Unidimensional continuous-variable quantum key distribution with discrete modulation

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Physics Letters A ••• (••••) ••••••

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Unidimensional continuous-variable quantum key distribution with discrete modulation Wei Zhao, Ronghua Shi, Yanyan Feng, Duan Huang ∗ School of Computer Science and Engineering, Central South University, Changsha 410083, China

a r t i c l e

i n f o

Article history: Received 17 July 2019 Received in revised form 12 October 2019 Accepted 12 October 2019 Available online xxxx Communicated by M.G.A. Paris Keywords: Continuous-variable quantum key distribution Discrete modulation Unidimensional Heisenberg uncertainty relation

a b s t r a c t In this paper, we proposed an unidimensional continuous-variable quantum key distribution (CV-QKD) protocol with discrete modulation, which waives the necessity in one of the quadrature modulations and further simplified the implementation of the CV-QKD protocol. On the basis of the Heisenberg uncertainty relation, we analyze the boundary between the unphysical and physical region. Besides, we utilize a novel proof approach to achieve a lower bound valid of transmission distance. This scheme shows an available method to further simplified the implementation of the QKD. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Quantum key distribution (QKD), which promising unconditional security in quantum physics, is one of the prospective and feasible technology to date [1–5]. Particularly, discrete-variable (DV) QKD [1,2] and continuous-variable (CV) QKD [6–8] are two main implementation ways of QKD. Both in DV-QKD and CV-QKD, several significant achievements have been achieved. For implementing DV-QKD, Pan et al. launched a low-Earth-orbit satellite, which key rate is around 20 orders of magnitudes greater than optical fiber [1]. Meanwhile, the Gaussian-modulated coherent-stated (GMCS) scheme was achieved through an 80 km optical fiber [7]. In terms of technological implementations and applications, the secret key of DV-QKD is encoded on the polarization of a single photon. Due to the employment of single-photon detection, imperfections of such devices can be exploited to attack the system [9,10]. Consequently, recent developments in the QKD field are interested in the CV coding bits without the requirement of singlephoton detection. The CV-QKD has two main modulation approaches, namely the Gaussian modulation and discrete modulation. The Gaussian modulation can achieve higher secret key rate, moreover, most experiments are adopted with Gaussian modulated coherent states [7]. In fact, the Gaussian modulated coherent state is difficult to be perfectly achieved in the practice. Furthermore, the finite constellation

*

Corresponding author. E-mail address: [email protected] (D. Huang).

https://doi.org/10.1016/j.physleta.2019.126061 0375-9601/© 2019 Elsevier B.V. All rights reserved.

of finite energy is necessary to approximate such a Gaussian in the GMCS protocol [11]. Compared with Gaussian modulation, the discrete modulation has several primary advantages. For one thing, the discrete modulation has characterized with a small constellation, which dramatically simplified the error correction procedure [12]. For another, the discrete modulation can simplify the state preparation procedure [13–15]. Therefore, the discrete modulation can achieve the high reconciliation efficiency and low signal-tonoise ratio. In consequence, we proposed an unidimensional CV-QKD protocol with discrete modulation. The proposed protocol can further simplify the implementation of the QKD. The unidimensional CVQKD has been proposed by Vladyslav et al. [16]. Subsequently, the finite-size [17] and composable security [18] of the unidimensional CV-QKD have been proved in the asymptotic regime. Besides, an unidimensional CV-QKD protocol has been achieved in an experiment [19]. Even though several unidimensional CV-QKD have been proposed [20–22], all related works are not aware of the discrete modulation, which is essential to the implementation in unidimensional CV-QKD. In what follows, we illustrate the unidimensional CV-QKD with three types of discrete modulation, namely the two-state, fourstate and eight-state CV-QKD. First of all, we review the discrete modulation, as well as its covariance matrix and density matrix. Secondly, we calculated the secret key rate in the linear channel under the asymptomatic regime, in other words, we restrict the Eve’s attack performed to a linear quantum channel. Subsequently, we utilize a novel proof approach to achieve a lower bound valid

