Quantum secure direct communication based on four-particle cluster state grouping

Accepted Manuscript

Quantum Secure Direct Communication Based on Four-particle Cluster State Grouping Zhengwen Cao, Dan Song, Geng Chai, Chen He, Guang Zhao PII: DOI: Reference:

S0577-9073(18)31028-1 https://doi.org/10.1016/j.cjph.2019.01.019 CJPH 794

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

25 July 2018 22 November 2018 17 January 2019

Please cite this article as: Zhengwen Cao, Dan Song, Geng Chai, Chen He, Guang Zhao, Quantum Secure Direct Communication Based on Four-particle Cluster State Grouping, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.01.019

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Highlights • The technical design enhances the communication efficiency and the qubit-

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utilization ratio. • 91.921% of information can be exchanged in lossy channel.

• Bandwidth or lower frequency of channel needs to be considered in practical implementations.

• Appropriate error-correcting codes of receiver make up for the channel

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deficiency.

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Quantum Secure Direct Communication Based on Four-particle Cluster State Grouping

a School

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Zhengwen Caoa,∗, Dan Songa , Geng Chaia , Chen Hea , Guang Zhaoa of Information Science and Technology, Northwest University, Xi’an 710127, China

Abstract

In this paper we present a quantum secure direct communication protocol based

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on four-particle cluster states. In our protocol both the sender and the receiver keep two particles of the cluster state, and we verify that our protocol can prevent the eavesdropper from intercepting valid messages. Meanwhile, we also analyze our protocol in a lossy channel under the attack of an eavesdropper. We show that both the communication efficiency and the qubit-utilization ratio

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are improved compared with other existing schemes.

Keywords: Quantum secure direct communication, cluster state, lossy channel

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2010 MSC: 81P45

1. Introduction

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Quantum secure direct communication (QSDC) is an emerging type of quantum communication which is different from Quantum Key Distribution. A fundamental difference is that the secret information is directly transmitted through the quantum channel. QSDC was first proposed in [1], [2], [3], and then [4], [5]

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developed the idea of quantum dense-coding and proposed a scheme which em-

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ployed quantum entangled states to achieve approximate security. The use of various quantum states as the informational carrier in the QSDC

✩ This work is supported by the Natural Science Basic Research Program of Shaanxi Province (No. 2018JM6123), Project of Graduate Independent Innovative of Northwest University, China (Grant No. YZZ17173, No. YZZ17175). ∗ Corresponding author Email address: [email protected] (Zhengwen Cao)

Preprint submitted to Journal of LATEX Templates

February 27, 2019

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protocol includes a single photon state [6], the Bell state [5], [7], [8], the GHZ 10

state [9], [10], [11], the W state [12], [13] and the cluster state. Many QSDC protocols have been proposed using the cluster state to improve the perfor-

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mance. In [14] the cluster state was proposed as a new entangled state, and [15] defined the maximal multi-entangled state (MMES) for the first time. [16], [17], [18] experimentally implemented the preparation of the cluster state. Sub15

sequently, [19] and [20] designed quantum teleportation protocols to transmit

single photon and EPR particles via cluster state channels. The authors in [21] and [22] loaded classical information into cluster state particles to realize QSDC.

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The security of the quantum communication protocols has been gradually improved.

Based on the property of the cluster state, in this paper we propose a new

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approach to further improve the performance of QSDC. The proposed protocol has some nice properties. First, the cluster states are just one-time transmitted particles in the quantum channel, so that the communication efficiency is im-

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proved. Second, by analyzing the security bound of our protocol, we show that our work can be employed for error-correction in a lossy channel.

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The rest of this paper is organized as follows. In Sec. 2 we introduce our protocol based on four-particle cluster states. In Sec. 3 we show the security verification of the protocol under three major attacks, and in Sec. 4 we give some

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analysis in the case of a lossy channel. We compute the quantum communication transmission efficiency and the qubit-utilization ratio of this protocol in Sec. 5.

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Finally we conclude this work in Sec. 6.

