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Quantum catalysis-based discrete modulation continuous variable quantum key distribution with eight states
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Quantum catalysis-based discrete modulation continuous variable quantum key distribution with eight states
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Ying Guo
a, b
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, Jianzhi Ding , Yun Mao , Wei Ye , Qin Liao
a, c
, Duan Huang
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School of Computer, Central South University, Changsha 410083, China b School of Automation, Central South University, Changsha 410083, China c School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore
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Article history: Received 9 December 2019 Received in revised form 28 January 2020 Accepted 13 February 2020 Available online xxxx Communicated by M.G.A. Paris Keywords: Quantum key distribution Continuous variable Photon catalysis Discrete modulation
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How to lengthen the maximum transmission of continuous variable quantum key distribution (CVQKD) has been a notorious hard problem in quantum communications. Here, we propose a simple solution to this problem, i.e., quantum catalyzing CVQKD for discrete modulation with eight states. The quantum catalysis, which can facilitate the conversion of the target ensemble, is used for not only tolerating more excess noise but also lengthening the maximum transmission distance. Security analysis shows that the zero-photon catalysis (ZPC), which is actually seen as a noiseless attenuation can be used as an elegant candidate for the performance improvement of discrete modulation (DM)-CVQKD. The numerical simulations show the ZPC-involved DM-CVQKD protocol outperforms the original DM-CVQKD in terms of maximum transmission distance as well as tolerable noise. Moreover, the ZPC-involved DM-CVQKD protocol can tolerate lower reconciliation efficiency and allow the lower detection efficiency to achieve the same performance. © 2020 Elsevier B.V. All rights reserved.
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1. Introduction
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Quantum key distribution (QKD) [1–5] is the most promising quantum information technology and it promotes the first quantum information task to reach the mature technology level. The aim of QKD systems is to share an unconditionally secure key between legal parties (Alice and Bob) due to the guarantee of fundamental laws of quantum mechanics [6,7]. At present, according to the different implementations of quantum key distribution, there are two traditional approaches, i.e. discrete variable quantum key distribution (DVQKD) [4,8,9] and continuous variable quantum key distribution (CVQKD) [10–13]. Compared with the former, the latter eliminates the difficult problem of preparing single photon, and thus promises higher secret key rates with the assistants of using homodyne or heterodyne detectors [14]. Particularly, the Gaussian modulation (GM) CVQKD using the coherent state was proposed theoretically and proven experimentally [15,16], which makes them more compelling and practical. The traditional CVQKD systems, however, have a limitation in the long distance safety communications. To eliminate this drawback, one of feasible approaches is to apply quantum operations, involving the non-Gaussian operations
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E-mail address: [email protected] (Y. Guo). https://doi.org/10.1016/j.physleta.2020.126340 0375-9601/© 2020 Elsevier B.V. All rights reserved.
and the linear amplifiers. For instance, the noiseless linear amplifiers [17–19] have been proposed to extend the transmission distance of both point to point CV-QKD and the measurement-deviceindependent CVQKD [20–23]. In addition, the photon-subtracted operation [24–27], which can be simulated by non-Gaussian postprocessing, has been theoretically proven to improve the performance of GM-CVQKD systems effectively, but its success probability is lower than 0.25 even though the modulation variance is optimized, which may result in the limited performance improvement of CVQKD. In order to solve this problem, the quantum catalysis (QC) [28,29], which is a feasible and successful scheme, has presented the better performances of GM-CVQKD in contrast to the photon-subtracted operation. Due to the fact that in the GM protocols, the reconciliation efficiency is quite low, A. Leverrier provided an initial discrete modulation (DM) scheme and proved its unconditional security [30]. Subsequently, an eight-state modulation (ESM) scheme was proposed in CVQKD [31], theoretically realizing the long-distance safe communications of more than 100 kilometers. In this paper, we propose the QC-based DM-CVQKD with eight states, which can not only solve the problem of low reconciliation efficiency of GM-CVQKD, but also result in the performance improvement of the traditional DM-CVQKD. The DM-CVQKD utilizes the QC scheme to prevent the loss of information effectively before
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Fig. 1. (a) EB scheme of the DM-CVQKD protocol. (b) EB scheme of the QC-based DM-CVQKD protocol, where the dotted line box represents zero-photon catalysis operation. Het: heterodyne detection; Hom: homodyne detection; B(T): BS operator with transmittance T; T c : transmission effciency; ε : excess noise; η : detection efficiency; v el : electronic noise; ZPS: zero-photon catalysis.
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the sender sends the modulated coherent state to the receiver, resulting in the performance improvement in terms of the maximal transmission distance. For the security analysis, we show the success probability of the ZPC. Moreover, we consider its maximum transmission distance and tolerable noise. We investigate its tolerance for both reconciliation efficiency and detection efficiency. We find that the ZPC-involved DM-CVQKD scheme has better performance when comparing with the traditional DM-CVQKD protocol. The structure of this paper is as follows. In section 2, we propose the QC-based DM-CVQKD system with eight states. In section 3, we show the performance simulation results of the proposed protocol and compare the performance improvement with traditional DM-CVQKD protocol. Finally, our main conclusions are given in section 4.
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2. The QC-based DM-CVQKD protocol
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In this section, we mainly describe the proposed QC-based scheme for DM-CVQKD. In order to explain our proposed protocol well, it is necessary for us to briefly introduce the original eightstate protocol.
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2.1. The DM-CVQKD with eight states
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The eight-state protocol was first proposed by ref. [31]. Here we briefly introduce the preparationand-measurement (PM) and entangled-based (EB) versions. The specific derivation process is shown in ref. [31]. In the PM model, Alice randomly selects one of the eight coherent states αk8 = α e ikπ /4 , k ∈ {0, 1, . . . 7}, with α being a real positive number, and then sends to Bob through the quantum channel. Bob takes homodyne or heterodyne detection to obtain his own data. Subsequently, Alice and Bob need to perform the post-processing, such as reconciliation algorithms and privacy amplification, before distilling secret keys [14]. Although the PM scheme is easy to be implemented in practice, it is not conducive to security analysis. In fact, the PM scheme is equivalent to the EB scheme, which is usually used for security proofs [32,33]. As shown in Fig. 1(a), Alice prepares a bipartite entangled state |8 ,
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| 8 =
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|ψk αk8 ,
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where
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i (4k+1)mπ 4
|φm ,
(2)
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|φm = √ λm
|8n + m . √ (8n + m)! n =0
Z 8 σz YI
XI Z 8 σz
(3)
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λ0,4 = e−α 4 1
λ1,5 = e−α
2
σz = diag (1, −1), X = Y = 1 + 2α , and
sinh(α 2 ) + sin(α 2 ) ±
√ α2 α2 ± 2 sin( √ ) cosh( √ )
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√
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2 cos( √ ) sinh( √ ) 2 2
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√ α2 α2 ± 2 sin( √ ) cosh( √ ) , 2
where V A = 2α 2 represents modulation variance.
