Performance improvement of eight-state continuous-variable quantum key distribution with an optical amplifier

Performance improvement of eight-state continuous-variable quantum key distribution with an optical amplifier

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Performance improvement of eight-state continuous-variable quantum key distribution with an optical amplifier Ying Guo, Renjie Li, Qin Liao ∗ , Jian Zhou, Duan Huang School of Information Science and Engineering, Central South University, Changsha 410083, China

a r t i c l e

i n f o

Article history: Received 29 June 2017 Received in revised form 4 December 2017 Accepted 6 December 2017 Available online xxxx Communicated by A. Eisfeld Keywords: Quantum key distribution Discrete modulation Continuous variable Optical amplifier

a b s t r a c t Discrete modulation is proven to be beneficial to improving the performance of continuous-variable quantum key distribution (CVQKD) in long-distance transmission. In this paper, we suggest a construct to improve the maximal generated secret key rate of discretely modulated eight-state CVQKD using an optical amplifier (OA) with a slight cost of transmission distance. In the proposed scheme, an optical amplifier is exploited to compensate imperfection of Bob’s apparatus, so that the generated secret key rate of eight-state protocol is enhanced. Specifically, we investigate two types of optical amplifiers, phase-insensitive amplifier (PIA) and phase-sensitive amplifier (PSA), and thereby obtain approximately equivalent improved performance for eight-state CVQKD system when applying these two different amplifiers. Numeric simulation shows that the proposed scheme can well improve the generated secret key rate of eight-state CVQKD in both asymptotic limit and finite-size regime. We also show that the proposed scheme can achieve the relatively high-rate transmission at long-distance communication system. © 2017 Elsevier B.V. All rights reserved.

1. Introduction As quantum technology develops, more and more real-world applications of quantum technology have been emerging. One of the most advanced applications is quantum cryptography [1]. Quantum key distribution (QKD) [2,3] is a branch of quantum cryptography that allows two distant legitimate partners, Alice and Bob, to share an one-time pad sequence of bits, namely a random secure key, over an insecure quantum channel controlled by an eavesdropper Eve. Generally, QKD can be classified into two types, i.e. discretevariable (DV) QKD, e.g. the Bennett–Brassard 1984 (B B84) protocol [4], and continuous-variable (CV) QKD [5,6]. DVQKD encodes information in properties of single photon pulses while CVQKD encodes key bits in the quadratures (xˆ and pˆ ) of the optical field. Compared with the DVQKD, CVQKD offers higher secret key rate, thus it spotlights a huge number of researchers. Up till now, various CVQKD protocols are proposed such as G G02 protocol [7], squeezed-state protocol [8], unidimensional protocol [9], no-switching protocol [10], entangle source-in-themiddle (ESIM) protocol [11] and measurement-device-independent (MDI) [12,13] protocol, as well as several experiments are real-

*

Corresponding author. E-mail address: [email protected] (Q. Liao).

https://doi.org/10.1016/j.physleta.2017.12.011 0375-9601/© 2017 Elsevier B.V. All rights reserved.

ized [14–16]. Most of the above-mentioned protocols belong to Gaussian-modulated CVQKD [17–19], which is the most extensively applied in CVQKD and its unconditional security proofs in both collective attacks and coherent attacks [20,21] has been proposed [7]. However, Gaussian-modulated CVQKD is now facing the problem of transmission in the long-distance range compared with its DVQKD counterpart [22]. Theoretically, the overall signal-to-noise ratio (SNR) drops rapidly as transmission distance increases. Thus, this deterioration of channel conditions directly results in a rapid reduction of reconciliation efficiency. Unfortunately, Gaussian-modulated CVQKD can not break the limitation of low SNR or obtain high reconciliation efficiency under low SNR [22]. To break this limitation, discretely-modulated CVQKD protocol has been proposed [23] and its unconditional security proof has been shown in [23,24]. Thereinto, four-state CVQKD protocol has been implemented theoretically and experimentally [25]. Recently, a better discretely-modulated scheme with higher secret key rate and longer transmission distance, called eight-state protocol, was proposed [22], which improves the secret key rate and its transmission distance achieves more than 100 kilometers [26]. Since the discretely-modulated eight-state protocol exhibits excellent performance at low SNR, we further improve its capability by applying an optical amplifier (OA) [27]. The proposed scheme (eight-state protocol with an optical amplifier) can enhance the generated secret key rate of eight-state protocol by compensating

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the imperfection of the detector at Bob’s side with only slight cost of transmission distance. The performance of proposed scheme in asymptotic limit and finite-size regime is investigated. In asymptotic limit, we mainly focus on improvement of secret key rate and the trend of transmission distance reduction. In detail, we quantify and analyze the improvement of secret key rate by defining a novel parameter, improvement ratio, and thereby find that the OA has totally different effects on secret key rate under different transmission distance. As a result, a parameter called critical transmission distance is used to define the maximum transmission distance where OA exerts positive influence upon the performance of eight-state protocol. In finite-size regime, we obtain more practical result of the proposed scheme. This paper is organized as follows: in Sec. 2 we give the description of eight-state protocol and detail the proposed eight-state scheme with an optic amplifier. Numerical simulation and discussion are shown in Sec. 3. Conclusion is drawn in Sec. 4. Detailed derivation of equations is included in the Appendix.

