Quantum secure direct communication against the ...

Prog. Theor. Exp. Phys. 2015, 123A02 (12 pages) DOI: 10.1093/ptep/ptv161

Quantum secure direct communication against the collective noise with polarization-entangled Bell states Li Dong1,∗ , Jun-Xi Wang1 , Qing-Yang Li1 , Hong-Zhi Shen2 , Hai-Kuan Dong1 , Xiao-Ming Xiu1,3,∗ , Yuan-Peng Ren4 , and Ya-Jun Gao1 1

College of Mathematics and Physics, Bohai University, Jinzhou 121013, P. R. China School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, P. R. China 3 Centre for Quantum Technologies, National University of Singapore, 117543, Singapore 4 College of Engineering, Bohai University, Jinzhou 121013, P. R. China ∗ E-mail: [email protected], [email protected] 2

............................................................................... We propose a quantum secure direct communication protocol via a collective noise channel, exploiting polarization-entangled Bell states and the nondemolition parity analysis based on weak cross-Kerr nonlinearities. The participant Bob, who will receive the secret information, sends one of two photons in a polarization-entangled Bell state exploiting the transmission circuit against the collective noise to the participant Alice, who will send the secret information, by the means of photon block transmission. If the first security check employing the nondemolition parity analysis is passed, the task of securely distributing the quantum channel is fulfilled. Encoding secret information on the photons sent from Bob by performing single-photon unitary transformation operations, Alice resends these photons to Bob through the transmission circuit against the collective noise. Exploiting the nondemolition parity analysis to distinguish Bell states, Bob can obtain the secret information from Alice after the second security check is passed, and the resulting Bell states can be applied to other tasks of quantum information processing. Under the condition of the secure quantum channel being confirmed, the photons that are utilized in the role of the security check can be applied to the function of secure direct communication, thus enhancing the efficiency of transmitting secret information and saving a lot of resources.

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1.

A61

Introduction

Quantum mechanics embodies different properties from classical mechanics and creates a new discipline field of quantum information, accompanied by information science and computer science. Quantum information includes two aspects; one is quantum computation and the other is quantum communication. Quantum computation possesses a powerful computation capability, which cannot be approached by classical computers based on classical physics. Quantum communication provides the possibility of transmission of secret information based on the quantum no-cloning theorem [1–3], in which many quantum-key distribution (QKD) protocols are proposed. QKD allows two legitimate participants to share beforehand a string of bits as a key with a secret, and encrypt and decrypt secret information according to the shared key. Generally, QKD protocols can be divided into two classes according to the modes of applications. The first class of QKD protocols uses individual information carriers, for instance, the BB84-QKD © The Author(s) 2015. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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Received September 1, 2015; Revised October 11, 2015; Accepted October 12, 2015; Published December 1, 2015