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of transmission distance [23]. Thirdly, on the basis of the Heisenberg uncertainty relation, we calculate and analyze the boundary between the unphysical region and physical region. The paper is structured as follows. In Sec. 2.1, we first review the characterize of discrete modulation, as well as its initial state and the covariance matrix. To be specific, we insist on three instances of such protocol, namely the two-state, four-state and eight-state protocol. In Sec. 2.2, we demonstrate the structure of the proposed protocol with discrete modulation, and derive the relationship between the unmodulated quadrature. In Sec. 3, we calculate and simplify the secret key rate. Moreover, we also analyze the uncertainty relations of unmodulated quadrature. Besides, we calculate the boundary between the unphysical and physical region on account of the Heisenberg uncertainty relation. In particular, a novel proof approach is applied to establish a lower bound of distance in the appendix. Finally conclusions are drawn in Sec. 4. 2. Unidimensional CV-QKD protocol The discrete modulation, which exists high tolerance against the excess noise, is more suitable to achieve longer distance than Gaussian modulation. In what follows, we propose an unidimensional CV-QKD protocol using discrete modulation, and we insist on three instances of such protocols for which good error correction schemes are known: the two-state protocol, the four-state protocol and as well as the eight-state protocol [12,24,25].

and

|φ0  = √ |φ1  = √

∞ 

1

sinhα 2 n=0



Γ2 =

V I2

Z 2σ Z

Z 2σ Z

V I2

1

ρ4 =



|γ γ | + |γ ∗ γ ∗ | + | − γ −γ | + | − γ ∗ −γ ∗ |

4

3 

λi |φi φi |,

(8)

i =0

(1)

and

S4 = {|α e i π /4 , |α e 3i π /4 , |α e 5i π /4 , |α e 7i π /4 },

(2)

|φk  = √

1

λ0,2 = e−α (coshα 2 ± cosα 2 ),

S8 = {|α , |α e i π /4 , |α e i π /2 , |α e 3i π /4 , |α e i π , |α e 5i π /4 , (3)

respectively, where α is chosen to be a positive real number. After Bob performs homodyne detection, he obtains the real random variable y i for i ∈ {1, 2, · · · , n}. Subsequently, Alice and Bob use a reverse reconciliation, and the sign b i of y i encodes the raw key bit: we note b i = 1 if y i ≥ 0 and b i = 0 if y i < 0. Alice must then recover the value of the string b = {b1 , b2 , · · · , bn }. To help her, Bob sends some side-information consisting of the quadrature measured, the absolute value of y i as well as the syndrome of b for a linear error correcting code Alice and Bob agreed on beforehand. Alice then proceeds by decoding her word where x = {x1 , x2 , · · · , xn } corresponds to the sign of the quadrature Bob measured for the state she sent. Two-state protocol Alice sends n coherent states drawn from S2 = {|γ , | − γ } with probability 1/2 to the quantum channel. Then, Bob obtains a mixture state, which density matrix ρ2 is symbolized with

2

2 1 2 λ1,3 = e −α (sinhα 2 ± sinα 2 ), 2 2 ∞ e −α /2  (−1)n α 4n+k

and

2

(7)

,

of variance V = 1 + V M , where V M is the modulation variance satisfies with V M = 2α 2 . Four-state protocol Alice sends n coherent states drawn from S4 = {|γ , |γ ∗ , | − γ , | − γ ∗ } with probability 1/4 to the quantum channel. Hence, Bob gets a mixture state ρ4 with

S2 = {|α e −i π /4 , |α e 3i π /4 },

1



σ Z = diag(1, −1) and the covariance Z 2 = 1/ 2 3/2 −1/2 α 2 (λ30/2 λ− + λ λ ). To be specific, Alice prepares the state 1 1 0

with the notation

ρ2 =

4

(6)

(−i )n α 2n+1 |2n + 1. √ (2n + 1)!

where I 2 = diag(1, 1),

In what follows, we describe several CV-QKD protocols with discrete modulation, meanwhile, give some mathematical definitions, which will be used in the rest of the section. In the twostate, four-state and eight-state protocol, Alice sends n random coherent states drawn from

|α e 3i π /2 , |α e 7i π /4 },

e

−i π

Subsequently, Bob performs homodyne detection and classical post-precessing procedures. Next, we compute the covariance matrix Γ2 of the initial bipartite state, which can be defined as

=

2.1. Discrete modulation CV-QKD

∞  (−i )n α 2n |2n, √ (2n)! coshα 2 n=0

1

√

λk

n =0

(4n + k)!

(9)

|4n + k , k ∈ {0, 1, 2, 3} .

