S B is converted into (Ω inj n B n 1, Ω knl n B n ) ((Θ inj n B n 1, Θ knl n B n )). Then, Alice detects the state of (Ω inj n B n 1, Ω knl n B n )((Θ inj n B n 1, Θ knl n B n )) by imposing two Bell state measurements on the 1st and the 3rd qubits and on the nd and the 4th qubits, respectively, and publishes her measurement result on it to Bob. According to Alice s announcement and his own logical unitary operation Ω knl n (Θ knl n ), Bob can read out Alice s two-bit (i n, j n ). At the same time, Alice can infer out Bob two-bit (k n, l n ) from her own logical unitary operation Ω inj n (Θ inj n ). This concludes the description of the proposed QD protocols. The proposed QD protocol under the case of collective-dephasing (collective-rotation) noise takes the entanglement swapping between two logical Bell states both in the state of Φ + dp AB( Φ + r AB) for example. Actually, any two logical Bell states are qualified. On the other hand, as the two-step QD protocol of Ref. [35] uses the entanglement swapping between two original Bell states, the proposed QD protocols can be regarded as the generalization of the two-step QD protocol of Ref. [35] into the cases of collective-dephasing noise and collective-rotation noise, respectively. In order to further illustrate the proposed protocols, it is necessary to give out the physical pictures. Without loss of generality, only the physical picture of the proposed protocol in the case of collective-dephasing noise is given, which is shown in Fig. 1. As the security checks can be regarded to be relatively independent from the the bidirectional communication process, the physical picture ignores the security checks. Here, the first two logical Bell states (A 1, B 1 ) and (A, B ) are taken for example. For clarity, the first two logical Bell states (A 1, B 1 ) and (A, B ) are also taken to concretely explain the bidirectional communication process. Suppose that Alice s first two-bit (i 1, j 1 ) is 00, and Bob s first two-bit (k 1, l 1 ) is 10. Without loss of generality, the case of collective-dephasing noise is merely considered here. Suppose that Alice s measurement result of (A 1, A ) is Φ + dp A 1A. According to Eq. (7), Alice can refer out that Bob s measurement result of (B 1, B ) is Φ + dp B 1B. Accordingly, with Bob and Alice s encoding operations, the new Φ + dp B 1B undergoes the following transformations: Φ + dp B 1B Ω B1 I ΩB X Φ+ dp B 1B Ψ + dp B 1B. (8) The Ψ + dp B 1B is published by Alice in Step 5. According to three known pieces of information: Φ + dp B 1B, Ω B X, and Ψ+ dp B 1B, Bob can read out that (i 1, j 1 ) is 00. Likewise, according to three known pieces of information: Φ + dp B 1B, Ω B1 I and Ψ + dp B 1B, Alice can deduce that (k 1, l 1 ) is Security Analysis Against Eve s Active Attacks In each of the proposed QD protocols, Bob sends two sequences S A and S B to Alice one after another. Accordingly, two security checks with the decoy photon technology [54 55] are employed to guarantee their transmission security. Eve wishes to launch her attacks exclusively on the traveling logical qubits from logical Bell states. However, she will inevitably leave her trace on decoy logical qubits, due to no knowledge about their positions in the traveling sequences. Without loss of generality, the case of collective-dephasing noise is taken to conduct the security analysis here. The transmission of S A from Bob to Alice is considered at first. (i) The intercept-resend attack Eve prepares fake decoy logical qubits also randomly in one of the four states { 0 dp, 1 dp, + dp, dp } beforehand. Then, she intercepts S A and sends the fake decoy logical qubits to Bob. As Alice s measurement results on the fake decoy logical qubits are not always identical to the genuine ones, Eve is discovered with a probability of 50%. [53] (ii) The measure-resend attack Eve intercepts S A and randomly uses one of the two sets of measuring bases Z dp and X dp to measure it before resending it to Alice. As Eve does not always choose the right bases Alice used to prepare decoy logical qubits, she is discovered with a probability of 5%. [53] (iii) The entangle-measure attack Eve entangles her auxiliary photons E = { E 1, E,..., E i,...} with the logical qubits in S A through a unitary operation U E. Accordingly, the quantum system evolves as follows: [51]