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protocol [4] is the first unconditionally secure key-distribution protocol which uses two nonorthogonal groups of orthogonal single-photon states; the B92-QKD protocol [5] exploits two nonorthogonal single-photon states, thus reducing the experimental requirements but halving the efficiency at the same time. The second class of QKD protocols relies on quantum entanglement, such as the E91-QKD protocol [6], which uses Einstein–Podolsky–Rosen (EPR) pairs as the quantum channel, where the generalized Bell’s inequality [7] and Clauser–Horne–Shimony–Holt inequality [8] are applied to check the security of the key distribution; the BBM92-QKD protocol [9] exploits the security check bases composed of two nonorthogonal sets of orthogonal bases, affording facilities for the realization of secret key distribution. So far, numerous researchers have paid extensive attention to QKD protocols [10–12]. Besides QKD protocols, deterministic secure quantum communication (DSQC) protocols [13– 19] with classical bit transmission have been proposed; these are also QKD protocols, as a matter of fact. However, quantum secure direct communication (QSDC) protocols [20–27] are genuinely direct communication protocols with security, which allow direct information transmission between legitimate participants without the assistance of a key, which is necessary in DSQC protocols and QKD protocols [28]. Employing a two-step security check and dense coding [29], Deng et al. proposed the first QSDC protocol [22] to realize direct quantum communication, where single-qubit measurement is performed on the check photons which have to be discarded after measurement. Exploiting parity check, Bouwmeester put forth a probabilistic QKD scheme [30]. Assisted by a fast polarization modulator (Pockels cells), Kalamidas [31] presented deterministic single-photon error rejection in the theoretical and experimental aspects. In order to overcome the disadvantageous effect of arbitrary collective unitary noise, Wang [32] proposed a QKD scheme by adopting the odd-parity security check. Applying parity analysis, the tasks of information transmission can be fulfilled with or without control [33,34]. With regard to cross-Kerr nonlinearities providing indirect interaction between two individual photons, it stimulates the discussion of researchers [35–45], and the progress in theoretical and experimental aspects predicts the feasibility of applying cross-Kerr nonlinearities [45–50]. To overcome the influence of collective noise and enhance the efficiency of QSDC protocols based on Bell-state measurement, we present a QSDC protocol employing photonic systems, where the method of nondemolition [51–56] parity analysis is adopted in the security check and the process of obtaining secret information. In this protocol, the transmitted photons can be sent to the target position without being affected by collective noise. Discarding the conventional parity analysis, the nondemolition parity analysis is exploited, by which the check photons can be exploited in the communication process and hence enhance the efficiency of communication. Considering its practical implementation, double crossKerr nonlinearities and a minus phase shifter [57,58] are applied instead of negative phase-shift Kerr media, so the experimental requirements and resource cost are reduced. To lower the error probability, a photon-number measurement is performed to replace the conventional homodyne measurement. Based on the application of a two-step security check and dense coding, the present protocol provides high efficiency and security in secret-information transmission. The construction of the rest of the paper is as follows. The secure distribution of polarizationentangled Bell states and the explicit QSDC protocol are presented in Sects. 2 and 3, respectively. The efficiency and security of the present QSDC protocol is analyzed and discussed in Sect. 4. Finally, in the last section, Sect. 5, our work is concluded.

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The secure distribution of polarization-entangled Bell states via the collective noise channel

Photons transmit with the maximal velocity in nature, and can feasibly be operated with the currently available techniques. Moreover, they possess lower decoherence due to their weak interaction with the environment. All these features make them a promising candidate for an information carrier transmitting over long-distance quantum communication. A Bell state is the simplest entangled state, which can embody the nonlocal property of quantum mechanics, and is one of the best quantum channels for implementing quantum communication protocols. Any one of four polarization-entangled Bell states can be used as the quantum channel to realize communication protocols, which can be expressed as follows under the bases of {|H , |V } or {|+, |−} (|± = √1 (|H ± |V )): 2

1 = √ (| + + + | − −) Ai ,Bi , 2 1 = √ (| − + + | + −) Ai ,Bi , 2 1 = √ (| + + − | − −) Ai ,Bi , 2 1 = √ (| − + − | + −) Ai ,Bi , 2

(1)

where the subscript Ai (Bi ) represents the photon label and sequence order. Here, the legitimate participants select the state | − Ai ,Bi as the initial quantum channel to fulfill the two-step QSDC protocol. At first, the secret-information receiver, Bob, prepares a large number of singlet Bell states (| − Ai ,Bi ), which can be realized with numerous methods [59]. By adopting the method of photon block transmission [22], where the failure of one block cannot affect the others, thus guaranteeing efficiency, Bob selects the photon Ai in the singlet state | − Ai ,Bi , and sends it to the secret-information sender, Alice, while the photon Bi is kept . At his location, Bob needs a quantum memory to store the photon (e.g., Bi ) from each Bell state. After the photon Ai is returned from Alice, Bob performs nondemolition parity analyses on the photon Ai and the photon Bi to obtain the secret information that Alice wants to transmit to him. As shown in the left-hand box in Fig. 1, in Bob’s site, the photon Bi will pass through paths B1 and B2 accompanied by the probe-coherent states |α and |α, where the potential photons in paths (B1 , B2 ) enable the probe-coherent states (|α, |α) to accumulate the phase shifts (θ, θ ) on them due to the presence of cross-Kerr nonlinearities. Subsequently, the probe-coherent states and the photon Ai are sent to Alice via paths (P1 , P2 ) and (A1 , A2 ). As for the transmission of the photon Ai , in the practical distribution process, there are many noise factors in the environment to disturb the distribution of the polarization-entangled singlet state | − Ai ,Bi , such as atmospheric fluctuation and inhomogeneity, thermal and mechanical fluctuation, and the birefringence of the optical fiber. For the sake of simplicity, if the transmitted photons are close enough to each other in the space or time interval, we can suppose that they suffer from the same noise, i.e., the collective noise channel. In the practical implementation, there are two approaches for transmitting photons, i.e., the free space and the optical fiber. If the photon passes through the 3/12