We now proceed with evaluating the covariance matrix Γ4 of the initial bipartite four-state. Algebraic manipulations show that it has the following form



Γ4 =

V I2

Z 4σ Z

Z 4σ Z

V I2

3



(11)

, 3/ 2

−1/2

where Z 4 = 2α 2 k=0 λk−1 λk , the definition of V , I 2 and Z 4 have been defined in Eq. (11). Eight-state protocol Alice sends n coherent states drawn from S8 = {|γ1 , |γ2 , |γ1∗ , |γ2∗ , | − γ1 , | − γ2 , | − γ1∗ , | − γ2∗ } with probability 1/8 to Bob. Hence, Bob sees a mixture ρ8 given by

ρ8 =

=

1

|γ1 γ1 | + |γ2 γ2 | + | − γ1 −γ1 | + | − γ2 −γ2 | (12)  + |γ1∗ γ1∗ | + |γ2∗ γ2∗ | + | − γ1∗ −γ1∗ | + | − γ2∗ −γ2∗ |

8

3 

λi |φi φi |,

i =0

where

(|γ γ | + | − γ −γ |) = λ0 |φ0 φ0 | + λ1 |φ1 φ1 |,

(4)

λ0 = e −α coshα 2 , λ1 = e −α sinhα 2 , 2

1

α2

2

1

(5)

α2

λ0,4 = e−α (coshα 2 + cosα 2 ± 2cos √ cosh √ ), 2

4

with the notation

(10)

2

λ1,5 = e−α (sinhα 2 + sinα 2 ± 4

2

√

α

2

2

α2

2cos √ sinh √ 2 2

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±

√

α2

3

α2

2sin √ cosh √ ), 2 2

1

α2

α2

λ2,6 = e −α (coshα 2 − cosα 2 ± 2sin √ sinh √ ), 2

4

2

1

λ3,7 = e −α (sinhα 2 − sinα 2 ∓ 2

√

4

±

√

α2

α

2

2

α2

2cos √ sinh √ 2 2

Fig. 1. The unidimensional CV-QKD protocol with discrete modulation. BS, beam splitter; Sqz, squeeze; EPR, Einstein-Podolsky-Rosen; QM, quantum memory.

α2

2sin √ cosh √ ), 2 2

(13)

∞ e −α /2 

|φk  = √

λk

√

i =0

α 8n+k (8n + k)!

Γ8 =

V I2

Z 8σ Z

Z 8σ Z

V I2

(15)

, 7

3/ 2

−1/2

k=0 λk−1 λk

0



V ( V 2 − 1)

V

0 V2

0

2 − V V−1

0

V I2

ZσZ

ZσZ

V I2

0

⎞

⎟ 2 ⎟ − V V−1 ⎟ ⎟, ⎟ 0 ⎟ ⎠

0

1



(17)

,

γ A0 A1

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

√1

V

0

0

V

0

0

1

√1

V

0

Z

√ − VZ

V

Z

0

⎞

0

(18)

V2

Subsequently, as depicted in Fig. 1, Alice sends mode A 1 to the linear channel. Here, the transmittance and excess noise are symbolized as T x, p and x, p , respectively. Therefore, matrix γ A 0 B 1 has the following form

⎛

γ A0 B 1

V

⎜ ⎜ 0 =⎜ ⎜ B1 ⎝ Cx 0

B

0

Cx 1

V

0

0

Vx 1

B

0

0

T p ( V 2 + χlinep )

 − TpV Z

0

 − TpV Z

⎞

⎟ ⎟ ⎟, ⎟ ⎠

⎞

V

0

0

Vx 0

B

0

0

η T p ( V 2 + χtotp )

 − ηT p V Z

0

√

0

 − ηT p V Z



η C xB 1 =

ηT x V

(20)

Z and V xB 0 = η( V xB 1 +

3. Security analysis 3.1. Secret key rate Afterwards, we analyze the security of the proposed protocol. The secret key rate K in the asymptotic regime has the following form

(21)

where β represents the reconciliation efficiency, I A B (or χ B E ) symbolizes the mutual information between Bob and Alice (or Eve). The mutual information I A B is easily calculated from the following equation

I A B = log2

VA V A|B

(22)

,

where V A denotes the amplitude quadrature and V A | B can be calculated from the elements of matrice γ A | B . The above matrice is defined as follows

γ A | B = γ A − σ A B [ X γ B X]MP σ AT B 

V 2 [T x x +χhom +1]− T x Z 2 V [T x x +χhom +1]

0

0

V



(23)

,

T AB

where γ A , γ B , γ A B and γ are the submatrices of γ A 0 B 0 . Moreover, X = diag(1, 0) and MP represents the operation of pseudoinverse. Accordingly, we have

V A = V , V A|B =

V 2 [T x x + χhom + 1] − T x Z 2 V [T x x + χhom + 1]

(24)

.