The state |φk  could be described as follows,

e−

|φk  = √

Alice first prepares eight coherent displaced states

|βk  = |α e

kπ 4

 , k ∈ Z,

(1)

where α is chosen to be a positive real number and Z = {0, 1, 2, ..., 7}. Alice then randomly sends one of these eight coherent states to Bob with equal probability through an insecure quantum channel controlled by an eavesdropper called Eve. The quantum channel is characterized by the excess noise  and the transmission efficiency T . The total noise added to Bob’s input by effects of quantum channel can be expressed by χline = 1/ T +  − 1. After receiving the state sent by Alice, Bob subsequently performs homodyne detection or heterodyne detection. Since Bob’s apparatus is imperfect, detection efficiency η can hardly achieve 1 which denotes the perfect detection. Moreover, Bob’s detector also introduces some thermal noise υel when Bob measures the received state. Taking effect of detection efficiency η and thermal noise υel into account, one can derive a conclusive quantity χh , where χhom = [(1 − η) + υel ]/η and χhet = [1 + (1 − η) + 2υel ]/η are the case of homodyne detection and heterodyne detection, respectively. Consequently, the total quantity of noise χtot can be described as χtot = χline + χh / T . Finally, Alice and Bob share a string of the secret key by using error correction and privacy amplification. Since the PM version is equal to the entanglement-based (EB) version, which is more convenient for security analysis. In EB version, Alice prepares a pure two-mode state,

1 7

| 8  =

4

|ψk |βk ,

(2)

k =0

where the states

|ψk  =

7 1

2

e

i (4k+1)mπ 4

|φk , k ∈ Z,

m =0

are orthogonal non-Gaussian states.

λk

(3)

∞ 

e

α (8n+k) √ (8n+k)!

|8n + k, k ∈ Z

(4)

n =0

1

α2

α2

2

2

λ0(4) = e−α [cosh(α 2 ) + cos(α 2 ) ± 2cos( √ )cosh( √ )], 2

4

1 α2 α2 2 λ1(5) = e−α [sinh(α 2 ) + sin(α 2 ) ± 2cos( √ )sinh( √ ) 4 2 2

±



α2

α2

2sin( √ )cosh( √ )], 2 2

1 α2 α2 2 λ2(6) = e−α [cosh(α 2 ) − cos(α 2 ) ± 2sin( √ )sinh( √ )], 4 2 2 1

α2

α2

2

2

(5)

λ3(7) = e−α [sinh(α 2 ) − sin(α 2 ) ∓ 2cos( √ )sinh( √ ) 2

4

±

2.1. Eight-state CVQKD protocol

2

where

2. Eight-state CVQKD with an optical amplifier In this section, we mainly elaborate discretely-modulated eightstate CVQKD protocol with an optical amplifier. To make the derivation self-contained, we briefly introduce the original eightstate CVQKD protocol first.

α2



α2

α2

2sin( √ )cosh( √ )]. 2 2

Alice prepares the entangled state | 8  with variance V = V a + 1 and V a = 2α 2 , where she implements projective measurements on one of the set |ψk ψk | for k ∈ Z to the first half of | 8  and projects the second half of set |ψk ψk | on one of the eight coherent states |βk . After modulation, the modulated state is sent to Bob through an untrusted quantum channel. The covariance matrix of modulated state can be expressed by



γ8 =

V I2 Z 8σ Z

Z 8σ Z V I2



(6)

,

where V = V a + 1, I 2 = diag[1, 1], variance Z 8 = V a

7 

k=0

3 2

λk−1 λk

− 12

σ Z = diag[1, −1], and the co-

.

Bob’s detection efficiency is modeled by a beam splitter with transmission efficiency η . An EPR state with variance N d is used to model the thermal noise υel that is introduced by the process of Bob’s detector, where N d = ηχhom /(1 − η) is for homodyne detection and N d = (ηχhet − 1)/(1 − η) is for heterodyne detection. After that, Bob implements the reverse reconciliation and privacy amplification to generate the final bit string of secret key shared with Alice. 2.2. The improved eight-state scheme with an optical amplifier Due to the imperfection in Bob’s apparatus, the detection process cannot be ideal, thus final secret key rate is lower than expectation. Fortunately, the impact of imperfect apparatus can be reduced by applying an optical amplifier. In what follows, we elaborate the proposed eight-state CVQKD with an optical amplifier placed at Bob’s side. Fig. 1 shows the Entanglement-based (EB) version of the proposed scheme against Eve’s collective attacks. Firstly, Alice prepares an entangled state | 8 . After modulation, one mode of modulated state with variance V = V a + 1 is sent to Bob through an insecure quantum channel. After the quantum channel, the mode go through an optical amplifier placed in input of Bob’s apparatus and is detected by Bob’s apparatus. In the viewpoint of calculating secret key rate, this can be deemed trusted detection noise [28]. Here we mainly take two types of OA: phase-sensitive amplifier (PSA) and phaseinsensitive amplifier (PIA) into consideration, as shown in the light green box of Fig. 1, these two types of optical amplifiers are described as follows.