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√ 1 α2 α2 2 λ3,7 = e−α sinh(α 2 ) − sin(α 2 ) ∓ 2 cos( √ ) sinh( √ ) 4 2 2 2
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(4)
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λk−1 √ , with the parameters λk
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where I = diag (1, 1), Z 8 = 2α 2
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For the security analysis, it is necessary to give the covariance matrix of the state |8 given by
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are orthogonal non-Gaussian states, and the state |φm is given by
8 =
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(5)
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Fig. 2. Comparison of the correlations Z G (GM) and Z 8 (DM), where Z G = with V = V A + 1.
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After the generation of the two-mode entangled state |8 , half of the one in mode A is retained to make projection measurements, and then another half of the one is sent to Bob through the quantum channel. Bob measures the received state by using homodyne or heterodyne detection. Finally, Alice and Bob obtain the secret keys after post-processing. It should be noted that the DM scheme is different from the GM one because the correlation term reflected by the covariance matrix is not the maximum entanglement, that is, the existence relationship Z G ≥ Z 8 . Therefore, it is difficult to be derived since the prepared state |8 is a nonGaussian state. In order to solve this problem, we compare the covariance correlated terms between the DM and GM schemes, as shown in Fig. 2. We find that it is difficult to distinguish Z 8 and Z G when V A is small enough (V A ≤ 0.5), which means that the mutual information of the DM scheme between Bob and Eve is very similar to that of the GM scheme. Therefore, we can get the security bounds for the DM-CVQKD protocol. 2.2. The QC-based DM-CVQKD with eight states
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In this part we introduce the scheme of the QC-based DMCVQKD. Because of the difficulty of single photon preparation, here we only consider zero photon catalysis (ZPC). The detailed description of the scheme’s PM model is as follows:
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step 1: Alice randomly selects one of the eight coherent states with equal probability, and then injects the selected quan tum state αk8 into one of the input ports of the BS with transmittance T, while injecting an zero-photon Fock state |0 in auxiliary C at the another input port. We detect |0 at the corresponding output port at the same time, and catalyzed quantum state |ψout can be obtained from another output port, which is the so-called ZPC operation [28,34], as shown in the dotted line box in Fig. 1(b). step 2: The catalyzed quantum state |ψout is transmitted to Bob via the Eve-controlled channel which is characterized by ε and transmission efficiency T c = 10−μL /10 , where μ = 0.2 dB/km is the loss coefficient for the standard optical fibers and L is the length of the fiber optics, and Eve may eavesdrop on the communication process. step 3: Bob performs homodyne or heterodyne measurement on the received quantum state, where Bob’s detection efficiency is modeled by a beam splitter with transmittance
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η. It is inevitable that electronic noise v el will be intro-
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Fig. 3. The success probability of implementing ZPC for different transmittance T and modulation variance V A .
duced during Bob’s detection process. So the noise introduced by homodyne (heterodyne) detection is defined as X hom = [(1 − η) + v el ] /η( X het = [1 + (1 − η) + 2v el ] /η). step 4: Bob shares the same key string with Alice through postprocessing. The above process is equivalent to the EB scheme, as shown in Fig. 1(b). Alice detects one half of the quantum state by heterodyne detections while another half is sent to Bob through quantum channel controlled by Eve after performing the ZPC operation. The above ZPC process can often be equivalent to operator O 0 given by
O 0 = T r [B ( T ) |0 C 0|] √ =: exp T − 1 b† b : =
√ b† b
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√
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represents a normally ordering of operator. Therefore, for any input state |ϕ in in mode B, the output state |ψout can be expressed as
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(8)
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with the parameters
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(9)
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So the covariance matrix of state A B 1 is derived as
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The success probability of implementing ZPC is shown in Fig. 3. In our proposed protocol, the role of the ZPC operation is to promote the transition between modes B and B 1 , thus giving birth to the state
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where P 0 represents the success probability for achieving ZPC operation given by
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Fig. 4. (a) The relationship between the secret key rate and the transmission distance of the ZPC-involved DM-CVQKD protocol. (b) Corresponding to (a), the transmittance T varies with the transmission distance. Here β = 0.9, V A = 0.5, v el = 0.05 and η = 0.6.
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3. Security analysis
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The derivation of the asymptotic key rate is given in the Appendix A. As we all know, the covariance matrix after the quantum state passes through the channel is the key to security analysis, so we first give the covariance matrix of the original eight-state protocol and the ZPC-involved DM-CVQKD protocol in order to analyze the performance improvement of the proposed protocol on the traditional DM-CVQKD. In the original eight-state protocol, when the state is emitted by Alice through the quantum channel, the covariance matrix A B 2 is expressed as
⎡
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T c 1 + 2α 2 + Xline I
In the ZPC-involved DM-CVQKD protocol, the covariance matrix
A B is expressed as
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Fig. 5. The maximal tolerable excess noise of the ZPC-involved DM-CVQKD. Here β = 0.9, V A = 0.5, η = 0.6 and v el = 0.05.
σz ⎥
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(13) where Xline represents the total channel-added noise is defined as Xline = 1/ T c + ε − 1. It can be seen from the comparison between Eq. (12) and Eq. (13) that the ZPC operation, which is actually seen as a noiseless attenuation, only affects the performance of the protocol by the transmittance T of the beam splitter. Therefore, the optimization of T plays an extremely important role in the performance analysis process. We first analyze the numerical simulation results of the ZPC operation success probability, as shown in Fig. 3. We find that ZPC
can maintain the success probability greater than 0.75 when the modulation variance is less than 0.5, which indicates that it is a feasible solution to improve the performance of the DM-CVQKD protocol by using ZPC operation. In order to better illustrate the achievability and superiority of ZPC operation, here we take the smallest success probability to perform subsequent performance analysis, that is, V A = 0.5. In Fig. 4(a), it shows that the relationship between the secret key rate and the transmission distance of the ZPC-involved DM-CVQKD. Intuitively, the DM-CVQKD can lengthen transmission distances by applying ZPC operation. Note that the solid line represents the homodyne detection and the dotted line represents the heterodyne detection. From the comparison we find that the performances of homodyne detection and heterodyne detection are basically same. Therefore, we use homodyne detection for the subsequent security analysis. ZPC operation can be regarded as a noiseless attenuator, so its performance improvement depends on the value of T. If T = 1, there is no catalytic effect, and the proposed protocol is equivalent to the original eight-state protocol. As shown in Fig. 4(b), we obtained the maximum key rate by optimizing the transmittance T with the ZPC operation, which shows the change curve of transmittance with transmission distance. From the figure, we can see that when L > 65.85, T = 1, the catalytic
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Fig. 6. (a) The relationship between the secret key rate and the reconciliation efficiency of the ZPC-involved DM-CVQKD protocol. (b) Corresponding to (a), the transmittance T varies with the reconciliation efficiency. Here V A = 0.5, η = 0.6, v el = 0.05 and = 0.005.