6 436 Communications in Theoretical Physics Vol. 63 U E 01 E i = α e 0 e 0 + α e 0 e 1 + α e 1 e 0 + α e 1 e 1, U E 10 E i = β e 0e 0 + β e 0e 1 + β e 1e 0 + β e 1e 1, φ + (α 00 e 0 e 0 + α 11 e 1 e 1 + β 00 e 0 e 0 + β 11 e 1 e 1 ) U E ψ + E i = 1 (U E 01 E i + U E 10 E i ) = 1 φ (α 00 e 0 e 0 α 11 e 1 e 1 + β 00 e 0e 0 β 11 e 1e 1 ) ψ + (α 01 e 0 e 1 + α 10 e 1 e 0 + β 01 e 0 e 1 + β 10 e 1 e 0 ), ψ (α 01 e 0 e 1 α 10 e 1 e 0 + β 01 e 0e 1 β 10 e 1e 0 ) φ + (α 00 e 0 e 0 + α 11 e 1 e 1 β 00 e 0e 0 β 11 e 1e 1 ) U E ψ E i = 1 (U E 01 E i U E 10 E i ) = 1 φ (α 00 e 0 e 0 α 11 e 1 e 1 β 00 e 0 e 0 + β 11 e 1 e 1 ) ψ + (α 01 e 0 e 1 + α 10 e 1 e 0 β 01 e 0e 1 β 10 e 1e 0 ), (9) ψ (α 01 e 0 e 1 α 10 e 1 e 0 β 01 e 0 e 1 + β 10 e 1 e 0 ) where α 00 + α 01 + α 10 + α 11 = β 00 + β 01 + β 10 + β 11 = 1, and e 0 e 0, e 0 e 1, e 1 e 0 and e 1 e 1 are Eve s probe states. In order to pass the security check, Eve must let α 00 = α 10 = α 11 = β 00 = β 01 = β 11 = 0, (10) if the decoy logical qubits are 0 dp or 1 dp ; Eve must make α 01 e 0 e 1 α 10 e 1 e 0 + β 01 e 0 e 1 β 10 e 1 e 0 = α 01 e 0 e 1 + α 10 e 1 e 0 β 01 e 0 e 1 β 10 e 1 e 0 = 0, (11) where 0 is a zero vector, if the decoy logical qubits are + dp or dp. Apparently, only when both Eqs. (10) and (11) hold is Eve not discovered. After inserting Eq. (10) into Eq. (11), it follows α 01 e 0 e 1 = β 10 e 1 e 0. (1) Equation (1) means that Eve is unable to distinguish α 01 e 0 e 1 from β 10 e 1 e 0. In other words, Eve is unable to discriminate between 0 dp and 1 dp. Therefore, Eve obtains nothing useful by measuring her auxiliary photons. Another case is that in order to get partial information useful, Eve may make α 01 e 0 e 1 β 10 e 1e 0 to discriminate her auxiliary photons. However, Eve will be discovered as the decoy logical qubits + dp and dp are disturbed by her, according to Eq. (11). [51] (iv) The Trojan horse attacks As the traveling logical qubits are transmitted in a single-direction way, the Trojan horse attack strategies, i.e., the invisible photon eavesdropping attack [56] and the delay-photon Trojan horse attack, [57] are invalid. On the other hand, as the second security check adopts the same decoy photon technology as the first one, the security analysis on the transmission of S B from Bob to Alice can be demonstrated in the same way. 6 Discussions and Conclusions (i) The information leakage problem After the entanglement swapping of two adjacent Bell states (A n 1, B n 1 ) and (A n, B n ), Alice can easily deduce Bob s measurement result of the collapsed logical Bell state at his site. Therefore, it is not necessary for Bob to announce the state of (B n 1, B n ) to Alice publicly. As a result, Eve has no access to the state of (B n 1, B n ), which totally has four possibilities. As for Eve, the announcement on the measurement result of (Ω inj n B n 1, Ω knl n B n )((Θ inj n B n 1, Θ knl n B n )) involves sixteen kinds of Alice and Bob s secret messages. That is to say, according to Shannon s information theory, [58] the quantum channel contains 16 p i log p i = log 1 16 = 4 i=1 bit information for Eve, just equal to the total amount of Alice and Bob s secret messages. Therefore, no information is leaked out to Eve. (ii) The information-theoretical efficiency The information-theoretical efficiency defined by Cabello [3] is η = b s /(q t + b t ), where b s, q t, and b t are the expected secret bits received, the qubits used and the classical bits exchanged between Alice and Bob. In the proposed QD protocol against collective-dephasing (collective-rotation) noise, the two adjacent Bell states (A n 1, B n 1 ) and (A n, B n ) can be used to transmit Alice s two-bit and Bob s two-bit with two bits consumed for the announcement on the measurement result of (Ω inj n B n 1, Ω knl n B n )((Θ inj n B n 1, Θ knl n B n )). Therefore, Cabello s information-theoretical efficiency of the proposed QD protocol against collective-dephasing (collective-rotation) noise is η = ( + )/(8 + ) = 40%. (iii) Comparisons with previous fault tolerant QD protocols without information leakage Apparently, the QD protocols in Refs. [51,53] are both fault tolerant and information leakage resistant. It is necessary to compare the proposed protocols with them. The comparisons, summarized in Table 3, are concentrated on four aspects including the initial quantum resource, the quantum measurement, the channel capacity, and the