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Fig. 1. (Color online.) Schematic diagram of the eight-state CVQKD with an optical amplifier. Alice randomly prepares one of eight-state discrete-modulated states and sends it to Bob through the untrusted quantum channel controlled by an eavesdropper Eve. Additionally, an optical amplifier is placed in the input of Bob’s apparatus. Bob detects the received mode to derive a sequence of bits shared with Alice by using homodyne detector or heterodyne detector.

2.2.1. Phase-sensitive amplifier PSA is a degenerate amplifier that only magnifies the chosen quadrature’s amplitude and reduces another quadrature’s amplitude. Generally its properties is modeled by a matrix,

√

γ

PSA

=

g

0

0

1 g



(7)

,



Symbol

Parameter description

Value

T

Transmission efficiency Excess noise Gain of optical amplifier Alice’s modulated variance Detection efficiency Thermal noise Inherent noise of PIA Reconciliation efficiency

Independent variable Changed Changed 0.25 0 .6 0.05 1.5 0 .8

g Va

2.2.2. Phase-insensitive amplifier Different from the phase-sensitive amplifier, PIA is a nondegenerate amplifier that is able to magnify both quadratures symmetrically. However, because of its practical imperfection, it is inevitable to introduce the inherent noise. The numerical model for PIA can be written as:



−αd

sivity T is T = 10 10 , where d represents the length of the quantum channel and α = 0.2 dB/km means the loss coefficient of optical fibers. See Appendix A for more details about V a .



where g means the gain of amplifier, and g ≥ 1.

g I2 γ PIA = √ g − 1σ z

Table 1 Parameter setting. The relationship between transmission distance d and transmis-



η υel N0

β



g − 1σz √ , g I2

(8)

where g represents the gain of amplifier similarly as the PSA. Inherent noise introduced by PIA is expressed as follows:



γnoise =

N0 I2 N 0 2 − 1σ Z



N 0 2 − 1σ Z N0 I2



,

(9)

where N 0 means variance of noise. As shown in Fig. 1, after the amplifier exerts influence upon the modulated mode, this mode is detected by Bob’s apparatus. Bob’s detection efficiency is modeled by a beam splitter with transmission efficiency η . An EPR state with variance N d is used to model the thermal noise υel that is introduced in process of Bob’s detection, where N d = ηχhom /(1 − η) is for homodyne detection and N d = (ηχhet − 1)/(1 − η) is for heterodyne detection. After that, Bob implements reverse reconciliation and privacy amplification to generate the final bit string of secret key shared with Alice. In this section, we introduce the EB version of the proposed scheme. By utilizing an optical amplifier, performance of eightstate CVQKD can be enhanced especially in terms of secret key rate. 3. Numerical simulation and analysis 3.1. Asymptotic case We show the performance of the proposed CVQKD scheme in asymptotic case. Note that, for simplification, here we only consider the heterodyne detection applied by Bob since homodyne

Fig. 2. (Color online.) Comparison of secret key rate under heterodyne detection without PIA and with a PIA where  = 0.005 and g = 1, 20, where g = 1 stands for original eight-state protocol and g = 20 corresponds to the proposed scheme with PIA whose gain is 20. Point Q is the intersection point of these two curves. The blue line stands for original eight-state protocol and the red line represents the proposed scheme.

detection shows the approximate performance with heterodyne detection in eight-state CVQKD. Table 1 shows the parameters setting that used in the simulation. Fig. 2 depicts the secret key rate of the proposed scheme as a function of transmission distance in configuration of heterodyne detection with different values of g. The secret key rate K increases within a finite distance because of the PIA placed in the input of Bob’s apparatus. However, the proposed scheme is outperformed by the original eight-state CVQKD protocol in terms of

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Fig. 3. Improvement ratio of heterodyne detection with a PIA,  = 0.005 and g = 20. There are two dashed lines intersected, and the intersection point is defined as P . Two dashed straight lines divide the whole region into four parts, where actually only two parts are meaningful. In the upper-left side of point P, it is the reinforced region, and reversely, the lower-right side represents the degraded region.

maximal transmission distance. The intersection point Q ( Q x , Q y ) represents both two scheme exist identical secret key rate when the abscissa is Q (x). Furthermore, we propose a new parameter called improvement ratio IR to quantify the improvement of secret key rate, which is given by

IR :=

K | g = g 0 − K | g =1 K | g =1

,

(10)

where g 0 denotes the value of gain of amplifier, which is specified in each simulation. As depicted in Fig. 3, the improvement ratio exceeds 0.6, which means that secret key rate is changed to 1.6 times when compared with the original eight-state protocol. Similarly as Fig. 2, there also exists an intersection point P in Fig. 3. Note that the intersection point P ( P x , 0) is equivalent to the point Q ( Q x , Q y ). In details, Q means equivalent key rate of scenario with a PIA and without PIA, and the point P stands for the condition that improvement ratio is zero, i.e. K | g = g0 = K | g =1 . As illustration of Fig. 3, two meaningful sections divided by point P are reinforced region and the degraded region. The former region stands for the area where the performance of secret key rate is enhanced, and the latter region represents the area where the performance is reversely degenerated. Theoretically, the value of gain g and excess noise  can affect the abscissa movement of point P . Fig. 4 (Top) shows the relationship between transmission distance and improvement ratio, with  = 0.005 and different g varies from 20 to ∞. As the gain of PIA g increases, improvement ratio becomes larger while the abscissa of point P , namely P x , becomes smaller. In particular, P x decreases to 145 km when the gain of PIA g approaches infinity. Moreover, Fig. 4 (Bottom) illustrates the situation of movement of point P under different excess noise  . As expected, P x moves right with reduction of excess noise  . Although our proposed scheme can improve the performance of eight-state CVQKD protocol within a certain range, however, if we do not notice the overlong length of fiber, results might be totally different. Despite a slight reduction of transmission distance, distance of 150 km is capable of satisfying requirements of communication. Therefore it is promising that the improved scheme is applied to long-distance with relatively high key rate communication system. To avoid the above-mentioned worse case that optical amplifier affects performance negatively, paying more attention to the length of fiber is critical. In order to make the