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2. The proposed protocol based on four-particle cluster states Our QSDC protocol in this paper is based on four-particle cluster states. A

nice property of a cluster state is that it can maintain persistent entanglement

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and a stable self-associated structure, based on which the entanglement of the cluster state is maintained despite the remote distance. Even if it is measured, the entanglement characteristic cannot be easily destroyed and disturbed by

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the de-coherence effect. As is shown in Fig. 1, at the beginning the cluster state consists of four particles a1 , a2 , a3 , and a4 , which are entangled before 40

measurement. Assuming that particle a2 is separated from the cluster state

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via measurement, the remaining particles are still entangled. Hence, the fourparticle cluster state is transformed into a three-particle cluster state.

After the measurement

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Before the measurement

Figure 1: The measurement of one particle of the cluster state.

2.1. The main idea of the proposed protocol

We employ the four-qubit cluster state to prepare the quantum communication channel:

1 (|0000i + |0011i + |1100i − |1111i)1234 , 2

(1)

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|ψi1234 =

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45

where the subscripts 1, 2, 3 and 4 indicate the four correlated particles, |0i and

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|1i are the computational basis of the single particle. Alice

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a1

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a3

a2

Entangled Channel

a4

Bob

Alice

Bob

a2

a1

PM

a3

Collapsed Channel a4

Figure 2: The PM of two particles of the cluster in our protocol.

The entangled channel of our protocol is illustrated in Fig. 2, where Alice and

Bob are the legitimate sender and receiver, respectively. At the very beginning, 50

a channel security test is performed: a single photon is randomly inserted in

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the particle sequences which are then sent from Alice to Bob. Bob returns the measurement results of the positions of the photons to Alice for the comparison test. Alice loads the encrypted information via local unitary operations. Then

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the cluster state is measured with the projective measurement (PM) by Alice and Bob, respectively. Finally, Bob recovers the classical initial cryptographs with Alice’s PM results. 2.2. The flow chart of the QSDC protocol

In this section we describe the details of the protocol, which include the

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following steps.

Step 1 (particle preparation): Alice prepares N groups of four-particle cluster

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states and records the qubits information. She divides these four-particle cluster groups into two sequences, namely, S1 (a1 , a3 ) and S2 (a2 , a4 ). She keeps one of the sequences S1 (a1 , a3 ) and sends the other S2 (a2 , a4 ) to Bob.

Step 2 (channel detection): Alice randomly inserts some single photons in the sequence as a detection sequence S2 0 and records the positions of the photons.

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She preserves the measurement basis sequence used for preparing the single

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photons and sends the detecting sequence S2 0 to Bob via the quantum channel. Bob receives the whole sequence of S2 0 . Meanwhile, Alice transmits the position information of the photons to Bob through the classical channel. Bob selects the measurement basis randomly, X-basis or Z-basis, for the single photon of

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the corresponding position based on the position information of the photons

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sent by Alice. The X-basis can be expressed as: 1 |−i = √ (|0i − |1i) . 2

1 |+i = √ (|0i + |1i) , 2

(2)

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Then Bob returns the measurement basis sequence to Alice by the classical channel. Alice compares the measurement basis sequence from Bob to ensure

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the channel’s safety. If the error rate is higher than the security threshold, the channel is unsafe and there exists an eavesdropper. In this case the communication should be terminated, otherwise, the channel is safe and we can implement step 3. 5

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Step 3 (encoding): If the channel is safe, the original information sequence 80

M can be transmitted from Alice to Bob. The different local unitary operations are selected by Alice to encode the original information sequence M on the

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quantum states sequence S1 (a1 , a3 ). The local unitary operation is expressed as: I = |0i h0| + |1i h1| ,

X = |0i h1| + |1i h0| ,

(3)

where I and X belong to the Pauli matrices [23]. The relationship between 85

the unitary transformations and the classical information sequence of the cor-

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responding particles a1 , a3 is shown in Table 1.

Table 1: Unitary transformation corresponding to the coding regulation.

a1

a3

00

I

I

01

X

X

10

X

I

11

I

X

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Information sequence

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Step 4 (measurement): Alice performs a PM on the two particles that she has reserved. Bob performs a PM on the two particles a2 and a4 which he holds and records the measurement results. After that, Alice sends her measurement results as a classical message to Bob via the classical channel.

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Step 5 (message recovery): Bob compares his measurement results with Alice’s measurement results to determine which unitary operation is selected

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by Alice. Thereby he can recover the original information sequence M . The corresponding encoding regulation is shown in Table 2. For instance, if the four-qubit cluster state is |0011i, then Alice’s measurement will be |11i because

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of the encoding operation of XI. If Bob gets his PM as |01i, he will obtain the classical information sequence ‘M ’ as ‘10’.