7 No. 4 Communications in Theoretical Physics 437 information-theoretical efficiency. According to Table 3, the proposed QD protocols exceed the QD protocols of Refs. [51,53] in channel capacity. Moreover, the proposed QD protocols have higher information-theoretical efficiency than the QD protocols of Ref. [53]. However, the proposed QD protocols do not take any advantage over the QD protocols of Refs. [51,53] in both initial quantum resource and quantum measurement. Table 3 Comparisons with previous fault tolerant QD protocols without information leakage. The protocols in Ref. [51] The protocols in Ref. [53] The protocols in this paper Initial quantum resource Product states of two Bell states Logical qubits Logical Bell states Quantum measurement Bell state measurements Single-particle measurements Bell state measurements Channel capacity classical bits per round classical bits per round 4 classical bits per round Information-theoretical efficiency 40% 33.3% 40% To sum up, two anti-noise QD protocols without information leakage are constructed in this paper by using the entanglement swapping technology for two logical Bell states. One works well over a collective-dephasing noise channel, while the other takes effect over a collective-rotation noise channel. The proposed QD protocols have the following features: (i) They use logical Bell states as the traveling quantum states to remove the disturbance of collective noise; (ii) They prevent the information leakage problem by swapping entanglement between two logical Bell states; (iii) They only need Bell state measurements for decoding, rather than four-qubit joint measurements; (iv) They are able to resist Eve s active attacks, such as the intercept-resend attack, the measure-resend attack, and the entangle-measure attack. Especially, they have perfect security towards the Trojan horse attacks, as the traveling logical qubits are transmitted in a single-direction way. References [1] C.H. Bennett, G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, IEEE Press, Bangalore (1984) 177. [] C.H. Bennett, G. Brassard, and N.D. Mermin, Phys. Rev. Lett. 68 (199) 557. [3] A. Cabello, Phys. Rev. Lett. 85 (000) [4] F.G. Deng and G.L. Long, Phys. Rev. A 68 (003) [5] F.G. Deng and G.L. Long, Phys. Rev. A 70 (004) [6] G.L. Long and X.S. Liu, Phys. Rev. A 65 (00) [7] K. Bostrom and T. Felbinger, Phys. Rev. Lett. 89 (00) [8] F.G. Deng, G.L. Long, and X.S. Liu, Phys. Rev. A 68 (003) [9] F.G. Deng and G.L. Long, Phys. Rev. A 69 (004) [10] C. Wang, F.G. Deng, Y.S. Li, X.S. Liu, and G.L. Long, Phys. Rev. A 71 (005) [11] C. Wang, F.G. Deng, and G.L. Long, Opt. Commun. 53 (005) 15. [1] X.H. Li, C.Y. Li, F.G. Deng, et al., Chin. Phys. 16 (007) 149. [13] X.B. Chen, Q.Y. Wen, F.Z. Guo, Y. Sun, G. Xu, and F.C. Zhu, Int. J. Quant Inform. 6 (008) 899. [14] W.F. Cao, Y.G. Yang, and Q.Y. Wen, Sci. China Ser. G-Phys. Mech. Astron. 53 (010) 171. [15] G. Gao, M. Fang, and R.M. Yang, Int. J. Theor. Phys, 50 (011) 88. [16] Z.W. Sun, R.G. Du, and D.Y. Long, Int. J. Theor. Phys. 51 (01) [17] B.C. Ren, H.R. Wei, M. Hua, T. Li, and F.G. Deng, Eur. Phys. J. D 67 (013) 30. [18] B.A. Nguyen, Phys. Lett. A 38 (004) 6. [19] Z.J. Zhang and Z.X. Man, (004) /pdf/quant-ph/ pdf. [0] Z.J. Zhang and Z.X. Man, (004) /pdf/quant-ph/ pdf. [1] Z.X. Man, Z.J. Zhang, and Y. Li, Chin. Phys. Lett. (005). [] X.R. Jin, X. Ji, Y.Q. Zhang, S. Zhang, et al., Phys. Lett. A 354 (006) 67. [3] Z.X. Man and Y.J. Xia, Chin. Phys. Lett. 3 (006) [4] X. Ji and S. Zhang, Chin. Phys. 15 (006) [5] Z.X. Man, Y.J. Xia, and B.A. Nguyen, J. Phys. B-At. Mol. Opt. Phys. 39 (006) [6] Z.X. Man and Y.J. Xia, Chin. Phys. Lett. 4 (007) 15. [7] Y. Chen, Z.X. Man, and Y.J. Xia, Chin. Phys. Lett. 4 (007) 19. [8] Y.G. Yang and Q.Y. Wen, Sci. China Ser. G-Phys. Mech. Astron. 50 (007) 558. [9] C.J. Shan, J.B. Liu, W.W. Cheng, et al., Mod. Phys. Lett. B 3 (009) 35. [30] T.Y. Ye and L.Z. Jiang, Chin. Phys. Lett. 30 (013) [31] F. Gao, S.J. Qin, Q.Y. Wen, and F.C. Zhu, Phys. Lett. A 37 (008) [3] F. Gao, F.Z. Guo, Q.Y. Wen, and F.C. Zhu, Sci. China Ser. G-Phys. Mech. Astron. 51 (008) 559. [33] Y.G. Tan and Q.Y. Cai, Int. J. Quant. Inform. 6 (008) 35. [34] G.F. Shi, Opt. Commun. 83 (010) 575. [35] G. Gao, Opt. Commun. 83 (010) 88.

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