Fig. 4. (Color online.) (Top) Relationship between transmission distance and improvement ratio with  = 0.005 and different g varies from 20 to ∞. (Bottom) Relationship between transmission distance and improvement ratio with g = 3 and different  varies from 0.003 to 0.01.

boundary clearer, here we propose a new parameter critical distance. It denotes the longest transmission distance, corresponding to different excess noise  , where the usage of optical amplifier still work positively. In other words, when improvement ratio is fixed to 0, corresponding distance is critical distance. Fig. 5 illustrates that the critical distance decreases as the value of excess noise  increases. 3.2. Security analysis in finite-size regime In order to simplify the theoretical QKD security proofs, researchers usually expand their theories based on some assumptions. For example, several proofs assume that there is no side channel in a communication system. Unfortunately, we cannot proof that the assumption described as above is true in exception of using device-independent QKD protocol. However, realization of device-independent QKD protocol is nearly impossible in physical at present. Another assumption that number of exchanged signals between Alice and Bob approaches to ∞ is usually used in asymptotic scenario. In other words, it assumes that Alice and Bob can use countless signals to derive secret key rate. However, the number of exchanged signals cannot be infinite, Alice and Bob need some signals to parameter estimation so that they can ensure the accuracy of QKD. We here consider the secret key rate K f that is derived from heterodyne detection in finite-size scenario where number of exchanged signals is confined to a finite value [29]. The finite-size scenario includes the loss of parameter estimation, so the analysis can be more realistic than asymptotic scenario [30].

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Fig. 5. Relationship between excess noise and critical distance (maximum transmission distance in case of improvement) with the gain of PIA g approaching to ∞ and the variance of modulated state V a = 0.25.

Fig. 6. (Color online.) Discretely-modulated eight-state protocol with PIA in finitesize scenario in the case of heterodyne detection where V a = 0.25, η = 0.6, υel = 0.05, β = 0.8, g = 1, 20 (g = 1 means original eight-state protocol). Solid curves mean the original eight-state protocol while the dashed curves represent the scenario with an PIA. From left to right, the first eight curves correspond to finite-size scenario of block length N = 108 , 1010 , 1012 , 1014 respectively and the last two curves stand for the asymptotic scenario.

We only consider the secret key rate K f in the condition that

the estimation of t¯ and ξ¯2 is equivalent to their expected values. For more detail about the calculation can be seen in Appendix B.2. In the simulation, the value of ε = 0.005 is quite conservative, since the fluctuation of value of ε exerts small influence on the final value of secret key rate K f [31]. Assuming that one half of exchanged signals m are used to implement the parameter estimation and the rest of exchanged signals n are used for derivation of quantum key. Although this situation can be considered as the worst, influence resulting from ratio of m and n, is so negligible and consequently K f we even can ignore the influence of ratio for final secret key rate. Moreover, the reconciliation efficiency β = 80% and the detection efficiency η = 60%. As shown in the Fig. 6, we notice that whether PIA is placed at Bob’s detection apparatus or not, the longest transmission distance in the asymptotic scenario is far more than transmission distance in finite-size scenario. Moreover, as the number of exchanged signals N increases, curves become closer to the curve of asymptotic scenario. It is because the bigger number of exchanged signals is, the more signals parameter estimation can use and therefore the

5

Fig. 7. (Color online.) Number of exchanged signals and secret key rate in the case of homodyne detection where V a = 0.25, η = 0.6, υel = 0.05, β = 0.8, g = 1, 20. In addition to the black dashed curve, from left to right, transmission distance d = 10 km, 13 km, 15 km, 18 km with g = 1 i.e. original eight-state protocol. The black dashed line means the finite-size scenario of CVQKD protocol with PIA (g = 20).