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Table 2: The correspondence of the measurement results and the unitary transformations.

Transform

Alice’s result

Bob’s result

|0i |0i

I ⊗I

|0i |0i

|0i |0i

|0i |1i

I ⊗I

|0i |1i

|0i |1i

|1i |0i

I ⊗I

|1i |0i

|1i |0i

|1i |1i

I ⊗I

|1i |1i

|1i |1i

|0i |0i

X ⊗X

|1i |1i

|0i |0i

|0i |1i

X ⊗X

|1i |0i

|0i |1i

|1i |0i

X ⊗X

|0i |1i

|1i |0i

|1i |1i

X ⊗X

|0i |0i

|1i |1i

|0i |0i

X ⊗I

|1i |0i

|0i |0i

|0i |1i

X ⊗I

|1i |1i

|0i |1i

|1i |0i

X ⊗I

|0i |0i

|1i |0i

|1i |1i

X ⊗I

|0i |1i

|1i |1i

|0i |0i

I ⊗X

|0i |1i

|0i |0i

|0i |1i

I ⊗X

|0i |0i

|0i |1i

|1i |0i

I ⊗X

|1i |1i

|1i |0i

|1i |1i

I ⊗X

|1i |0i

|1i |1i

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PM bases

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00

01

10

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3. Security analysis We verify the security of our protocol under three major attacks and two situations in a noisy channel of QSDC: the intercept-resend attack, the entangled CNOT attack and the assisted particle attack.

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The interception and resend attack: Eve can capture part of the par-

ticles and send her own quantum state to Bob in advance. Since there are four possible outcomes in the measurement of the PM measured quantum states, 105

Eve intercepts these two particles and gets the right measurement results with the probability of

1 4.

Let us assume that Alice sends N qubits to Bob in each

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communication. Eve needs to recover all the N qubits and sends the N decoy

qubits to Bob. The probability will be ( 41 )N and it will approach to 0 with an increase of N . That is to say, the more qubits exchanged between Alice and 110

Bob in one communicating process, the more unlikely it will be that Eve obtains all the information of the qubits. Therefore, the eavesdropping attack can be

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found during Bob’s detection process.

The entangled CNOT attack: Eve can perform a CNOT operation when Alice is sending particles to Bob. Suppose the particles a5 and a6 are generated by Eve which are entangled with a2 and a4 of the cluster state. Eve can group

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a2 and a4 with a5 , a6 , respectively. The particles (a2 and a4 ) are the controlled qubits, and the particles (a5 and a6 ) are the target qubits. The quantum system

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will become:

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|Ψi123456

1 (|000000i123456 + |110001i123456 + 2 |001110i123456 − |111111i123456 ) 1 = (|00i13 |0000i2456 + |10i13 |1001i2456 + 2 |01i13 |0110i2456 − |11i13 |1111i2456 )  
1 −  +  +  = [ φ 13 φ 24 φ 56 + φ− 24 φ− 56 + 2 + + −  
φ φ φ + φ− 24 φ+ 56 + 13 24 56 + + +  
ϕ ϕ ϕ + ϕ− 24 ϕ− 56 + 13 24 56 − + − − +
ϕ ϕ ϕ ϕ ϕ ]. + 13 24 56 24 56 =

8

(4)

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We can obviously determine that Eve is correspondingly equal to Bob. How120

ever, Eve can be discovered during the single photon detection process, since Alice and Bob will get 50% incorrect results when they analyze the classical

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results. Meanwhile, they deem that the channel is not safe. Therefore, the implementation of the entanglement attack is ineffective.

The assisted particle attack: Let us assume that Eve steals some of 125

the information. She performs an operation and creates an entangled state

with the two particles in the sequence S2 . According to the Stinespring dilation theorem [4], Eve’s eavesdropping is equivalent to performing a unitary operation

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ˆ on the Hilbert space, which consists of quantum signals and auxiliary systems. E

The auxiliary state of Eve is |ei. We assume the entanglement between a2 and 130

a4 can be shown as |φ+ i = |ϕiEve

√1 2



(|0i |0i + |1i |1i), and then

 |0i ⊗ |0ei + |1i ⊗ |1ei √ 2 α |00e00 i + β |01e01 i + α0 |11e11 i + β 0 |10e10 i √ , 2

ˆ⊗ E

=

(5)

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=

where α, α0 , β and β 0 are complex amplitudes of the singlet states and |α|2 +

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|β|2 = 1, and the same for α0 and β 0 . The error rate of Eve’s attack is 2

2

2

PError = |β| = 1 − |α| = 1 − |α0 | .