1 |+ Ai ,Bi = √ (|H H + |V V ) Ai ,Bi 2 1 |− Ai ,Bi = √ (|H H − |V V ) Ai ,Bi 2 1 | + Ai ,Bi = √ (|H V + |V H ) Ai ,Bi 2 1 | − Ai ,Bi = √ (|H V − |V H ) Ai ,Bi 2

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free space, the two paths (the reflected path and the transmitted path of the beam splitter) that the photon potentially passes through should be small enough in spatial distance. In this situation, the photon passing through the two paths can be thought of as suffering from collective noise. In another situation, if the optical fiber is applied, the two paths should be combined into one path in the same fiber. The photon passing through the reflected path and the transmitted path of the beam splitter will be distinguished by the temporal degree of freedom, which can be realized by setting different lengths in the two different paths, and the path difference is modulated when needed. By means of this method, the time interval needs to be set short enough, and, as a consequence, the photon can be supposed to undergo collective noise. Here, “small spatial distance” and “short time interval” mean that the transmitted photons are so close physically that they can be considered to suffer from collective noise. Although it passes through the collective noise channel, the photon Ai is sent to Alice without being affected, due to the application of the transmission circuit, as shown in the lower part of Fig. 1, which can overcome the effect of collective noise [60,61]. Bob sends the photon Ai to Alice through path A. Passing through the polarization beam splitter PBSb+ and the half wave plate HWPb1 45◦ , the polarization-entangled Bell state | − A,B , where the subscripts denote the photon paths, evolves as follows: 1 PBSb+ , HWPb1 45◦ | − A,B −−−−−−−−−−−→ √ (|H A2 |V B − |H A1 |H B ). 2 4/12

(2)


Fig. 1. A plot illustrating the secure transmission of the photon Ai from Bob to Alice and the first security check. In the lower part of this plot, we can see that the transmission circuit of the photon Ai includes two polarization beam splitters that reflect the vertical polarization state |V and transmit the horizontal polarization state |H , PBSa+ , PBSb+ ; and three half wave plates HWP 45◦ , where an angle of 45◦ between the optical axis and the horizontal polarization mode |H is set, HWPa1 45◦ , HWPa2 45◦ , and HWPb1 45◦ ; this functions as a NOT gate to exchange the horizontal polarization state |H and the vertical polarization state |V . The rest of the plot represents the nondemolition parity analysis to perform the first security check. The photon Bi passes through path B1 or path B2 accompanied by the probe-coherent state |α. The components PBSa1+(×) , PBSa2+(×) , PBSb1+(×) , and PBSb2+(×) can be employed to reflect the vertical polarization photon state |V (−45◦ polarization photon state |−) and transmit the horizontal polarization state |H (45◦ polarization state |+). After photon-number measurement is performed, the parity of check photons along the basis of {|H , |V } ({|+, |−}) can be distinguished. The probe-coherent states function as quantum buses linking two photons in the same Bell state, by which the nondemolition parity analysis can be completed.