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Fig. 7. (a) Comparison of secret key rates of the DM-CVQKD protocol and the ZPC-involved DM-CVQKD under different detection efficiency and electronic noise for L = 50 km. (b) Corresponding to (a), the transmittance T varies with the detection efficiency and electronic noise. (c) Comparison of secret key rates of the DM-CVQKD protocol and the ZPC-involved DM-CVQKD under different detection efficiency and electronic noise for L = 110 km. (d) Corresponding to (c), the transmittance T varies with the detection efficiency and electronic noise. Here β = 0.9, V A = 0.5 and = 0.005.
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effect occurs in the case of = 0.01, but because the T value is close to 1, the catalytic effect is not obvious, as shown in the upper subgraph in Fig. 4 (a). There always exists excess noise in the practical quantum channel, and hence we have to show the impact of excess noise on the performance of the ZPC-involved DM-CVQKD protocol. Fig. 4(a) also demonstrates intuitively the change of the secret key rate of the DM-CVQKD with the tunable excess noises ( = 0.005 and = 0.01). Obviously, with the increase of excess noise, the trans-
mission distance is shortened significantly. Therefore, in the practical experiment, if long-distance quantum key distribution is to be realized, the excess noise of the system must be stably controlled within a small range. Fig. 5 shows the maximum excess noise that the protocol can tolerate. It can be seen over a long distance, the ZPC-involved DM-CVQKD protocol exceeds the original DM-CVQKD protocol in terms of maximum tolerable excess noise. As known, reconciliation efficiency is an important factor affecting the secret key rate. In Fig. 6, we show the relationship between
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Fig. 8. (a) The relationship between the secret key rate and the detection efficiency. (b) The relationship between the secret key rate and the electronic noise. Here β = 0.9, V A = 0.5 and = 0.005.
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the secret key rate and the reconciliation efficiency when optimizing over T . We find the ZPC-involved DM-CVQKD protocol can tolerate lower reconciliation efficiency. When L = 50 km, the traditional eight-state protocol can tolerate the lowest reconciliation efficiency of 0.617, and the ZPC-involved DM-CVQKD protocol can tolerate 0.553. When L = 80 km, the traditional eight-state protocol can tolerate 0.741, and the ZPC-involved DM-CVQKD protocol can tolerate 0.675, which further indicates that the ZPC operation has a positive effect on the performance of the DM-CVQKD protocol. Moreover, the imperfection of coherent detections is also an important factor affecting the performance of the DM-CVQKD protocols. Fig. 7 gives a three-dimensional analysis chart, which aims to analyze the combined effect of the two on the secret key rate. From Fig. 4, it can be seen that when L > 65.85 km, the catalysis starts to occur, so in order to analyze the effect of ZPC on the eight-state protocol, we take the values of L < 65.85 km and L > 65.85 km to analyze, respectively. When L = 50 km, there is no catalytic effect (T = 1, as shown in Fig. 7(b)). Therefore, the original DM-CVQKD protocol and the ZPC-involved DM-CVQKD protocol demonstrate the similar performance. As shown in Fig. 7(a), simulation charts of the above-mentioned two protocols coincide exactly. Whereas for L = 110 km, the catalytic effect occurs (the variation of T with the detection efficiency and electronic noise is shown in Fig. 7(d)), but because the degree of improvement is not particularly obvious, so we give two-dimensional simulation charts of detection efficiency-secret key rate (v el = 0.05) and electronic noise-secret key rate (η = 0.6), as shown in Fig. 8. From Fig. 8(a), we can see that to achieve the same secret key rate, the ZPC-involved DM-CVQKD protocol can tolerate lower detection efficiency than the traditional eight-state protocol. For example, we can see that when the secret key rate 1.71 × 10−4 is achieved, the detection efficiency of the traditional eight-state protocol needs to be 0.877, while the ZPC-involved DM-CVQKD protocol only needs 0.688 with other parameters fixed. Fig. 8(b) shows that with the secret key rate of 1.2 × 10−4 , the electronic noise that can be tolerated by the traditional eight-state protocol is 0.176, while the ZPC-involved DM-CVQKD protocol is 0.359. This shows that through ZPC operation, while achieving the same performance, it can slightly reduce the requirements for detection equipment and reduce costs.
noiseless attenuation and can be implemented with current experimental technologies. We analyze the secret key rate of the ZPCinvolved DM-CVQKD against the Gaussian collective attack. The numerical simulations show the ZPC-involved DM-CVQKD protocol has the advantage of increasing maximum transmission distance and tolerable excess noise, compared with the original DM-CVQKD protocol. It can also tolerate lower reconciliation efficiency. In order to further illustrate the superiority of the ZPC-involved scheme, we compare the effects of Bob’s detection efficiency and electronic noise on performance, and find the ZPC-involved DM-CVQKD protocol allows lower detection efficiency and higher electronic noise for achieving the same performance. It is worth noting that it is important to control the excess noise in practical implementations. If the long-distance quantum key distribution is to be realized, the excess noise of the system should be stably controlled to a small range since it has a significant impact on the performance of the protocol.
4. Conclusion
where I A B and χ E B correspond to mutual information between Alice-Bob and Eve-Bob, and β is the reconciliation efficiency. The mutual information between Alice and Bob I A B for homodyne detection is calculated by
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgements
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 61572529, 61871407).