(6)

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It can be seen that Eve is bound to interfere with the system and change the states under an auxiliary particle attack, which will be discovered by the legiti135

mate communication party in the channel security detection phase.

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In view of information theory, the accessible amount of information in the

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quantum system is limited to the Holevo bound [24], χ(ρ) = S(ρ) −

X

pi S(ρi ),

(7)

i

where S(ρ) is the von Neumann entropy of the state ρ and ρ =

P

i

pi ρi . ρi is the

quantum state which is prepared by the communicating party with probability

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pi . If the communicating party produces a state |0i or |1i with a probability of P 1 i pi log pi = 1 bits. 2 , the entropy of the photon in the sequence is H(P ) = − 9

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Eve can obtain a message, which satisfies the inequality IE ≤ S(ρ) −

X

pi S(ρi ) < H(P ).

(8)

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Since there are four encoding ways of unitary transformations between Alice and Bob, they can exchange 2-bit information via two particles. Assuming the 145

information is disclosed, Eve will deduce the corresponding unitary transforma-

tion which contains only 2 × (− 41 log2 41 ) = 1 bits of unknown information. It can be seen from Eq. (8) that Eve can get 1-bit of information at most. But in

the detection step Eve will be found out, because this cluster state is disturbed.

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Therefore, Eve will be discovered and the communication will be terminated. It

is worth mentioning that the single photon detection and the unitary operation selection should be random so that our protocol can identify a variety of attacks. There are three other attacks existing in quantum communications: the denial of service attack, the invisible photon-Trojan horse attack and the time

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delay-Trojan horse attack. First, the eavesdropper cannot perform unitary operations on photons, because each of them is determinate and after the operation the photon will be disturbed and the legal party will be aware of the disturbance.

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Second, the eavesdropper must insert other photons with different wavelengths, then Alice encodes these photons with a unitary operation. Eve can obtain the corresponding unitary operation but not the initial information, because the unitary operation is dependent on wavelength. Finally, the time delay-Trojan

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horse attack is described as Eve inserting a photon with the same wavelength

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but not synchronized. In this case, Eve cannot recover the valid information of the unitary operation.

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4. The communication in a lossy channel

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As we all know, QSDC can be implemented when the communication ar-

rives at a level of unconditional security theoretically. Let us consider a special circumstance where the communication channel is a lossy one. Meanwhile, we can regard this situation as the one where the eavesdropper controls the legal 10

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communication channel. We assume that A and B represent Alice and Bob, 170

respectively.

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4.1. The mutual information of the channel transmission probability The channel transmission probability is γ ∈ [0, 1]. Then, let us discuss the relation between the channel capacity and the channel transmission probability.

Assume that Alice can send a perfect state with probability P (A1 ), while Bob 175

can correctly receive the state with probability P (B1 |A1 ) = γ. On the other hand, the transmitting probability of an imperfect state of Alice and the prob-

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ability of corresponding reception by Bob are P (A2 ) and P (B1 |A2 ) = 1 − γ, respectively. Therefore, P (B2 |A1 ) = 1 − γ and P (B2 |A2 ) = γ. Meanwhile, Bob

will correctly receive the state with a probability of P (B1 ) and an incorrect 180

one with P (B2 ). Hence, we need to consider how P (Ai |Bi ) changes with the increase of channel transmission probability γ and i ∈ 1, 2.

Alice prepares a perfect state and sends it to Bob with probability of P (A1 ) =

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p and an imperfect one with P (A2 ) = 1 − p, where p ∈ [0, 1]. We can express P (B1 ) with the Law of Total Probability. X i

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P (Ai )P (B1 |Ai ) = 2pγ + 1 − γ − p.