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After that, the photon Ai transmits through the collective noise channel (path A1 or path A2 ) and suffers from an arbitrary collective noise, which can be denoted as Collective noise

|H A1 (A2 ) −−−−−−−−−→ α|H + β|V ,

(3)

where the coefficients α and β indicate the noise effect parameters of the collective noise, and path A1 or path A2 , where the photons suffer the same noise, is the path that the photon Ai potentially goes through. As a consequence, the singlet state is changed to 1 (4) √ [(α|H + β|V ) A2 |V B − (α|H + β|V ) A1 |H B ]. 2 Passing through HWPa1 45◦ , PBSa+ , and HWPa2 45◦ , the photon Ai is transmitted perfectly from Bob to Alice, and the related process can be denoted as 1 HWPa1 45◦ , PBSa+ , HWPa2 45◦ −−−−−−−−−−−−−−−−−→ √ α(|H V − |V H ) A1 B + β(|H V − |V H ) A2 B 2 (5)

Finally, Alice will obtain the photon Ai from path A1 with probability |α|2 or from path A2 with probability |β|2 , which is determined by the noise parameters. That is to say, a Bell state as the quantum channel is distributed between Bob and Alice via the collective noise channel. Before performing the parity analysis to check whether the distribution of the quantum channel is secure or not, Alice needs to check whether Trojan-horse photons are present or not. In order to circumvent Trojan-horse photon attack [62–64], Alice sets a wavelength filter to let photons with legitimate frequencies pass through, so that photons of other wavelengths cannot enter into Alice’s setup to undergo the same operations as the legitimate photons. By accurately controlling the open and close times of the receiving window, the participants can be kept from the attack of delay-time photons. Counting the photon number or adding a photon-number splitter is an effective method of checking for the presence of multiphoton signals. As a consequence, Alice’s information concerning encoding operations cannot be stolen after the attacker retrieves these photons and conducts the measurement if the above Trojan-horse photon attacks are used. Confirming no Trojan-horse photon, Alice and Bob perform the first security check. Unlike the previous protocols, the present protocol uses nondemolition parity analysis with two nonorthogonal sets of orthogonal polarization bases, i.e., the measurement basis of {|H , |V } and the measurement basis of {|+, |−}. Applying the parity analysis setup shown in Fig. 1, the first security check of the distribution of Bell states can be completed. PBSb1+(×) , PBSb2+(×) , PBSa1+(×) , and PBSa2+(×) should be adopted to perform the parity analysis on the photon pair along the identical bases of {|H , |V } or ({|+, |−}), and the situations along different measurement bases need to be discarded. Explicitly, the polarization beam splitters PBSb1+ , PBSb2+ , PBSa1+ , and PBSa2+ transmit |H states and reflect |V states, while PBSb1× , PBSb2× , PBSa1× , and PBSa2× can be applied to transmit |+ states and reflect |− states. Alice lets the photon Ai and the probe-coherent states (|α, |α) pass through the Kerr media to perform the same operation as Bob does, which is shown in the right-hand box of Fig. 1. The photon Ai also enables generation of the phase shifts (θ, θ ) on the probe-coherent states (|α, |α). Before entering the beam splitter (BS), the minus phase shifters (−θ, −θ ) are inserted into two lines on the probe-coherent states (|α, |α). Finally, the photon-number measurement is performed by Alice to obtain the parity information of the check photon pairs. 5/12


= α| − A1 B + β| − A2 B .

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Based on Eq. (1), if photons (Ai , Bi ) are in odd parity (|+− − |−+ or |H V − |V H ), the measurement result will indicate a signal of security, i.e., the secure distribution of Bell states is completed, and the next step of the communication task can be expected. If the parity of the check photons is even (|++ ± |−− or |H H ± |V V ), a signal of insecurity is presented, which may be stepped from a potential eavesdropper on the line; these photons should be discarded and everything restarted. All in all, some countermeasures should be adopted to overcome these obstacles. At this step, the secure distribution of polarization-entangled Bell states has been completed.

3.

The quantum secure direct communication protocol with dense coding and nondemolition parity analysis

I = |H H | + |V V |, X = |H V | + |V H |, Y = |H V | − |V H |, Z = |H H | − |V V |.