Hence, I A B is calculated as

I A B = log2 (19)

⎟ ⎟ ⎟, ⎟ ⎠

χhom ). Here, χlinep represents the channel-added noise with the notation χlinep = T1 + p − 1, χhom denotes the noise introduce by p homodyne detection with χhom = [(1 − η) + v el ], and χtotp is the total noise with χtotp = χlinep + χhom / T p .

=

⎟ √ − VZ ⎟ ⎟ ⎟. ⎟ 0 ⎠

B

K = β I A B − χB E , (16)

with Z ∈ { Z 2 , Z 4 , Z 8 }. After squeezing the phase quadrature, the covariance matrix with the discrete modulation has the form

⎛

B

and V x 1 = 1 + T x x . In particular,

Cx 0

B

where V = 1 + V M . Different with the traditional unidimensional modulation, the covariance matrix for unidimensional CV-QKD with discrete modulation can be described as



Tx Z V

0

with the notation C x 0 =

.

In the traditional unidimensional modulation, the coherent state is modulated by its phase quadrature ( P ) or amplitude quadrature ( X ) in the phase space. Therefore, the modulated quadrature obeys the Gaussian distribution, while the other quadrature normalized at the shot-noise unit (SUN). For traditional unidimensional modulation, squeezing the amplitude quadrature with √ squeezing parameter −log ( V ), the covariance matrix can be described as [16]

⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ V ( V 2 − 1) ⎝

B

⎝ Cx 0

2.2. An introduction to unidimensional CV-QKD

V

V

⎜ ⎜ 0

γ A0 B 0 = ⎜ ⎜



where the covariance Z 8 = 2α 2

⎛

⎛

|8n + k , k ∈ {0, 1, 2, · · · , 7} . (14)

Similarly, the covariance matrix Γ8 can be described as





the homodyne detection is deemed as the beam splitter (BS) with electronic noise v el . After the operation of BS, we have

and 2

B

with the notation C x 1 =

VA V A|B

=

V 2 [T x x + χhom + 1] V 2 [T x x + χhom + 1] − T x Z 2

In the Eq. (21), the Holevo bound

χ B E = S (ρ E ) − S (ρ Ep B ),

.

(25)

χ B E has the following form (26)

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where S (ρ E ) and S (ρ E B ) are the entropy and conditional enp tropy of eavesdropper’s state ρ E and ρ E B , respectively. As S (ρ E ) = pB pB S (ρ A 0 B 1 ) and S (ρ E ) = S (ρ A H G ), we can rewrite χ B E as 0

χ B E = S (ρ A 0 B 1 ) − S (ρ =

  2  λi − 1 G

2

i =1

pB A0 H G )

−

  5  λi − 1 G

2

i =3

(27)

,

with the notation G (x) = (x + 1)log2 (x + 1) − xlog2 x. Specially, λ1,2 and λ3,4,5 represent symplectic eigenvalues of matrix γ A 0 B 1 and γ A 0 H G , respectively. The analytical expression for λ1,2 of the matrix γ A 0 B 1 is

λ21,2 =

1 2

A±





A 2 − 4B ,

(28)

with the notation

A = λ21 + λ22 = detγ A 0 + detγ B 1 + 2detσ A 0 B 1 ,

(29)

B = λ21 λ22 = detγ A 0 B 1 .

Before we calculate the second part of Eq. (27), we need to consider the mixture state γ A 0 B 0 H G . The above state is derived by applying a BS operation on mode B 1 and H 0 , with





γ A 0 B 0 H G = B S T γ A 0 B 1 ⊕γ H 0 G B S ,

(30)

where

where

⎛

γ A0

T γ Ap0BH G = ⎜ ⎝ C A0 −H

C TA −G 0

C A0 −H

C A 0 −G

⎞

C H −G ⎟ ⎠.

γH C TH −G

(38)

γG

Besides, the matrix C i j −kl reads

B S = I2 ⊕ Y B1 H0 ⊕ I2, and



Y B1 H0 =

⎜

Y H0 G = ⎝

ηI2

√ − 1 − ηI2

⎛

(31)

√

1 − ηI2





V d I2

− V d2 − 1σ Z

(32)

,

ηI2



γB 0 A0 H G =

γB0 σ BT0 A 0 H G

σB 0 A0 H G γ A0 H G

Therefore, the mixture state ing equation

⎠.