parameter estimation is approaching to perfection. However, it is impossible for number of exchanged signals to reach infinity in practice. Moreover, although finite-size effect seems much closer to the asymptotic scenario when the number of exchanged signals N is set to 1014 , it is too difficult to realize the enormous information exchanging. Without exaggeration, the value of the block length of exchanged signals N = 108 is an astronomical number for a realistic CVQKD system not to mention 1014 . But fortunately, the secret key rate K f of eight-state protocol, whether PIA is placed in system or not, is still higher than Gaussian protocol in finite-size scenario [32]. In addition, it is interesting shows that the reinforced region and degraded region will disappear when number of exchanged signals N remains a relatively low value, but these two regions will appear as N grows. As shown in Fig. 7, there are obvious differences of number of exchanged signals between different curves corresponding to various transmission distance. It shows that this novel scheme can not only enhance the secret key rate K f , but also numerously decrease the number of exchanged signals. Thus the whole data size of CVQKD derivation procedure can be cut down and the system load also decreases correspondingly. Therefore it is confirmed that the proposed scheme is also a good idea to improve performance of system in practical. In this section, we mainly analyze performance of the proposed scheme in asymptotic limit and finite-size regime. By using an optical amplifier, performance of secret key rate is improved with slight cost of transmission distance. 4. Conclusion In this paper, we propose a scheme to improve performance of eight-state CVQKD protocol using an optical amplifier. In the proposed scheme, an optical amplifier is exploited to compensate imperfection of Bob’s apparatus, so that the generated secret key rate of eight-state protocol is enhanced. Specifically, we investigate two types of optical amplifiers, phase-insensitive amplifier (PIA) and phase-sensitive amplifier (PSA), and thereby obtain approximately equivalent improved performance for eight-state CVQKD system when applying these two different amplifiers. Numeric simulation shows that the proposed scheme can well improve the generated secret key rate of eight-state CVQKD in both asymptotic limit and finite-size regime. We also show that the proposed scheme can achieve the relatively high-rate transmission at long-distance communication system.

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Fig. 8. (Color online.) Heterodyne detection with a practical detector,  = 0.005, g = 1. It illustrates the compressed variation trend of V a ’s optimal interval as the transmission distance extends.

Fig. 10. (Color online.) Heterodyne detection with a practical detector, d = 100 km,  = 0.005. When the gain of amplifier varies from 1 to infinity, the secret key rate also increases gradually. Besides, while the gain of amplifier raises and approaches to ∞, the gain of amplifier approaches to a certain value rather than the infinity. Fortunately, the optimal interval also contains the value of variance V a = 0.25.

As shown in Fig. 9, we plot the secret key rate as a function of variance V a with different excess noise  . Although optimal intervals of all curves become narrow, a public interval including 0.25 still exist. Furthermore, the secret key rate is treated as a function of variance V a with different gain of amplifier g. Fig. 9 illustrates that the higher the gain of amplifier g is, the narrower the optimal interval of secret key rate is. Thus, we draw the same conclusion as above. As a result, we choose the variance of modulated state V a as a constant 0.25. A.2. The example values of g

Fig. 9. (Color online.) Heterodyne detection with a practical detector, d = 100 km, g = 1. As a function of variance V a , the secret key rate presents the same compressed trend as Fig. 8, while the value of excess noise raises.

Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant No. 61379153, No. 61572529), and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2017zzts147). Appendix A. Parameter optimization

For convenient, we choose an example values of g for the following simulation. Specifically, we plot a three-dimension function with two independent variables, i.e. gain of optical amplifier and transmission distance. Fig. 11 (Top) illustrates that the secret key rate rises steadily as the gain of PIA increases. Fig. 11 (Bottom) shows that the slope of the curve reaches the maximal value when g is around 3, meanwhile the curve becomes smoother as g increases to 20, i.e. the slope is approaching to 0. Thus, g = 3, 20 are used for the following simulation. Appendix B. Calculation of secret key rate B.1. Asymptotic secret key rate for eight-state protocol with an optical amplifier

A.1. The optimal variance V a An optimal V a is necessary to maximize the performance of CVQKD. As shown in Figs. 8, 9 and 10, we set different values of d, g and  respectively to look for a public value of V a that maximal the secret key rate of eight-state protocol. First, we use optimal interval to denote the interval of the highest value of secret key rate. Fig. 8 illustrates that the optimal interval becomes compressed as the transmission distance increases. While parameters g and  are fixed to legitimate values. Moreover, as transmission distance has been increasing, the secret key rate also becomes smaller and all curves have a public interval including 0.25 where the secret key rate can reach the highest value. We thus have V a = 0.25. On the basis of next simulation results, it is confirmed that V a = 0.25 is the optimal value for variance V a .

Assuming that Eve performs the optimal collective attacks, the information accessible to Eve is generally confined to the Holevo bound χ B E . Therefore, the definition of secret key rate in the case of reverse reconciliation under collective attacks can be expressed as

K ≥ β I A B − χB E ,

(B.1)

where β is reconciliation efficiency, I A B is the mutual information of Alice and Bob, and χ B E is Holevo bound. The mutual information between Alice and Bob I A B is derived from Bob’s measured variance V B = η T ( V + χtot ) and the conditional variance V B | A = η T (1 + χtot ) using Shannon’s equation for homodyne detection, namely

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1 where h(x) = ( x− )log2 ( x−2 1 ) − ( x+2 1 )log2 ( x+2 1 ), λ1,2 are symplectic 2 eigenvalues of covariance matrix