(6)

As a result, the Bell state shown in Eq. (5) is transformed to I

− → α| − A1 B + β| − A2 B , X

− → α|− A1 B + β|− A2 B , Y

− → α|+ A1 B + β|+ A2 B , Z

− → α| + A1 B + β| + A2 B .

(7)

Then Alice returns the encoded photon Ai to Bob through the transmission circuit against collective noise shown in Fig. 2. That is, the photon Ai passes through the collective noise channel, and finally arrives at Bob’s site. Owing to the application of the transmission circuit against collective noise, the photon Ai has not been affected. For simplicity, we take one of the four cases in Eq. (7) as an example to describe this process. If Alice wants to send bits 11 to Bob, after her encoding operation Y , the state of photons Ai and Bi is transformed to α|+ A1 B + β|+ A2 B . Passing through the elements of HWPa2 45◦ , PBSa+ , and HWPa1 45◦ , the state evolves as HWPa2 45◦ , PBSa+ , HWPa1 45◦

α|+ A1 B + β|+ A2 B −−−−−−−−−−−−−−−−−→ 1 √ [(α|H + β|V ) A2 |H B + (α|H + β|V ) A1 |V B ]. 2 6/12

(8)


After the first security check on the Bell-state distribution is passed, a quantum secure direct communication protocol can be realized with the following procedures. No matter which path the photon comes from, Alice encodes secret information needing to be transmitted to Bob by performing the single-photon unitary transformation U on both paths A1 and A2 according to the following rules. If the information needing to be transmitted is 00, Alice needs to perform identity transformation (i.e., I ) on the photon Ai ; the remaining corresponding rules are 01: Z , 10: X , and 11: Y , where the four single-photon unitary transformations can be denoted as

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In the returning process of the photon Ai , the collective noise is denoted as |H A1 (A2 ) → δ|H + γ |V , |V A1 (A2 ) → δ |H + γ |V ,

(9)

where the coefficients δ, γ and δ , γ indicate the different noise parameters of the collective noise on the polarization states |H and |V , respectively. So the whole state shown in Eq. (8) is transformed as 1 Collective noise −−−−−−−−−→ √ {[(αδ + βδ )|H + (αγ + βγ )|V ] A2 |H B 2 + [(αδ + βδ )|H + (αγ + βγ )|V ] A1 |V B }.

(10)

At Bob’s position, use of the composition HWPb1 45◦ , PBSb+ , and HWPb2 45◦ can recover the state of the photon coming from Alice, i.e., HWPb1 45◦ , PBSb+ , HWPb2 45◦

−−−−−−−−−−−−−−−−−−→ (αδ + βδ )|+ C1 B + (αγ + βγ )|+ C2 B .

(11)

Bob will obtain the photon Ai from paths C1 and C2 with the probabilities |αδ + βδ |2 and |αγ + βγ |2 , respectively. After receiving the photon block from Alice, Bob performs a second security check by analyzing the parity of photons Ai and Bi ; the details of the procedure are as follows. Bob performs a nondemolition parity analysis on photons Ai and Bi . As mentioned above, the parity analysis includes two models, which are distinguished by different measurement bases, {|H , |V } or {|+, |−}, and the analysis principle is the same as that in the first security check. Alice informs Bob of the positions of test qubits to evaluate whether the shared polarization-entangled Bell states carrying secret information are destroyed or not depending on Alice’s single-photon unitary transformations on the corresponding check photons. It needs to be noted that the conclusion of the security check needs to be judged by Alice, and, as a result, one of the two bits is exposed in the process. 7/12


Fig. 2. A plot illustrating the transmission of the photon Ai from Alice to Bob, the second security check, and the encoding–decoding process. Performing encoding operations U on the photon Ai , Alice sends it to Bob via the collective noise channel. The photon Ai is transmitted to Bob, and then Bob performs a parity analysis to fulfill the second security check and obtain the secret information. After completion of the parity analysis shown on the right-hand side of this illustration, another parity analysis needs to be performed to distinguish Bell states along the different basis, which is omitted because of the limitations of the page layout.