(34)

.

γ Ap0BH G is characterized by the follow-

γ Ap0BH G = γ A 0 H G − σ BT0 A 0 H G X γ B 0 X

M P

σB 0 A0 H G .

As stated above, eigenvalues λ3,4,5 of the matrix solutions of the third order polynomial

(35)

γ Ap0BH G being the

t 3 − Δ31 t 2 − Δ32 t − Δ33 = 0, which is calculated from the matrix

(39)

,

where αmn is the elementary submatrice describing the correlap tions between a part of modes of the covariance matrice γ A BH G .

(36)

γ Ap0BH G with

= detγ A 0 + detγ H + detγG + 2detC A 0 − H + 2detC A 0 −G + 2detC H −G , Δ32 = λ23 λ24 + λ24 λ25 + λ23 λ25 = detγ A 0 H + detγ H G + detγ A 0 G = 2detC A 0 H − H G + 2detC A 0 H − A 0 G + 2detC H G − A 0 H ,

As stated in Sec. 3.1, if we want to estimate the mutual inB B formation χ B E , we need to acquire the value of C x 1 and V x 1 . Nevertheless, we unable to estimate the T x and x , due to that no modulation has been performed in the X quadrature at the B transmitter. In spite of the unknown C x 1 , we can still estimate B1 the parameter V x through measuring the P quadrature at the receiving end. Based on the Heisenberg uncertainty principle, the unknown parameters need to satisfy the requirement of the physicality as

γ A 0 B 1 + i Ω ≥ 0,

(37)

(40)

where Ω is the symplectic form [26]

Ω=

2 



ω, ω =

i =1

Δ31 = λ23 + λ24 + λ25

= λ23 λ24 λ25 = detγ A 0 H G ,



(33)





C i j −kl =

αik αil α jk α jl

3.2. Uncertainty relations

V d2 − 1σ Z ⎟ V d I2



0

⎞

Here, V d is the variance satisfies with V d = 1 + v el /(1 − η). Particularly, γ B 0 A 0 H G can be derived by applying the permutation matrix on mode γ A 0 B 0 H G , and it can be described as

Δ33

Fig. 2. The physical region and unphysical region under different detection scenarios. The green line is the ideal condition with η = 1, v el = 0 and V M = 10. The red line denotes the realistic detection, with the efficiency of homodyne detection η = 0.6, the modulation variance V M = 1 and the electronic noise v el = 0.1. Here, we set the excess noise x, p = 0.01, and the transmittance T x, p = 0.2. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

0 −1

1 0

 .

(41)

Furthermore, considering the realistic scenario, when the mode B 1 is converted to B 0 , the new covariance matrix γ A 0 B 0 are constrained on another uncertainty relations. According to the uncertainty relation, the states need to satisfy with

γ A 0 B 1 + i Ω ≥ 0,

(42a)

γ A 0 B 0 + i Ω ≥ 0.

(42b)

Here, two uncertainty relations represent two parabolic curves respectively. Without loss of generality, we take the Eq. (42a) of the four-state unidimensional CV-QKD protocol as a case to analyze the uncertainty relation. As depicted in Fig. 2, the plane is

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Fig. 3. The influence of other related parameters on the proposed protocol. (a) Different modulation variance with T x, p = 0.2 and with V M = 10 and x, p = 0.01.

5

x, p = 0.01; (b) Different transmittance

B

Fig. 4. (a) The K versus V x 1 for different modulation modes, respectively. From right to left, the lines correspond to the protocol of two-state, four-state and eight-state B and Gaussian-state unidimensional CV-QKD protocol. Here, we set V M = 3, v el = 0.01, η = 0.6, β = 0.97, x, p = 0.01 and T x = 0.1. (b) The K versus C x 1 for different modulation modes, respectively. From right to left, the lines correspond to the protocol of Gaussian-state, eight-state, four-state and two-state unidimensional CV-QKD protocol, respectively. Here, we set V M = 3, v el = 0.01, η = 0.8, β = 0.97, x, p = 0.01 and T x = 0.5.