γ A B1 =







γ A σ A B1 VI T Z 8σ Z = √ 2 σ A B1 γB1 T Z 8 σ Z T ( V + χline ) I 2

 (B.4) X

and λ3,4,5 are symplectic eigenvalues of covariance matrix γ A FEG that represents the covariance matrix under condition of Bob’s projective measurement. It is well-known that when the state ρ A B 1 between Alice and Bob is Gaussian, the Holevo information between Eve and Bob can be maximized. Therefore, the Gaussian Holevo information can be regarded as the bound for that of eight-state protocol. As shown in Fig. 12, when V a is small enough, Z G and Z 8 are too close to distinguish. Hence, one has S 8 ( y : E ) ≈ S G ( y : E ) and I 8 ( y : E ) ≈ I G ( y : E ). Now we can calculate the Holevo information of eight-state protocol. Here we begin with the symplectic eigenvalues λ1,2 . According to the covariance matrix γ A B 1 , the symplectic eigenvalues λ1,2 are given by



λ1,2 =

1 2

(A +



A 2 − 4B ),

(B.5)

with



A = det γ A + det γ B 1 + 2det σ A B 1 ,

(B.6)

B = det γ A B 1 . X

It is evident that entropy S (ρ A FE G ) should be obtained by the symplectic eigenvalues λ3,4,5 of covariance matrix Fig. 11. (Color online.) (Top) Three-dimension figure of gain of PIA, transmission distance and secret key rate. (Bottom) Function of gain of PIA and secret key rate, where different colors represent different transmission distances and the transmission becomes longer and longer from the deep red region to the deep blue region. When g = 3, the slope seems to reach the medium value and when g = 20, although the secret key rate is improved, the slope of g = 20 approaches to 0.

I AB =

1 2

log 2

VB V B| A

1

V + χtot

2

1 + χtot

= log 2

(B.2)

,

where V B | A =  X B 2  −  X A X B 2 / X A 2 , is the conditional variance of Alice based on Bob’s measurement and χtot = χline + χh / T , χh = χhom . Meanwhile, the Holevo bound can be calculated as

χ B E = S (ρ A B1 ) − S (ρ

XE AF G)

=

2  i =1

h(λi ) −

5  i =3

XE AF G

have to acquire the covariance matrix γ derived from the covariance matrix γ A F G B under the condition of Bob’s projective measurement. After the modulated mode goes through the quantum channel, it reaches the input of an beam splitter with transmission efficiency η that is used to model the detection efficiency of Bob’s apparatus. In addition, we also use an EPR state of variance N to model the thermal noise introduced to the modulated mode by Bob’s detection. So the derivation of total covariance matrix γ A F G B can be written as

γ A B F G = γ BTS (γ A B 1 ⊕ γ F 0 G )γ B S , where

h(λi ),

(B.3)

γ AXFEG . We thus

γB S = I 2 ⊕



 √ √ 1 − ηI2 √ ηI2 √ ⊕ I2, − 1 − ηI2 ηI2

(B.7)

(B.8)

Fig. 12. Comparison of correlation Z 8 for eight-state protocol and Z G for Gaussian modulation protocol as a function of V a . (Left) Large value of V a . (Right) Small value of V a .

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8

and





γF0 G = ⎣ 

Nd I 2 N d2

N d2 − 1σ Z

− 1σ Z

PSA χhom =

⎤ ⎦.

(B.9)

Nd I 2

After an appropriate rearrangement of rows and columns of the covariance matrix γ A B F G , we can derive the covariance matrix γ A F G B . Then, the matrix γ AXFEG is written as follows,

γ AXFEG = γ A F G − σ AT F G B H σ A F G B ,

(B.10)

γ AF G , σ A F G B can be distilled from the covariance matrix γAF G σAF G B γAF G B = , and H hom = ( X T γ B X ) M P is for homoσAF G B γB dyne detection, H het = (γ B + I 2 )−1 is for heterodyne detection rewhere

spectively. We could obtain the rest three symplectic eigenvalues λ3,4,5 using Eq. (B.10),

λ3,4 =



1 (C 2

±



C 2 − 4D ),

λ5 = 1 ,

(B.11)

where for homodyne detection, C and D can be written as

A χhom + V

C hom =



B + T ( V + χline )

T ( V + χtot )

D hom =

,

√ √ V + B χhom B , T ( V + χtot )

(B.12)

for heterodyne detection, C and D could be given by

C het =

 D het =

V +



B χhet

B + T ( V + χline ) + 2T Z 8 ]

T 2 ( V + χtot )2

2 .

T ( V + χtot )



,

(B.14)

Then we introduce how to derive the secret key rate for the proposed eight-state protocol where the optical parametric amplifier is employed in Bob’s apparatus. For different circumstances, we mainly discuss two optical parametric amplifiers, an ideal phasesensitive amplifier (PSA) and a practical phase-insensitive amplifier (PIA). Besides, for general configurations, we should include two cases, e.g. the homodyne detection and heterodyne detection. So we utilize a PSA to compensate imperfection in the case of homodyne detection while we adopt a PIA to reduce the bad influence of detector’s imperfection in the case of heterodyne detection. By replacing parameters χhom and χhet with new parameters, PSA PIA χhom and χhet , the new secret key rate can be derived from Eq. (B.1). B.1.1. Phase-sensitive amplifier and homodyne detection PSA is a degenerate amplifier that only magnifies the amplitude of the chosen quadrature and reduces another quadrature’s amplitude. We therefore give a matrix to model for the phase-sensitive amplifier as follows,

√

γ

PSA

=

g

0

0

1 g

 ,



(B.15)

where g means the gain of amplifier, and g ≥ 1. We change the chosen quadrature’s amplitude by varying the gain of amplifier from 1 to ∞. In the case of homodyne detection, the new secret key rate is PSA derived from substituting χhom for the original parameter χhom . PSA The new parameter χhom is given by

(B.16)

,

υel denotes thermal noise.