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Appendix A. The derivation of asymptotic secret key rate
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We assume that Eve performs Gaussian collective attack to achieve the best results, and the information obtained is limited to the Holevo range χ E B . It should be noted that state | A B 1 can still be regarded as a Gaussian state through the ZPC so that we can directly calculate the secret key rate by using the results of the conventional Gaussian CVQKD protocol. Consequently, the secret key rate can be calculated as
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K = P 0 (β I A B − χ E B ) ,
(14)
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Declaration of competing interest
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We have proposed a method of enhancing the performance of DM-CVQKD by using ZPC operation, which is actually seen as a
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I AB =
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39 40
(16)
S ( A B ) = G [(λ1 − 1)/2] + G [(λ2 − 1)/2] ,
(17)
with G (x) = (x + 1) log (x + 1) − x log x and λ1,2 in the equation can be calculated by
λ1,2 =
1
( ±
2
2
− 4D 2 ),
(18) X , b
with D = ab − c and√ = a + b − 2c , where a = = T c (Y + Xline ) and c = T c Z 8 . The second term S ( A / B ) is a function of the symplectic eigenvalues λ3,4 of the convariance matrix of Alice mode after Bob performing homodyne (heterodyne) detection, which is given by 2
2
2
2
S ( A / B ) = G [(λ3 − 1)/2] + G [(λ4 − 1)/2] ,
(19)
where λ3,4 can be calculated by
λ3,4 =
1
(A ±
2
A 2 − 4B ).
(20)
For homodyne detection we have
A hom =
X hom + T ( V + Xline ) + V D
37 38
(15)
.
where the first term S ( A B ) is a function of the symplectic eigenvalues λ1,2 of A B 2 , which is given by
33 34
1 + X tot
= S ( A B ) − S ( A/B ) ,
18 19
V + X tot
χB E = S ( E ) − S ( E / B )
10 12
2
log2
X tot represents the total noise which can be calculated as X tot = Xline + X hom(het ) / T c . Note that in the case of heterodyne detection, the mutual information I A B is twice the Eq. (15). The Holevo bound is given by:
9 11
1
B
hom
=D
T ( V + X tot ) D X hom + V T ( V + X tot )
,
.
(21) (22)
For heterodyne detection we have
41 42 43
A het =
2 X het + 2 X het [V D + T ( V + Xline )] + D 2 + 1 + 2T Z 82
T 2 ( V + X tot )2
44
(23)
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
,
B het =
( D X het + V )2 T 2(V
+ X tot )
2
.
(24)
References [1] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74 (1) (2002) 145. [2] V. Scarani, H. Bechmann-Pasquinucci, N.J. Cerf, M. Dušek, N. Lütkenhaus, M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81 (3) (2009) 1301. [3] M.A. Nielsen, I. Chuang, Quantum computation and quantum information (2002). [4] A.K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67 (6) (1991) 661. [5] H.K. Lo, M. Curty, K. Tamaki, Secure quantum key distribution, Nat. Photonics 8 (8) (2014) 595. [6] J.Y. Bang, M.S. Berger, Quantum mechanics and the generalized uncertainty principle, Phys. Rev. D 74 (12) (2006) 125012. [7] W.K. Wootters, W.H. Zurek, A single quantum cannot be cloned, Nature 299 (5886) (1982) 802.
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[8] C.H. Bennett, F. Bessette, G. Brassard, L. Salvail, J. Smolin, Experimental quantum cryptography, J. Cryptol. 5 (1) (1992) 3–28. [9] P.W. Shor, J. Preskill, Simple proof of security of the BB84 quantum key distribution protocol, Phys. Rev. Lett. 85 (2) (2000) 441. [10] S.L. Braunstein, P. Van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77 (2) (2005) 513. [11] C. Weedbrook, S. Pirandola, R. García-Patrón, N.J. Cerf, T.C. Ralph, J.H. Shapiro, S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84 (2) (2012) 621. [12] C. Weedbrook, Continuous-variable quantum key distribution with entanglement in the middle, Phys. Rev. A 87 (2) (2013) 022308. [13] R. García-Patrón, N.J. Cerf, Unconditional optimality of Gaussian attacks against continuous-variable quantum key distribution, Phys. Rev. Lett. 97 (19) (2006) 190503. [14] F. Laudenbach, C. Pacher, C.H.F. Fung, A. Poppe, M. Peev, B. Schrenk, H. Hübel, Continuous-variable quantum key distribution with Gaussian modulation— the theory of practical implementations, Adv. Quantum Technol. 1 (1) (2018) 1800011. [15] P. Jouguet, S. Kunz-Jacques, A. Leverrier, Long-distance continuous-variable quantum key distribution with a Gaussian modulation, Phys. Rev. A 84 (6) (2011) 062317. [16] F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N.J. Cerf, P. Grangier, Quantum key distribution using Gaussian-modulated coherent states, Nature 421 (6920) (2003) 238. [17] R. Blandino, A. Leverrier, M. Barbieri, J. Etesse, P. Grangier, R. Tualle-Brouri, Improving the maximum transmission distance of continuous-variable quantum key distribution using a noiseless amplifier, Phys. Rev. A 86 (1) (2012) 012327. [18] B. Xu, C. Tang, H. Chen, W. Zhang, F. Zhu, Improving the maximum transmission distance of four-state continuous-variable quantum key distribution by using a noiseless linear amplifier, Phys. Rev. A 87 (6) (2013) 062311. [19] F.L. Yang, R.H. Shi, Y. Guo, J.J. Shi, G.H. Zeng, Continuous-variable quantum key distribution under the local oscillator intensity attack with noiseless linear amplifier, Quantum Inf. Process. 14 (8) (2015) 3041–3056. [20] Z. Li, Y.C. Zhang, F. Xu, X. Peng, H. Guo, Continuous-variable measurementdevice-independent quantum key distribution, Phys. Rev. A 89 (5) (2014) 052301. [21] Y.C. Zhang, Z. Li, S. Yu, W. Gu, X. Peng, H. Guo, Continuous-variable measurement-device-independent quantum key distribution using squeezed states, Phys. Rev. A 90 (5) (2014) 052325. [22] X.C. Ma, S.H. Sun, M.S. Jiang, M. Gui, L.M. Liang, Gaussian-modulated coherentstate measurement-device-independent quantum key distribution, Phys. Rev. A 89 (4) (2014) 042335. [23] H.X. Ma, P. Huang, D.Y. Bai, S.Y. Wang, W.S. Bao, G.H. Zeng, Continuous-variable measurement-device-independent quantum key distribution with photon subtraction, Phys. Rev. A 97 (4) (2018) 042329. [24] Z. Li, Y. Zhang, X. Wang, B. Xu, X. Peng, H. Guo, Non-Gaussian postselection and virtual photon subtraction in continuous-variable quantum key distribution, Phys. Rev. A 93 (1) (2016) 012310. [25] P. Huang, G. He, J. Fang, G. Zeng, Performance improvement of continuousvariable quantum key distribution via photon subtraction, Phys. Rev. A 87 (1) (2013) 012317. [26] Y. Guo, Q. Liao, Y. Wang, D. Huang, P. Huang, G. Zeng, Performance improvement of continuous-variable quantum key distribution with an entangled source in the middle via photon subtraction, Phys. Rev. A 95 (3) (2017) 032304. [27] Q. Liao, Y. Guo, D. Huang, P. Huang, G. Zeng, Long-distance