split into two parts: one is the unphysical region, another part is the physical region. On the one hand, in the unphysical region, two parameters cannot meat the above-mentioned requirement simultaneously, while the Heisenberg will be violated. On the other hand, the physical region subdivided into two regions, namely the real physical region R1 and the pseudo physical region R2. In the R1 region, the values of two parameters can ensure that the attack of Eve follows the physical criteria. Meanwhile, pseudo physical illustrate that, even if Eve’s attacks are unphysical, we can still achieve a physical state after some transformation and detection of Bob. Subsequently, we further consider the influence of other related parameters to its performance. In Fig. 3(a), as the modulation variance V M increases, the parabolic curve becomes larger and gradually moves towards left. Similarly, as depicted in Fig. 3(b), the parabolic curve becomes broader as the transmittance increases. In a word, increasing the modulation variance and the transmittance can effectively expand the physical region. B The curves for K versus the V x 1 under the different modulation modes are depicted in Fig. 4(a). In simulations, we can conB clude that the K decreases with the amplitude variance of V x 1 . In addition, Fig. 4(b) displays the relation between the K and the B covariance C x 1 . Unlike the Fig. 4(a), in this subgraph, we can find B that the K increases with the covariance of C x 1 . Fig. 5 displays a more intuitive description for the secure seB B cret key rate. In Fig. 5(a), independent variables V x 1 and C x 1 are B1 B1 set as V x ∈ [1, 2] and C x ∈ [0.3, 0.9]. From top to bottom, the surfaces correspond to the protocol of two-state, four-state and

eight-state unidimensional CV-QKD protocol. In simulations, the K B increase with the variance of C x 1 . While, whether there existing B1 the optimal value C xm for secret key rate? This question can be resolved from Fig. 5(b). In this subgraph, the independent variB B B ables V x 1 and C x 1 are in the range from V x 1 ∈ [0.995, 1.02] and B1 C x ∈ [0.1, 1.6]. From top to bottom, the surfaces correspond to the protocol of four-state and Gaussian unidimensional CV-QKD protoB col. We can see that there indeed exists a optimal value C xm1 to achieve a higher K . Besides, as shown in Fig. 5(b), we can conclude that the K for discrete modulation is better than Gaussian in this range of values. To further exploring the impact of T x and x on the K , the three-dimensional numerical simulation is carried out. As shown in Fig. 6, from top to bottom, the surfaces correspond to the protocol of two-state, four-state and eight-state unidimensional CV-QKD protocol, respectively. We can see that secure region decreases from the top to bottom, that is to say, the secure region of twostate is larger than eight-state unidimensional CV-QKD. Moreover, the K increases obviously as T x increases and x decreases. With the fixed parameter T p = 0.53, the K as a function of distance is presented with different modulation modes in Fig. 7(a). Moreover, what are the optimal value T p and the maximum transmission distance for two-state, four-state and eight-state? As depicted in Fig. 7(b), the optimal T p for eight-state, four-state and two-state are T p 8 = 0.4576, T p 4 = 0.4854 and T p 2 = 0.5061, respectively. Hence, we can conclude that Alice and Bob can achieve the secure secret key rate in the proposed protocol.

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B

Fig. 5. Three-dimensional plots for the secret key rate (bits/pulse) from the different range of parameters. (a) Different independent variables values with V x 1 ∈ [1, 2] and B C x 1 ∈ [0.3, 0.9]. From top to bottom, the surfaces correspond to the protocol of two-state, four-state and eight-state unidimensional CV-QKD protocol. (b) Comparison between the Gaussian modulation and four-state unidimensional CV-QKD protocol under different independent variables. From top to bottom, the surfaces correspond to the protocol B B B B of four-state and Gaussian unidimensional CV-QKD protocol. Here, V x 1 and C x 1 are in the range from V x 1 ∈ [0.995, 1.02] and C x 1 ∈ [0.1, 1.6]. The other parameters are set as V M = 3, T x, p = 0.5, η = 0.6.

implementation of the QKD. Specifically, we insist on three instances of such protocol, namely the two-state protocol, four-state protocol and eight-state protocol, as well as its initial state and the covariance matrix. For one thing, we analyze the boundary of physical region on the basis of the Heisenberg uncertainty relation. Particularly, the relationship of two parameters related to the unmodulated quadrature are derived on the realistic condition. For another, we calculated the secret key rate in a linear quantum channel under the asymptomatic regime. Besides, we utilize a novel proof approach to achieve a lower bound valid of transmission distance. This scheme shows an available method to further simplified the implementation of the QKD. Declaration of competing interest Fig. 6. The impact of T x and x on the K (bits/pulse). From top to bottom, the surfaces correspond to the protocol of two-state, four-state and eight-state unidimensional CV-QKD protocol, respectively. Here, we set V M = 3, v el = 0.01, η = 0.9, β = 0.9, p = 0.9 and T p = 0.7.