B.1.2. Phase-insensitive amplifier and heterodyne detection Different from the phase-sensitive amplifier, PIA is a nondegenerate amplifier which is able to magnify both quadratures symmetrically. However, because of its practical imperfection, it is inevitable to introduce some inherent noise that we use an Einstein– Podolsky–Rosen (EPR) state of variance N 0 to model. In EB scheme, one mode of the EPR state representing the inherent noise of PIA is injected into the second input of PIA and the other mode is stored. Thus the model for PIA can be written as





g I2 γ PIA = √ g − 1σ z





g − 1σ z √ , g I2

(B.17)

where g denotes the gain of amplifier similarly as the PSA. We can use different gains g to obtain the different quadratures’ amplitudes. Besides the matrix (B.17), we employ an EPR state to model the inherent noise as follows,



(B.13)

2 A χhet + B + 1 + 2χhet [ V

where

(1 − η) + υel

N I γnoise = 20 2 N 0 − 1σ Z



N 0 2 − 1σ Z



N0 I2

(B.18)

,

where N 0 denotes variance of noise. In the case of heterodyne detection, we use a new parameter PSA χhet to replace the original parameter χhet to have the new secret key rate. In contrast to PSA, what PIA introduces is not only the gain of amplifier g but also the variance of inherent noise N 0 . The detailed expression is written as PIA χhet =

1 + (1 − η) + 2υel + N 0 ( g − 1)η gη

.

(B.19)

PSA By replacing χhom with χhom , we obtain the secret key rate of the proposed eight-state CVQKD protocol in the case of homodyne detection and similarly acquire the secret key rate of that in the PIA case of heterodyne detection by substituting χhet for χhet .

B.2. Finite-size secret key rate for eight-state protocol with optical amplifier For simplicity, we focus the analysis on homodyne detection in the case of reverse reconciliation. N denotes number of exchanged signals between Alice and Bob, n is number of signals for derivation of QKD and rest of N will be used in parameter estimation procedure. Moreover, the x and y represent classic information hold by Alice and Bob respectively after they have finished the measurement procedure and the E corresponds to the quantum state of eavesdropper Eve. The finite-size secret key rate can be written as

Kf =

n N

(β I (x : y ) − S ε P E ( y | E ) − (n))

(B.20)

where β is efficiency of reconciliation, I (x : y ) represents mutual information between Alice and Bob, S ε P E ( y | E ) denotes the maximum of the Holevo information compatible with the statistics except with probability ε P E , and (n) is related to the security of the privacy amplification. For CVQKD protocol, value of secret key rate importantly depends on three parameters, Alice’s modulated variance V a , channel transmissivity T , and excess noise  . So one usually optimizes modulated variance V a to improve secret key rate as high as possible and let Alice and Bob estimate the value of T and  .

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In the beginning of protocol, Alice and Bob will make agreement on value of security parameter ε and procedure of protocol. This means that they know the value of failure probability of reconciliation ε EC in advance. Because virtual parameters ε¯ and ε P A always have optimized in advance, consequently other parameters’ initial optimization largely depends on ε P E whose estimation largely depends on sample number of parameter estimation m and some properties of quantum channel such as T and  etc. Consequently, we conclude sample number m corresponding to certain value of ε P E . Meanwhile, it offers an infimum of N corresponding to a certain value of ε . Before implementing QKD, Alice and Bob optimize the number of exchanged signals N, length of original secret key and Alice’s modulated variance V a , to acquire the highest final secret key rate compatible with ε -security by which finite-size scenario’s difference from asymptotic scenario is mainly reflected. Thereinto, security of finite-size scenario is mainly based on the probability ε P E , because other security parameter can be optimized. Generally we have ε = ε P E + ε EC + ε¯ + ε P A . Because these four parameters are really negligible, we usually have ε ≈ ε P E = ε EC = ε¯ = ε P A = 10−10 . The efficiency of reconciliation is set to 80%, and the mutual information I (x : y ) for homodyne detection, is generally obtained by

I (x : y ) =

1 2

log 2

VB V B| A

V + χtot

2

1 + χtot

(B.21)

,

χtot = χline + χt 2h , for homodyne detection,

where V = V a + 1,

χh = χhom =

1

= log 2

(1 − η) + υel

(B.22)





γ A B1εP E =

1 t2

+

ξ −1 t2

VB

− 1.

where V = V a + 1,

χh = χhet =

= log 2

V B| A



(B.29)

m m 1  1 xi y i 2   t = i = and = ( y i − txi )2 . ξ m 2 m

i =1 x i

(B.30)

i =1

ξ 2 are compatible with distribution as follows, Moreover,  t and 

ξ2  t N (t , m

) and 2

m ξ2

ξ2

m =1 x i

χ 2 m − 1,

(B.31)

where t and ξ 2 is true value of parameters. So, we can calculate the minimum of t and the maximum of ξ 2 .