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P (B1 ) =

(9)

So, P (B2 ) = 1 − P (B1 ) = −2pγ + γ + p. P (Ai |Bi ) can be calculated with Bayes’

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theorem as follows:

pγ , 1 − p − γ + 2pγ 1 − γ − p + pγ , P (A2 |B1 ) = 1 − p − r + 2pγ p − pγ P (A1 |B2 ) = , p + γ − 2pγ γ − pγ P (A2 |B2 ) = . p + γ − 2pγ

P (A1 |B1 ) =

(10) (11) (12) (13)

According to the definition of entropy, the mutual information I(A; B) can be represented in Eq. (14) and simulated in Fig. 3. I(A; B)

= H(Ai ) − H(Ai |Bi ) 11

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= −

X i

[−

P (Ai ) log2 P (Ai ) −

X

X i

P (Ai |Bj ) log2 P (Ai |Bj )].

(14)

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j

P (Bj )

Figure 3: A schematic plot of the relation among the channel transmission probability (γ), the source probability (P (A1 )) and the mutual information (I(A; B)) in a lossy channel. The

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‘∗’ depicts a coordinate (γ, P (A1 ), I(A; B)) when P (A1 ) = 0.5.

We present the relation between the channel transmission probability (γ), the source probability (P (A1 )) and the mutual information (I(A; B)) in Fig. 3.

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It is worth mentioning that we only consider the binary communication channel which has the supreme value of entropy at 1 bit/symbol, theoretically. The

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channel is symmetric and the mutual information reaches the maximum when P (A1 ) = 0.5.

According to the property of the von Neumann entropy [25], if P (Ai ) is the

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probability of the proportion of perfect cluster state, I(A; B) needs to satisfy

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an inequality, X S( P (Ai )ρi ) ≥ I(A; B),

(15)

i

where S is an entropy of the probability based on the different state sent by Alice.

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Assume that in our case Alice needs to transmit two particles S2 (a2 , a4 ) with

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the form of a reduced density matrix: = tr(|φ+ ihφ+ |) = 1.

trρi

(16)

S(

X

P (Ai )ρi ) = 1.

i

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The von Neumann entropy can be calculated in Eq. (17),

(17)

We can see that our protocol can transmit the maximum information Imax = 2 bits. From Fig. 3, there exists I0.5(A;B) = 0.91921 bits per unit. Therefore, we 205

can obtain I(A; B) = 2 ·

0.91921 1

= 1.83842 bits, which satisfies the inequality in

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Eq. (15). Therefore, we need to choose a communication channel with efficiency larger than 0.5. From the simulation, the maximum number needs to satisfy 0.5 ≤ γ ≤ 0.99 when P (A1 ) = 1, simultaneously. 4.2. Bit error-correcting in a lossy channel

If the information is transferred in an awful channel, like a Gaussian channel,

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let us discuss the measurement which is taken by Bob for a secure communication. We can clearly see that Eve can steal 1 bit of information at most from

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the above analysis, so there will be a 1 bit error in each unit of the four-particle cluster states. Therefore, Bob needs to correct the error bits in the sequence, if 215

he does not throw away those error bits. Let us seek a solution for this problem,

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which means that the error rate is higher than the threshold, i.e., an insecure communication.

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Our protocol can exchange 2 bits via 2 qubits, so Bob needs to detect these 2 error qubits and correct one of them. There exists four types of states which

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need to be exchanged, like |0i |0i, |0i |1i, |1i |0i and |1i |1i. After an lossy trans-

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mission, all the four will be changed in two situations, one is |0i |1i, |0i |0i, |1i |1i and |1i |0i, and the other one is |1i |0i, |1i |1i, |0i |0i and |0i |1i. According to the quantum error correcting condition [24], the error can be expressed as Pauli matrices [24]. Likewise, I, X, Y, Z represents no error, bit-flipped, bit&phase-

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flipped and phase-flipped errors occurring in any of the two indifferent qubits,

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respectively. We just need to consider the bit-flipped error and phase-flipped error in our protocol. The errors, which exist in one of two qubits, can be expressed as IX, XI,

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IZ and ZI. Bob needs to seek the corresponding anticommutation relation operators for these errors and construct the quantum error-correcting code for our protocol.

Table 3: The anticommutation relation operators with two errors.