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Table 1. The explicit relations among the encoding unitary transformations operated by Alice (Alice’s EUT), the resulting photon states after operation (RPS), the nondemolition parity analysis outcomes (Bob’s NPAO: I. the preceding measurement along the basis of {|H , |V } and the later measurement along the basis of {|+, |−}; II. the preceding measurement along the basis of {|+, |−} and the later measurement along the basis of {|H , |V }), and the secret information (SI) that is transmitted from Alice to Bob. Alice’s EUT I Z X Y

RPS −

| | + −|− |+

Bob’s NPAO (I; II)

SI

odd,odd; odd,odd odd,even; even,odd even,odd; odd,even even,even; even,even

00 01 10 11

4.

The efficiency and security of the present protocol

In conventional protocols, two security-check steps will cost 75% of all the photon pairs. No matter whether the selected measurement bases of the parity analysis are identical or not, the photon pairs used to check security will be discarded due to the application of a destructive measurement. So the efficiency of the conventional protocols is only 25%. In this protocol, since the random selection analysis basis is exploited in the measurement of photon Ai in Alice’s site and photon Bi in Bob’s location, half of the check photons are discarded in the statistical meaning because 50% of the check photon pairs are not in accord with each other in the measurement bases; i.e., photon pairs with wrong measurement bases occupy 25% of all the photon pairs and should be discarded. However, the other 50% check photon pairs (25% of all the photon pairs) utilized in checking security can be used in the communication process, if the first security check is passed. Including all the photon pairs that need not be discarded, 75% of all the photon pairs can be applied to the next communication process. Among this 75%, the second security check on the quantum channel will exploit half (their 50%). In the second security check, there is no situation of wrong selections on the check basis; this means that 75% of all the photon pairs will be applied to the communication process, although the amount of information on the check photons is reduced to one bit per (check) 8/12


The other one can be exploited in the next process and is obtained by Bob as a secret-information bit under the secure-channel condition. If the second security check is not passed, the communication process must be halted; two legitimate participants must inspect the process, find the reason for the failure, and resolve it. If the second security check is passed, another parity analysis along the different measurement basis from that in the preceding measurement that has been just completed needs to be performed. That is to say, if the preceding measurement is performed along the basis of {|H , |V }, the later measurement should be completed with the basis of {|+, |−}, and vice versa. On the basis of the outcomes of the two parity analyses, Bob can obtain the secret information that Alice wants to send him. The explicit corresponding relations among the encoding unitary transformations operated by Alice, the resulting states, the outcomes of the nondemolition parity analysis, and the secret information obtained by Bob can be found in Table 1. This completes the quantum secure direct communication between two legitimate participants.