continuous-variable quantum key distribution using non-Gaussian state-discrimination detection, New J. Phys. 20 (2) (2018) 023015. [28] Y. Guo, W. Ye, H. Zhong, Q. Liao, Continuous-variable quantum key distribution with non-Gaussian quantum catalysis, Phys. Rev. A 99 (3) (2019) 032327. [29] W. Ye, H. Zhong, Q. Liao, D. Huang, L. Hu, Y. Guo, Improvement of selfreferenced continuous-variable quantum key distribution with quantum photon catalysis, Opt. Express 27 (12) (2019) 17186–17198. [30] A. Leverrier, P. Grangier, Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation, Phys. Rev. Lett. 102 (18) (2009) 180504. [31] A. Becir, F.A.A. El-Orany, M.R.B. Wahiddin, Continuous-variable quantum key distribution protocols with eight-state discrete modulation, Int. J. Quantum Inf. 10 (01) (2012) 1250004. [32] F. Grosshans, Collective attacks and unconditional security in continuous variable quantum key distribution, Phys. Rev. Lett. 94 (2) (2005) 020504. [33] F. Grosshans, N.J. Cerf, J. Wenger, R. Tualle-Brouri, P. Grangier, Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables, arXiv preprint, arXiv:quant-ph/0306141, 2003. [34] L. Hu, Z. Liao, M.S. Zubairy, Continuous-variable entanglement via multiphoton catalysis, Phys. Rev. A 95 (1) (2017) 012310.
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Quantum catalysis-based discrete modulation continuous variable quantum key distribution with eight states
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Ying Guo
a, b
a
b
a
, Jianzhi Ding , Yun Mao , Wei Ye , Qin Liao
a, c
, Duan Huang
a
a
School of Computer, Central South University, Changsha 410083, China b School of Automation, Central South University, Changsha 410083, China c School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Singapore
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a r t i c l e
i n f o
Article history: Received 9 December 2019 Received in revised form 28 January 2020 Accepted 13 February 2020 Available online xxxx Communicated by M.G.A. Paris Keywords: Quantum key distribution Continuous variable Photon catalysis Discrete modulation
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a b s t r a c t
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How to lengthen the maximum transmission of continuous variable quantum key distribution (CVQKD) has been a notorious hard problem in quantum communications. Here, we propose a simple solution to this problem, i.e., quantum catalyzing CVQKD for discrete modulation with eight states. The quantum catalysis, which can facilitate the conversion of the target ensemble, is used for not only tolerating more excess noise but also lengthening the maximum transmission distance. Security analysis shows that the zero-photon catalysis (ZPC), which is actually seen as a noiseless attenuation can be used as an elegant candidate for the performance improvement of discrete modulation (DM)-CVQKD. The numerical simulations show the ZPC-involved DM-CVQKD protocol outperforms the original DM-CVQKD in terms of maximum transmission distance as well as tolerable noise. Moreover, the ZPC-involved DM-CVQKD protocol can tolerate lower reconciliation efficiency and allow the lower detection efficiency to achieve the same performance. © 2020 Elsevier B.V. All rights reserved.
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1. Introduction
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Quantum key distribution (QKD) [1–5] is the most promising quantum information technology and it promotes the first quantum information task to reach the mature technology level. The aim of QKD systems is to share an unconditionally secure key between legal parties (Alice and Bob) due to the guarantee of fundamental laws of quantum mechanics [6,7]. At present, according to the different implementations of quantum key distribution, there are two traditional approaches, i.e. discrete variable quantum key distribution (DVQKD) [4,8,9] and continuous variable quantum key distribution (CVQKD) [10–13]. Compared with the former, the latter eliminates the difficult problem of preparing single photon, and thus promises higher secret key rates with the assistants of using homodyne or heterodyne detectors [14]. Particularly, the Gaussian modulation (GM) CVQKD using the coherent state was proposed theoretically and proven experimentally [15,16], which makes them more compelling and practical. The traditional CVQKD systems, however, have a limitation in the long distance safety communications. To eliminate this drawback, one of feasible approaches is to apply quantum operations, involving the non-Gaussian operations
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E-mail address: [email protected] (Y. Guo). https://doi.org/10.1016/j.physleta.2020.126340 0375-9601/© 2020 Elsevier B.V. All rights reserved.
and the linear amplifiers. For instance, the noiseless linear amplifiers [17–19] have been proposed to extend the transmission distance of both point to point CV-QKD and the measurement-deviceindependent CVQKD [20–23]. In addition, the photon-subtracted operation [24–27], which can be simulated by non-Gaussian postprocessing, has been theoretically proven to improve the performance of GM-CVQKD systems effectively, but its success probability is lower than 0.25 even though the modulation variance is optimized, which may result in the limited performance improvement of CVQKD. In order to solve this problem, the quantum catalysis (QC) [28,29], which is a feasible and successful scheme, has presented the better performances of GM-CVQKD in contrast to the photon-subtracted operation. Due to the fact that in the GM protocols, the reconciliation efficiency is quite low, A. Leverrier provided an initial discrete modulation (DM) scheme and proved its unconditional security [30]. Subsequently, an eight-state modulation (ESM) scheme was proposed in CVQKD [31], theoretically realizing the long-distance safe communications of more than 100 kilometers. In this paper, we propose the QC-based DM-CVQKD with eight states, which can not only solve the problem of low reconciliation efficiency of GM-CVQKD, but also result in the performance improvement of the traditional DM-CVQKD. The DM-CVQKD utilizes the QC scheme to prevent the loss of information effectively before
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Fig. 1. (a) EB scheme of the DM-CVQKD protocol. (b) EB scheme of the QC-based DM-CVQKD protocol, where the dotted line box represents zero-photon catalysis operation. Het: heterodyne detection; Hom: homodyne detection; B(T): BS operator with transmittance T; T c : transmission effciency; ε : excess noise; η : detection efficiency; v el : electronic noise; ZPS: zero-photon catalysis.