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. Acknowledgements

4. Conclusion In this paper, we proposed an unidimensional CV-QKD with discrete modulation. The proposed protocol further simplified the

This work is supported by the National Natural Science Foundation of China (Grant No. 61801522), and National Nature Science Foundation of Hunan Province, China (Grant No. 2019JJ40352).

Fig. 7. Secret key rate K versus distance in the unidimensional CV-QKD protocols for (a) fixed parameter T p = 0.53, and (b) optimal values T po , respectively. Particularly, the optimal values for eight-state, four-state and two-state are T p 8 = 0.4576, T p 4 = 0.4854 and T p 2 = 0.5061, respectively. Here, we set V M = 3, v el = 0.01, η = 0.8, β = 0.97 and x, p = 0.01.

JID:PLA AID:126061 /SCO Doctopic: Quantum physics

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W. Zhao et al. / Physics Letters A ••• (••••) ••••••

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Appendix A

References

In section 3.1, we are working on the assumption that the quantum between Alice and Bob is a linear quantum channel [12,27,28]. In other words, we restrict the Eve’s attack performed to a linear quantum channel. In what follows, we utilize a novel proof approach to achieve a lower bound valid of distance [23]. Besides, the lower bound can resist collective attack. The asymptotic secret key rate of Devetak-Winter rate can be defined as [29]

[1] S. Liao, W. Cai, W. Liu, L. Zhang, Y. Li, J. Ren, J. Pan, Satellite-to-ground quantum key distribution, Nature 549 (2017) 43–47. [2] S.K. Chong, T.H. Wang, Quantum key agreement protocol based on BB84, Opt. Commun. 283 (2010) 1192–1195. [3] 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 (2009) 1301. [4] H.K. Lo, M. Curty, K. Tamaki, Secure quantum key distribution, Nat. Photonics 8 (2015) 595–604. [5] C. Weedbrook, S. Pirandola, R. Garcá-Patrón, N.J. Cerf, T.C. Ralph, J.H. Shapiro, S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84 (2012) 621. [6] L.B. Samuel, V.L. Peter, Quantum information with continuous variables, Rev. Mod. Phys. 77 (2005) 513. [7] F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. Cerf, P. Grangier, Quantum key distribution using gaussian-modulated coherent states, Nature 421 (2003) 238–241. [8] D. Huang, D. Lin, C. Wang, W. Liu, S. Fang, J. Peng, G. Zeng, Continuous-variable quantum key distribution with 1 Mbps secure key rate, Opt. Express 23 (2015) 17511–17519. [9] Y. Zhao, C.F. Fung, B. Qi, C. Chen, H. Lo, Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems, Phys. Rev. A 78 (2008) 042333. [10] F. Xu, B. Qi, H.K. Lo, Experimental demonstration of phase-remapping attack in a practical quantum key 118 distribution system, New J. Phys. 11 (2010) 66. [11] P. Jouguet, S. Kunz-Jacques, E. Diamanti, A. Leverrier, Analysis of imperfections in practical continuous-variable quantum key distribution, Phys. Rev. A 86 (2012) 032309. [12] A. Leverrier, P. Grangier, Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation, Phys. Rev. Lett. 102 (2009) 180504. [13] M.D. Reid, Quantum cryptography with a predetermined key, using continuousvariable Einstein-Podolsky-Rosen correlations, Phys. Rev. A 62 (2000) 062308. [14] T. Hirano, H. Yamanaka, M. Ashikaga, T. Konishi, R. Namiki, Quantum cryptography using pulsed homodyne detection, Phys. Rev. A 68 (2003) 042331. [15] S. Lorenz, N. Korolkova, G. Leuchs, Continuous-variable quantum key distribution using polarization encoding and post selection, Appl. Phys. B 79 (2004) 273–277. [16] V.C. Usenko, F. Grosshans, Unidimensional continuous-variable quantum key distribution, Phys. Rev. A 92 (2015). [17] P. Wang, X. Wang, J. Li, Y. Li, Finite-size analysis of unidimensional continuousvariable quantum key distribution under realistic conditions, Opt. Express 25 (2017) 27995. [18] Q. Liao, Y. Guo, C. Xie, D. Huang, P. Huang, G. Zeng, Composable security of unidimensional continuous-variable quantum key distribution, Quantum Inf. Process. 17 (2018) 113. [19] X. Wang, W. Liu, P. Wang, Y. Li, Experimental study on all-fiber-based unidimensional continuous-variable quantum key distribution, Phys. Rev. A 95 (2017) 062330. [20] H. Zhang, X. Ruan, X. Wu, L. Zhang, Y. Guo, D. Huang, Plug-and-play unidimensional continuous-variable quantum key distribution, Quantum Inf. Process. 18 (2019) 128. [21] W. Pu, W. Xuyang, L. Yongmin, Security analysis of unidimensional continuousvariable quantum key distribution using uncertainty relations, Entropy 20 (2018) 157. [22] D. Bai, P. Huang, Y. Zhu, H. Ma, T. Xiao, T. Wang, G. Zeng, Unidimensional continuous-variable measurement-device-independent quantum key distribution, arXiv: Quantum Physics, 2019. [23] S. Ghorai, P. Grangier, E. Diamanti, A. Leverrier, Asymptotic security of continuous-variable quantum key distribution with a discrete modulation, Phys. Rev. X 9 (2019) 021059. [24] A. Leverrier, P. Grangier, Long distance quantum key distribution with continuous variables, in: Conference on Quantum Computation, Communication, and Cryptography, Springer, Berlin, Heidelberg, 2011. [25] 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 (2012) 1250004. [26] A. Serafini, M.G.A. Paris, F. Illuminati, S. De Siena, Quantifying decoherence in continuous variable systems, J. Opt. B, Quantum Semiclass. Opt. 7 (2005). [27] M. Heid, N. Lütkenhaus, Security of coherent-state quantum cryptography in the presence of Gaussian noise, Phys. Rev. A 76 (2007) 022313. [28] D. Sych, G. Leuchs, Coherent state quantum key distribution with multi letter phase-shift keying, New J. Phys. 12 (2010) 053019. [29] I. Devetak, A. Winter, Distillation of secret key and entanglement from quantum states, Proc. R. Soc. A 461 (2005) 207–235.