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photon. So, if the second security check is confirmed, photon pairs playing the role of the second security check can be utilized in the subsequent communication process. All in all, the efficiency of the present protocol is 75%. The photon loss induced by the environment and its effect in the transmission channel is the main disadvantageous effect in any communication protocol. In the present protocol, if signal photons are lost, the setups do not respond at the corresponding time when the signal photons should arrive at the detectors, and the corresponding correlations do not exist. Combined with some errors from other sources, the overall error rate will be higher than the predetermined threshold value, so the communication has to restart with other photon transmission blocks. If photons of the coherent state are lost, this will reduce the intensity of the coherent state, and thus increase the error rate for distinguishing the phase shifts on the coherent state (i.e., Perr = 12 exp −2A|αθ |2 , where A is the rate of residual photons of the probe-coherent state). Enhancing the intensity of the initial coherent state can alleviate the disadvantageous effect resulting from the photon loss of the coherent state. Next, we analyze the communication process in the presence of an eavesdropper, i.e., security analysis on the information transmission of the present protocol. Due to the presence of the photon distribution process, an eavesdropper, Eve, has access to the transmitted photons and the probe-coherent states, by which she may obtain the available information in the distribution process. When Bob sends the signal photons and the probe-coherent states to Alice or Alice resends the signal photons back to Bob, Eve tries her best to access them to steal the information. There are some attack tricks on the transmitted photons, such as intercept attacks, intercept– measure–resend attacks, entangle–measure attacks, or other similar conventional attacks. However, the error introduced by these attacks will be detected in the parity analysis due to the adoption of the absolutely secure check method where two nonorthogonal sets of orthogonal bases are applied in the security check, stepping from the BBM92-QKD protocol, which is a secure protocol at the principal level of physics. Once Alice and Bob detect the eavesdropping action, Alice does not encode secret information on the signal photons, so Eve can obtain nothing from the intercepted signal photons. Except for the transmitted photons, the probe-coherent state can be obtained by Eve. She intercepts part of the probe-coherent state by using a beam splitter. In this situation, the eavesdropping action can escape detection by the two legitimate participants since they view it as photon loss of the probe-coherent state. After Eve performs a measurement on the intercepted part of the coherent state, the state of the signal photons collapses into one of the product states, |H V , |V H , | + −, or | − + . Then Eve intercepts the encoding-information photons from Alice to Bob and steals the state information according to her measurement outcomes on them. By this method, Eve can steal one bit from two-bit secret information, although her action will be discovered by Bob in the second security check later. To overcome this problem, the protocol should adopt a destructive parity analysis in the first security check, which exploits two nonorthogonal sets of orthogonal basis measurements to inspect Eve’s action. Explicitly, if the preceding parity analysis applies the basis of {|H , |V }, the later parity analysis will exploit the basis of {|+, |−}, and vice versa. For the instance of the security check using the first selection order, after Eve measures the probe-coherent state, the photons originally constituting the Bell state collapse into the state of |H V or the state of |V H . After the two legitimate participants perform the later parity analysis on them along the basis of {|+, |−}, it will give an incorrect result, embodying the absence of entanglement of the two photons, and thus they find the trace of the eavesdropper.

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As for the three modes of Trojan-horse photon attacks, the legitimate participants can adopt the corresponding countermeasures to overcome them, which are described in the previous paragraphs in Sect. 2.

5.

Summary

Acknowledgements The authors thank Bing He for helpful suggestions. This study was supported by the National Natural Science Foundation of China (Grant Nos. 11305016, 61301133, 11271055), the Program of the Educational Office of Liaoning Province of China (Grant No. L2013425), and the Program for Liaoning Excellent Talents in University of China (LNET Grant No. LJQ2014124).

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With regard to the transmission of photons, the noise in the channel should be considered in the practical application. In this protocol, the transmission circuit against collective noise is adopted, by which the transmitted photons can be sent to the target place without influence resulting from collective noise. The employment of the nondemolition parity analysis facilitates the security check, the distinction of four Bell states, and the receipt of the secret information from Alice. The nondemolition parity analysis means that the states that undergo measurement are not destroyed, and, as a consequence, these photon states can be exploited for other tasks of quantum information processing, which will save a lot of physics resources and enhance the efficiency of the communication protocol. The nondemolition parity analysis can be realized with cross-Kerr nonlinearities and is applied in many tasks of quantum information processing [35,58,65]. The application of the high-intensity probe-coherent state enhances the efficiency, and reduces the harmful influence of imperfect detectors and its intensity degradation. Moreover, the experimental requirement is simplified due to the employment of a minus phase shifter outside the Kerr media, rather than a negative phase shift in the Kerr media, which is a technical difficulty in the experimental realization. To sum up, in the collective noise channel, a quantum secure direct communication protocol exploiting polarization-entangled Bell states with the nondemolition parity analysis based on weak cross-Kerr nonlinearities is presented. To prevent destruction stepping from the collective noise, a transmission circuit against collective noise is adopted to enable photons encoded with secret information to transmit between two legitimate participants free of its disadvantageous influence. By means of the nondemolition parity analysis and photon-number measurement with the help of cross-Kerr nonlinearities, the efficiency of transmission of the secret information is enhanced and a lot of resources are saved.

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