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the sender sends the modulated coherent state to the receiver, resulting in the performance improvement in terms of the maximal transmission distance. For the security analysis, we show the success probability of the ZPC. Moreover, we consider its maximum transmission distance and tolerable noise. We investigate its tolerance for both reconciliation efficiency and detection efficiency. We find that the ZPC-involved DM-CVQKD scheme has better performance when comparing with the traditional DM-CVQKD protocol. The structure of this paper is as follows. In section 2, we propose the QC-based DM-CVQKD system with eight states. In section 3, we show the performance simulation results of the proposed protocol and compare the performance improvement with traditional DM-CVQKD protocol. Finally, our main conclusions are given in section 4.
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2. The QC-based DM-CVQKD protocol
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In this section, we mainly describe the proposed QC-based scheme for DM-CVQKD. In order to explain our proposed protocol well, it is necessary for us to briefly introduce the original eightstate protocol.
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2.1. The DM-CVQKD with eight states
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
88 89 90
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The eight-state protocol was first proposed by ref. [31]. Here we briefly introduce the preparationand-measurement (PM) and entangled-based (EB) versions. The specific derivation process is shown in ref. [31]. In the PM model, Alice randomly selects one of the eight coherent states αk8 = α e ikπ /4 , k ∈ {0, 1, . . . 7}, with α being a real positive number, and then sends to Bob through the quantum channel. Bob takes homodyne or heterodyne detection to obtain his own data. Subsequently, Alice and Bob need to perform the post-processing, such as reconciliation algorithms and privacy amplification, before distilling secret keys [14]. Although the PM scheme is easy to be implemented in practice, it is not conducive to security analysis. In fact, the PM scheme is equivalent to the EB scheme, which is usually used for security proofs [32,33]. As shown in Fig. 1(a), Alice prepares a bipartite entangled state |8 ,
7
| 8 =
4
91
|ψk αk8 ,
1
(1)
k =0
93 94
where
95
7 1
|ψk =
2
96
e
i (4k+1)mπ 4
|φm ,
(2)
m =0
2 ∞ e −α /2
|φm = √ λm
|8n + m . √ (8n + m)! n =0
Z 8 σz YI
XI Z 8 σz
(3)
7
k=0
1
λ0,4 = e−α 4 1
λ1,5 = e−α
2
σz = diag (1, −1), X = Y = 1 + 2α , and
sinh(α 2 ) + sin(α 2 ) ±
√ α2 α2 ± 2 sin( √ ) cosh( √ )
2
√
2
2
2
2
α
2
106 108 110 111 113
α
114 115 116 117
2
2 cos( √ ) sinh( √ ) 2 2
2
1 α2 α2 2 λ2,6 = e−α sinh(α 2 ) − sin(α 2 ) ± 2 sin( √ ) sinh( √ ) 4 2 2
√ α2 α2 ± 2 sin( √ ) cosh( √ ) , 2
where V A = 2α 2 represents modulation variance.
118 119 120 121
√ 1 α2 α2 2 λ3,7 = e−α sinh(α 2 ) − sin(α 2 ) ∓ 2 cos( √ ) sinh( √ ) 4 2 2 2
105
112
α α cosh(α 2 ) + cos(α 2 ) ± 2 cos( √ ) cosh( √ )
4
103
109 2
3/ 2
2
102
107
(4)
,
λk−1 √ , with the parameters λk
100
104
where I = diag (1, 1), Z 8 = 2α 2
98
101
α 8n+m
For the security analysis, it is necessary to give the covariance matrix of the state |8 given by
97 99
are orthogonal non-Gaussian states, and the state |φm is given by
8 =
92
(5)
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Fig. 2. Comparison of the correlations Z G (GM) and Z 8 (DM), where Z G = with V = V A + 1.
√
V2 −1
After the generation of the two-mode entangled state |8 , half of the one in mode A is retained to make projection measurements, and then another half of the one is sent to Bob through the quantum channel. Bob measures the received state by using homodyne or heterodyne detection. Finally, Alice and Bob obtain the secret keys after post-processing. It should be noted that the DM scheme is different from the GM one because the correlation term reflected by the covariance matrix is not the maximum entanglement, that is, the existence relationship Z G ≥ Z 8 . Therefore, it is difficult to be derived since the prepared state |8 is a nonGaussian state. In order to solve this problem, we compare the covariance correlated terms between the DM and GM schemes, as shown in Fig. 2. We find that it is difficult to distinguish Z 8 and Z G when V A is small enough (V A ≤ 0.5), which means that the mutual information of the DM scheme between Bob and Eve is very similar to that of the GM scheme. Therefore, we can get the security bounds for the DM-CVQKD protocol. 2.2. The QC-based DM-CVQKD with eight states
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In this part we introduce the scheme of the QC-based DMCVQKD. Because of the difficulty of single photon preparation, here we only consider zero photon catalysis (ZPC). The detailed description of the scheme’s PM model is as follows:
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step 1: Alice randomly selects one of the eight coherent states with equal probability, and then injects the selected quan tum state αk8 into one of the input ports of the BS with transmittance T, while injecting an zero-photon Fock state |0 in auxiliary C at the another input port. We detect |0 at the corresponding output port at the same time, and catalyzed quantum state |ψout can be obtained from another output port, which is the so-called ZPC operation [28,34], as shown in the dotted line box in Fig. 1(b). step 2: The catalyzed quantum state |ψout is transmitted to Bob via the Eve-controlled channel which is characterized by ε and transmission efficiency T c = 10−μL /10 , where μ = 0.2 dB/km is the loss coefficient for the standard optical fibers and L is the length of the fiber optics, and Eve may eavesdrop on the communication process. step 3: Bob performs homodyne or heterodyne measurement on the received quantum state, where Bob’s detection efficiency is modeled by a beam splitter with transmittance
85 86 87
η. It is inevitable that electronic noise v el will be intro-
41 42
Fig. 3. The success probability of implementing ZPC for different transmittance T and modulation variance V A .
duced during Bob’s detection process. So the noise introduced by homodyne (heterodyne) detection is defined as X hom = [(1 − η) + v el ] /η( X het = [1 + (1 − η) + 2v el ] /η). step 4: Bob shares the same key string with Alice through postprocessing. The above process is equivalent to the EB scheme, as shown in Fig. 1(b). Alice detects one half of the quantum state by heterodyne detections while another half is sent to Bob through quantum channel controlled by Eve after performing the ZPC operation. The above ZPC process can often be equivalent to operator O 0 given by
O 0 = T r [B ( T ) |0 C 0|] √ =: exp T − 1 b† b : =
√ b† b
88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103
(6)
104 105
T
106
√
√ where B ( T ) =: exp[ T − 1)(b† b + c † c ) +(c † b + cb† ) 1 − T ]: and : ·:
107
represents a normally ordering of operator. Therefore, for any input state |ϕ in in mode B, the output state |ψout can be expressed as
109
|ψout = O0
(7)
(8)
.