S D W = β I A B − supχ E ,

(43)

where the definitions of β and I A B are coincided with Eq. (21), and χ E is the Eve’s Holevo information in the Devetak-Winter rate. The channel is described via Kraus operators { E i }. The opera † tors satisfy i = E i E i = I A . Without loss of generality, we take the four-state unidimensional CV-QKD protocol into consideration. Accordingly, the quantum state ςab is expressed as

ςa,b =

3 1 

4

|ψk |ψl  ⊗ σk,l ,

(44)

k,l=0

where 3 1

|ψk  =

σk,l =

2

e i (1+2k)mπ /4 |φm ,

m =0



(45) †

E i |αk αl | E i .

i

Besides, the covariance matrix of state ςab has been described in the Eq. (11). In what follows, we need to calculate the lower bound on parameters Z 4 . First of all, we define parameters as fol3 lows  = k=0 |ψk ψk | and C = a ⊗ b + a†  ⊗ b† , where  represents the orthogonal projector onto the space spanned. Accordingly, we have Z 4 = Tr(C X ). For concreteness, the positive semidefinite matrice X represents the state ςab . Moreover, the matrice X need to satisfy the following linear constraints



tr( B 1 X ) = 1 + 2T α 2 ,

√

(46)

tr( B 2 X ) = 2 T α , with the notation

B 1 =  ⊗ (1 + 2b† b), B 2 = (|ψ0 ψ0 | − |ψ2 ψ2 |) ⊗ qˆ + (|ψ1 ψ1 | − |ψ3 ψ3 |) ⊗ pˆ . (47) Consequently, the lower bound of Z 4∗ satisfies with the following linear constraints

Z 4∗ := min tr(C X ),

⎧ Tr( B 1 X ) = 1 + 2T α 2 ⎪ ⎪ ⎪ ⎨ Tr( B X ) = 2√ T α 2 such that . ⎪ Tr((|ψl ψk |) X ) = 14 αl |αk  ⎪ ⎪ ⎩ X ≥0

(48)

By utilizing the optimum bound on Z 4∗ , we can obtain the final whole covariance matrix ∗4 . Subsequently, the lower bound on supχ E can be calculated from the state ςab with covariance matrix



∗

Γ4 =

V I2 Z 4∗ σ Z

Z 4∗ σ Z V I2



.

(49)