IX

XI

IZ

ZI

Operators

IZ

ZI

IX

XI

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Errors

During the channel detection stage, both legal sides of the communication will detect the error, which means that there exists an eavesdropper. In this case, Bob receives the error information and needs to recover and correct it. Alice 235

encodes the secret information into two qubits, the quantum error-correcting

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code can be constructed in Eq. (18), where Mi is the row of the matrix of the quantum error-correcting code.

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g1 :

M1 = XX

g2 :

M2 = XZ

g3 :

M3 = ZX

(18)

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Assuming that in one of the two qubits a bit-flipping error occurs, such as the second qubit, EX2 = IX. Here, I means that the first qubit has no error. Bob can implement his quantum error-correcting code on this error EX2 .

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That is to say, there exists a gi satisfying the anticommutation relationship, i.e.

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{Mi , EX2 } = 0. It can be calculated as follows: M2 EX2

= (X ⊗ Z)(I ⊗ X) = (I ⊗ X)(X ⊗ Z) = X ⊗ ZX = −(X ⊗ XZ) = −EX2 M2 .

(19)

Then Bob can detect that the error occurred in the second qubit via the quantum

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error-correcting code {g1 , g2 , g3 }, and correct it via the corresponding anticom245

mutation relation operators.

The transmission efficiency of our protocol is ξ =

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5. Efficiency analysis bs qt +bt

=

2 4+2

= 0.33, where

bs is the useful classical bits which should be exchanged, qt is the total number

of qubits, bt is the total number of classical bits. The encoding regulation is 1 250

qubit corresponding to 1 bit information. There are only two particles involving the encoding of the message. The qubit-utilization ratio is η =

qu qt

=

2 4

= 0.5,

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where qu is the useful number of qubits which are transmitted in the protocol. A comparison with other related protocols is shown in Table 4.

[26] designed a QSDC protocol based on five-particle cluster state. We can 255

see that 2 bit classical information was transmitted via a 5 qubit, which had a lower qubit utilization ratio than our work. [27] and [28] proposed two QSDC

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protocols based on a four-particle cluster state. In [27], there was 1 bit exchanged in communication, which is less than 2 bits. Meanwhile, an XORoperation needs to be engaged in the receiver and controller in [28]. This is redundant, since the quantum property can guarantee the security of commu-

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nication, and the efficiency will be decreased with more information exchanged.

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Therefore, it is clear that the efficiency and qubit utilization ratio of the suggested protocol are higher than the protocols of [26], [27] and [28].

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Table 4: The efficiencies of different protocols.

Protocol

Efficiency ξ

Qubit utilization ratio η

Wang et al. [27]

0.10

0.20

Chang et al. [26]

0.14

0.25

Nanvakenari et al. [28]

0.18

0.33

Proposed protocol

0.33

0.50

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6. Conclusion and outlook In recent years, QSDC has been experimentally realized in only a few labo-

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ratories. [29] reported the first experimental demonstration of quantum secure

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direct communication with single photons, which was based on the DL04 pro-

tocol equipped with a simple frequency coding. It has the advantage of being robust against channel noise and loss. In this case, we need to consider a fi270

nite maximum number N of frequency channels, which is a value changing with

the maximum and minimum modulation frequencies and the channel frequency spacing. Quantum memory has been used to implement QSDC in [30], which

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takes a fundamental step toward practical QSDC and demonstrates a potential application for long-distance quantum communication in a quantum network. 275

Regarding the coincidence rate, the bit rate is approximately 2.5 per second and the error rate is approximately 0.1, which results in the 90% fidelity. All in all, we need to consider the channel frequency and bit rate to improve the fidelity of

quantum source.

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communication. Theoretically, we need to choose a big bandwidth and superior

Our QSDC protocol based on a four-particle cluster is theoretically proved,

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and it is shown that it can withstand the attack from eavesdropping, for instance, the intercept-resend attack, the entangled CNOT attack and the as-

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sisted particle attack. In addition, we analyzed our protocol in a lossy channel, meanwhile, the receiver can correct the error bit according to his quantum errorcorrecting code without throwing it away.

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References

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distribution scheme, Physical Review A 65 (3) (2002) 032302.

[2] A. Beige, B. G. Englert, C. Kurtsiefer, H. Weinfurter, Secure communi-

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cation with a publicly known key, Acta Physica Polonica 101 (3) (2001) 357. [3] A. Beige, B. G. Englert, C. Kurtsiefer, H. Weinfurter, Secure communica-

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