P0
√ T αk8 .
A B1 =
Z σ
8 z
117
Z σ
8 z
Y I
with the parameters
119 120 121 122 123 124
(9)
125 126 127
So the covariance matrix of state A B 1 is derived as
X I
115
118
2 ( T −1)αk8
114 116
The success probability of implementing ZPC is shown in Fig. 3. In our proposed protocol, the role of the ZPC operation is to promote the transition between modes B and B 1 , thus giving birth to the state
A B = e √ 1
112 113
where P 0 represents the success probability for achieving ZPC operation given by
P 0 =e
110 111
P 0 |ψ , in
2 ( T −1)αk8
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,
(10)
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Fig. 4. (a) The relationship between the secret key rate and the transmission distance of the ZPC-involved DM-CVQKD protocol. (b) Corresponding to (a), the transmittance T varies with the transmission distance. Here β = 0.9, V A = 0.5, v el = 0.05 and η = 0.6.
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X = Y = 1 + 2T α 2 Z
8
= 2T α
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3/2 λk−1 , λk k =0
(11)
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α is re-
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3. Security analysis
94 95
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The derivation of the asymptotic key rate is given in the Appendix A. As we all know, the covariance matrix after the quantum state passes through the channel is the key to security analysis, so we first give the covariance matrix of the original eight-state protocol and the ZPC-involved DM-CVQKD protocol in order to analyze the performance improvement of the proposed protocol on the traditional DM-CVQKD. In the original eight-state protocol, when the state is emitted by Alice through the quantum channel, the covariance matrix A B 2 is expressed as
⎡
40 41 42 43
A B2
1 + 2α 2 I ⎢ ⎢ =⎢ ⎢ √ 2 λ3/2 ⎣ k − 1 2 √ 2 Tcα
44 45
λk
k =0
√
2 T c α2
σz
√
2 k =0
3/2
λk−1 √ λk
⎤
48 49
⎥ ⎥ ⎥ ⎦
2
⎡
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
A B 2
2
1 + 2T α I ⎢ ⎢ =⎢ ⎢ √ 3/2 2 λk−1 ⎣ 2 2 Tc T α
k =0
λk
2
σz
√
√
2
3/2 λk−1 T c T α2 λk k =0
⎤
σz ⎥ ⎥ ⎥ ⎥ ⎦
T c 1 + 2T α + Xline I 2
98 99 100 101 102 103 104 105 106 107 109
T c 1 + 2α 2 + Xline I
In the ZPC-involved DM-CVQKD protocol, the covariance matrix
A B is expressed as
51
97
108
(12)
50
96
Fig. 5. The maximal tolerable excess noise of the ZPC-involved DM-CVQKD. Here β = 0.9, V A = 0.5, η = 0.6 and v el = 0.05.
σz ⎥
46 47
84 85
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(13) where Xline represents the total channel-added noise is defined as Xline = 1/ T c + ε − 1. It can be seen from the comparison between Eq. (12) and Eq. (13) that the ZPC operation, which is actually seen as a noiseless attenuation, only affects the performance of the protocol by the transmittance T of the beam splitter. Therefore, the optimization of T plays an extremely important role in the performance analysis process. We first analyze the numerical simulation results of the ZPC operation success probability, as shown in Fig. 3. We find that ZPC
can maintain the success probability greater than 0.75 when the modulation variance is less than 0.5, which indicates that it is a feasible solution to improve the performance of the DM-CVQKD protocol by using ZPC operation. In order to better illustrate the achievability and superiority of ZPC operation, here we take the smallest success probability to perform subsequent performance analysis, that is, V A = 0.5. In Fig. 4(a), it shows that the relationship between the secret key rate and the transmission distance of the ZPC-involved DM-CVQKD. Intuitively, the DM-CVQKD can lengthen transmission distances by applying ZPC operation. Note that the solid line represents the homodyne detection and the dotted line represents the heterodyne detection. From the comparison we find that the performances of homodyne detection and heterodyne detection are basically same. Therefore, we use homodyne detection for the subsequent security analysis. ZPC operation can be regarded as a noiseless attenuator, so its performance improvement depends on the value of T. If T = 1, there is no catalytic effect, and the proposed protocol is equivalent to the original eight-state protocol. As shown in Fig. 4(b), we obtained the maximum key rate by optimizing the transmittance T with the ZPC operation, which shows the change curve of transmittance with transmission distance. From the figure, we can see that when L > 65.85, T = 1, the catalytic
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Fig. 6. (a) The relationship between the secret key rate and the reconciliation efficiency of the ZPC-involved DM-CVQKD protocol. (b) Corresponding to (a), the transmittance T varies with the reconciliation efficiency. Here V A = 0.5, η = 0.6, v el = 0.05 and = 0.005.
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Fig. 7. (a) Comparison of secret key rates of the DM-CVQKD protocol and the ZPC-involved DM-CVQKD under different detection efficiency and electronic noise for L = 50 km. (b) Corresponding to (a), the transmittance T varies with the detection efficiency and electronic noise. (c) Comparison of secret key rates of the DM-CVQKD protocol and the ZPC-involved DM-CVQKD under different detection efficiency and electronic noise for L = 110 km. (d) Corresponding to (c), the transmittance T varies with the detection efficiency and electronic noise. Here β = 0.9, V A = 0.5 and = 0.005.