⎧  ⎪  ⎪ ξ2 ⎪ ⎪ t − zε P E /2 , ⎨ tmin ≈ mV a √ ⎪  ⎪ ξ2 2 ⎪ 2 ⎪ ξ2 ≈ ξ + z , √ ⎩ ε P E /2 max

(B.32)

m

x



ξ 2 whose expected vale −t dt. Theoretically, we take  t and  π 0

(B.23)

In the case of heterodyne detection, we have,

I (x : y ) = log 2

tmin Z 8 ξ Z ( V a + 1) I 2 2 tmin Z 8 ξ Z (tmin V a + ξmax )I2,

2 where tmin and ξmax represent minimum of t and maximum of ξ 2 compatible with sample couples. We simplify the computation of γε P E as a computation under only 2-dimensional confidence interval through parameter estimation of t and ξ 2 . By maximum-likelihood estimators for the normal liner model, we obtain

√2

χline =

V + χtot 1 + χtot

χtot = χline + χh /t 2 ,

1 + (1 − η) + 2υel + N 0 ( g − 1)η gη

ues are,

E[ t] =



T

(B.33)

E[ξ ] = 1 + T  . (B.24)

,

2

2

(B.25)

and

Using all of derived equations as above, we can calculate tmin and 2 ξmax

⎧  ⎪ √ 1 + T ⎪ ⎪ ⎪ ⎨ tmin ≈ T − zε P E /2 mV , a

(B.34)

√ ⎪ ⎪ (1 + T  ) 2 ⎪ 2 ⎪ . √ ⎩ ξmax ≈ 1 + T  + zε P E /2 m

χline =

1 t2

+

ξ2 − 1 t2

− 1.

(B.26)

In order to calculate S ε P E ( y | E ), first we have a matrix representing the shared state ρ A B 1 ,



γρ A B 1



where t = T ∈ R and z follows a centered normal distribution with variance ξ 2 = 1 + T  . So we can find a covariance matrix γε P E which successfully minimize the secret key rate K f ,

where zε P E /2 has (1 − er f ( zε P E /2 / 2))/2 = zε P E /2, and er f (x) =

and 2

9

γ Aε P E σ A B 1 ε P E = σ A B1ε P E γB1ε P E 

=

(√ V a + 1) I 2 T Z 8σ Z







T Z 8σ Z . (T V a + 1 + T  ) I 2

A = det γ A ε P E + det γ B 1 ε P E + 2det σ A B 1 ε P E , B = det γ A B 1 ε P E , C=

(B.27) D=

We need to evaluate the covariance matrix γρ A B compatible 1 with the exchanged data except with failure probability of parameter estimation ε P E . Parameters estimation of covariance matrix γε P E can be implemented by sampling of m ≡ N − n couples of correlated variables (xi , y i )i =1...m . Moreover we take a model into our consideration for these correlated variables, where the data of Alice and Bob are related through the following relation,

y = tx + z,

After substitute Eq. (B.34) into Eq. (B.29), for homodyne detection, we obtain A, B, C , and D to derive λ1−5 for homodyne detection.

(B.28)

A χh + V



B + T ( V + χline )

T ( V + χtot )



B

V +



B χh

T ( V + χtot )

(B.35)

,

, (1−η)+υ

ξ 2 −1

el where χh = χ hom = and χline = t12 + t 2 − 1. For hetgη erodyne detection, A and B is the same as case of homodyne detection, we have C and D,

C=

2 A χhet + B + 1 + 2χhet [ V

 D=

V +



B χhet

T ( V + χtot )

T 2(V

2 ,



B + T ( V + χline ) + 2T Z 8 ]

+ χtot )2

, (B.36)

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10 1+(1−η)+2υ + N ( g −1)η

ξ 2 −1

0 el where χh = χ het = and χline = t12 + t 2 −1. gη According to Eqs. (B.35) and (B.36), we can obtain symplectic eigenvalues λ1,2,3,4,5 as follows, A, B, C , and D to derive S ε P E ( y | E ).



λ1,2 =



λ3,4 =

1 2



(A ±

A 2 − 4B ), (B.37)



1

5

(C ± C 2 − 4D ), λ = 1 2 S ε P E ( y | E ) is then obtained by S ε P E ( y| E ) =

2 

h(λi ) −

i =1

5 

h(λi ).

(B.38)

i =3

The term (n) related to security of privacy amplification can be written as



(n) ≡ (2dimH X + 3)

log 2 ( ε2¯ ) n

2

1

n

εP A

+ log 2 (

)

(B.39)

where HY is Hilbert space corresponding to the variable y used in the raw key, ε¯ is the smoothing parameter, ε P A is the failure probability of privacy amplification procedure. Generally, we have H X = 2 since for all continuous-variable protocols. Moreover, the second term can be neglected because the convergence speed of 2/n is fast than log 2 ( ε 1 ) largely. So, we have (n) as a function PA of n,



(n) 7

log 2 (2/¯ε ) n

(B.40)

Finally, the final secret key rate K f can be derived using Eqs. (B.21) (or Eq. (B.24)), (B.38), (B.40). References [1] C.H. Bennett, F. Bessette, G. Brassard, L. Salvail, J. Smolin, J. Cryptol. 5 (1) (1992) 3.

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