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effect occurs in the case of = 0.01, but because the T value is close to 1, the catalytic effect is not obvious, as shown in the upper subgraph in Fig. 4 (a). There always exists excess noise in the practical quantum channel, and hence we have to show the impact of excess noise on the performance of the ZPC-involved DM-CVQKD protocol. Fig. 4(a) also demonstrates intuitively the change of the secret key rate of the DM-CVQKD with the tunable excess noises ( = 0.005 and = 0.01). Obviously, with the increase of excess noise, the trans-
mission distance is shortened significantly. Therefore, in the practical experiment, if long-distance quantum key distribution is to be realized, the excess noise of the system must be stably controlled within a small range. Fig. 5 shows the maximum excess noise that the protocol can tolerate. It can be seen over a long distance, the ZPC-involved DM-CVQKD protocol exceeds the original DM-CVQKD protocol in terms of maximum tolerable excess noise. As known, reconciliation efficiency is an important factor affecting the secret key rate. In Fig. 6, we show the relationship between
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Fig. 8. (a) The relationship between the secret key rate and the detection efficiency. (b) The relationship between the secret key rate and the electronic noise. Here β = 0.9, V A = 0.5 and = 0.005.
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the secret key rate and the reconciliation efficiency when optimizing over T . We find the ZPC-involved DM-CVQKD protocol can tolerate lower reconciliation efficiency. When L = 50 km, the traditional eight-state protocol can tolerate the lowest reconciliation efficiency of 0.617, and the ZPC-involved DM-CVQKD protocol can tolerate 0.553. When L = 80 km, the traditional eight-state protocol can tolerate 0.741, and the ZPC-involved DM-CVQKD protocol can tolerate 0.675, which further indicates that the ZPC operation has a positive effect on the performance of the DM-CVQKD protocol. Moreover, the imperfection of coherent detections is also an important factor affecting the performance of the DM-CVQKD protocols. Fig. 7 gives a three-dimensional analysis chart, which aims to analyze the combined effect of the two on the secret key rate. From Fig. 4, it can be seen that when L > 65.85 km, the catalysis starts to occur, so in order to analyze the effect of ZPC on the eight-state protocol, we take the values of L < 65.85 km and L > 65.85 km to analyze, respectively. When L = 50 km, there is no catalytic effect (T = 1, as shown in Fig. 7(b)). Therefore, the original DM-CVQKD protocol and the ZPC-involved DM-CVQKD protocol demonstrate the similar performance. As shown in Fig. 7(a), simulation charts of the above-mentioned two protocols coincide exactly. Whereas for L = 110 km, the catalytic effect occurs (the variation of T with the detection efficiency and electronic noise is shown in Fig. 7(d)), but because the degree of improvement is not particularly obvious, so we give two-dimensional simulation charts of detection efficiency-secret key rate (v el = 0.05) and electronic noise-secret key rate (η = 0.6), as shown in Fig. 8. From Fig. 8(a), we can see that to achieve the same secret key rate, the ZPC-involved DM-CVQKD protocol can tolerate lower detection efficiency than the traditional eight-state protocol. For example, we can see that when the secret key rate 1.71 × 10−4 is achieved, the detection efficiency of the traditional eight-state protocol needs to be 0.877, while the ZPC-involved DM-CVQKD protocol only needs 0.688 with other parameters fixed. Fig. 8(b) shows that with the secret key rate of 1.2 × 10−4 , the electronic noise that can be tolerated by the traditional eight-state protocol is 0.176, while the ZPC-involved DM-CVQKD protocol is 0.359. This shows that through ZPC operation, while achieving the same performance, it can slightly reduce the requirements for detection equipment and reduce costs.
noiseless attenuation and can be implemented with current experimental technologies. We analyze the secret key rate of the ZPCinvolved DM-CVQKD against the Gaussian collective attack. The numerical simulations show the ZPC-involved DM-CVQKD protocol has the advantage of increasing maximum transmission distance and tolerable excess noise, compared with the original DM-CVQKD protocol. It can also tolerate lower reconciliation efficiency. In order to further illustrate the superiority of the ZPC-involved scheme, we compare the effects of Bob’s detection efficiency and electronic noise on performance, and find the ZPC-involved DM-CVQKD protocol allows lower detection efficiency and higher electronic noise for achieving the same performance. It is worth noting that it is important to control the excess noise in practical implementations. If the long-distance quantum key distribution is to be realized, the excess noise of the system should be stably controlled to a small range since it has a significant impact on the performance of the protocol.
4. Conclusion
where I A B and χ E B correspond to mutual information between Alice-Bob and Eve-Bob, and β is the reconciliation efficiency. The mutual information between Alice and Bob I A B for homodyne detection is calculated by
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Acknowledgements
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 61572529, 61871407).
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Appendix A. The derivation of asymptotic secret key rate
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We assume that Eve performs Gaussian collective attack to achieve the best results, and the information obtained is limited to the Holevo range χ E B . It should be noted that state | A B 1 can still be regarded as a Gaussian state through the ZPC so that we can directly calculate the secret key rate by using the results of the conventional Gaussian CVQKD protocol. Consequently, the secret key rate can be calculated as
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K = P 0 (β I A B − χ E B ) ,
(14)
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Declaration of competing interest
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We have proposed a method of enhancing the performance of DM-CVQKD by using ZPC operation, which is actually seen as a
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I AB =
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(16)
S ( A B ) = G [(λ1 − 1)/2] + G [(λ2 − 1)/2] ,
(17)
with G (x) = (x + 1) log (x + 1) − x log x and λ1,2 in the equation can be calculated by
λ1,2 =
1
( ±
2
2
− 4D 2 ),
(18) X , b
with D = ab − c and√ = a + b − 2c , where a = = T c (Y + Xline ) and c = T c Z 8 . The second term S ( A / B ) is a function of the symplectic eigenvalues λ3,4 of the convariance matrix of Alice mode after Bob performing homodyne (heterodyne) detection, which is given by 2
2
2
2
S ( A / B ) = G [(λ3 − 1)/2] + G [(λ4 − 1)/2] ,
(19)
where λ3,4 can be calculated by
λ3,4 =
1
(A ±
2
A 2 − 4B ).
(20)
For homodyne detection we have
A hom =
X hom + T ( V + Xline ) + V D
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(15)
.
where the first term S ( A B ) is a function of the symplectic eigenvalues λ1,2 of A B 2 , which is given by
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1 + X tot
= S ( A B ) − S ( A/B ) ,
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V + X tot
χB E = S ( E ) − S ( E / B )
10 12
2
log2
X tot represents the total noise which can be calculated as X tot = Xline + X hom(het ) / T c . Note that in the case of heterodyne detection, the mutual information I A B is twice the Eq. (15). The Holevo bound is given by:
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1
B
hom
=D
T ( V + X tot ) D X hom + V T ( V + X tot )
,
.
(21) (22)
For heterodyne detection we have
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A het =
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T 2 ( V + X tot )2
44
(23)
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,
B het =
( D X het + V )2 T 2(V
+ X tot )
2
.
(24)
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