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Journals >Laser & Optoelectronics Progress >Volume 58 >Issue 7 >Page 0727004 > Article

Laser & Optoelectronics Progress
Vol. 58, Issue 7, 0727004 (2021)

Security Analysis of Three-State Continuous Variable Quantum Key Distribution Protocol Based on Discrete Modulation

Youdong Sun, Yongxin Xing, and Tianyi Wang^*

Author Affiliations

College of Big Data and Information Engineering, Guizhou University, Guiyang , Guizhou 550025, China

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DOI: 10.3788/LOP202158.0727004 Cite this Article Set citation alerts

Youdong Sun, Yongxin Xing, Tianyi Wang. Security Analysis of Three-State Continuous Variable Quantum Key Distribution Protocol Based on Discrete Modulation[J]. Laser & Optoelectronics Progress, 2021, 58(7): 0727004 Copy Citation Text

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Abstract

At present, the continuous-variable quantum key distribution (CV-QKD) protocol based on discrete modulation has received more attention. In this paper, we propose a three-state CV-QKD protocol based on post-selection. The security key rate of the proposed protocol is calculated under collective attack and reverse reconciliation and compared with that of the four-state protocol. The simulation results show that the proposed three-state protocol can outperform the four-state protocol in security rate if the transmission distance is not too long, indicating that the three-state protocol is more feasible for a short-range application, such as metropolitan area networks.

Keywords

continuous variable discrete modulation post-selection quantum key distribution quantum optics three-state protocol

\{\begin{array}{l} \hat{x} = \frac{\hat{a} + {\hat{a}}^{†}}{2} \\ \hat{p} = \frac{\hat{a} - {\hat{a}}^{†}}{2 i} \end{array}

。（1）

x (ϕ_{B}) = x c o s ϕ_{B} + p s i n ϕ_{B}

，（2）

P (m| S) = \sqrt[]{\frac{2}{π (1 + ξ)}} e x p [- 2 \frac{{(m - \sqrt[]{η} S)}^{2}}{1 + ξ}]

，（3）

ε = {[1 + e x p (8 \frac{\sqrt[]{η}}{1 + ξ} |m| α)]}^{- 1}

。（4）

I_{A B} = 1 - h (ε)

，（5）

|E_{E P R}〉 = \sqrt[]{\frac{π}{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x_{1} d x_{2} e x p (- V x_{1}^{2} - x_{2}^{2} / V) {|\frac{x_{1} + x_{2}}{\sqrt[]{2}}〉}_{E_{1}} {|\frac{x_{1} - x_{2}}{\sqrt[]{2}}〉}_{E_{2}}

，（6）

\frac{1}{2} (V + \frac{1}{V}) = \frac{1 - η + ξ}{1 - η} 。

（7）

|S〉 = {(\frac{2}{π})}^{\frac{1}{4}} \int_{- \infty}^{\infty} d x e x p [- {(x - s)}^{2}] |x〉

。（8）

{|x〉}_{A} {|x_{2}〉}_{E_{2}} \to {|\sqrt[]{η} x - \sqrt[]{1 - η} x_{2}〉}_{A} {|\sqrt[]{1 - η} x + \sqrt[]{η} x_{2}〉}_{E_{2}}

，（9）

{|x〉}_{A} {|\frac{x_{1} - x_{2}}{\sqrt[]{2}}〉}_{E_{2}} \to {|m〉}_{A} {|\sqrt[]{\frac{1 - η}{η}} m + \frac{x_{1} - x_{2}}{\sqrt[]{2 η}}〉}_{E_{2}}

。（10）

|φ (S, m)〉 = {(\frac{8}{π^{3} η^{2}})}^{\frac{1}{4}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x_{1} d x_{2} φ (S, m) {|\frac{x_{1} + x_{2}}{\sqrt[]{2}}〉}_{E_{1}} {|\sqrt[]{\frac{1 - η}{η}} m + \frac{x_{1} - x_{2}}{\sqrt[]{2 η}}〉}_{E_{2}}

，（11）

φ (S, m) = e x p \{- {[\sqrt[]{\frac{1 - η}{η} (x_{1} - x_{2})} + \frac{m}{\sqrt[]{η}} - S]}^{2} - V x_{1}^{2} - x_{2}^{2} / V\}

。（12）

〈φ (S, m)| φ (S, m)〉 = P (m / S) 。

（13）

|e_{i j}〉 = N |φ [{(- 1)}^{i} |S|, φ {(- 1)}^{j} |m|]〉

，（14）

\{\begin{array}{l} |e_{00}〉 = N \{{(\frac{8}{π^{3} η^{2}})}^{\frac{1}{4}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x_{1} d x_{2} e x p \{- {[\sqrt[]{\frac{1 - η}{2 η}} (x_{1} - x_{2}) + \frac{|m|}{\sqrt[]{η}} - |s|]}^{2} - V x_{1}^{2} - x_{2}^{2} / V\} \times \\ {|m〉}_{A} {|\frac{x_{1} + x_{2}}{\sqrt[]{2}}〉}_{E_{1}} {|\sqrt[]{\frac{1 - η}{η}} m + \frac{x_{1} - x_{2}}{\sqrt[]{2 η}}〉}_{E_{2}}\} \\ |e_{01}〉 = N \{{(\frac{8}{π^{3} η^{2}})}^{\frac{1}{4}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x_{1} d x_{2} e x p \{- {[\sqrt[]{\frac{1 - η}{2 η}} (x_{1} - x_{2}) + \frac{- |m|}{\sqrt[]{η}} - |s|]}^{2} - V x_{1}^{2} - x_{2}^{2} / V\} \times \\ {|m〉}_{A} {|\frac{x_{1} + x_{2}}{\sqrt[]{2}}〉}_{E_{1}} {|\sqrt[]{\frac{1 - η}{η}} m + \frac{x_{1} - x_{2}}{\sqrt[]{2 η}}〉}_{E_{2}}\} \\ |e_{10}〉 = N \{{(\frac{8}{π^{3} η^{2}})}^{\frac{1}{4}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x_{1} d x_{2} e x p \{- {[\sqrt[]{\frac{1 - η}{2 η}} (x_{1} - x_{2}) + \frac{|m|}{\sqrt[]{η}} + |s|]}^{2} - V x_{1}^{2} - x_{2}^{2} / V\} \times \\ {|m〉}_{A} {|\frac{x_{1} + x_{2}}{\sqrt[]{2}}〉}_{E_{1}} {|\sqrt[]{\frac{1 - η}{η}} m + \frac{x_{1} - x_{2}}{\sqrt[]{2 η}}〉}_{E_{2}}\} \\ |e_{11}〉 = N \{{(\frac{8}{π^{3} η^{2}})}^{\frac{1}{4}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d x_{1} d x_{2} e x p \{- {[\sqrt[]{\frac{1 - η}{2 η}} (x_{1} - x_{2}) + \frac{- |m|}{\sqrt[]{η}} + |s|]}^{2} - V x_{1}^{2} - x_{2}^{2} / V\} \times \\ {|m〉}_{A} {|\frac{x_{1} + x_{2}}{\sqrt[]{2}}〉}_{E_{1}} {|\sqrt[]{\frac{1 - η}{η}} m + \frac{x_{1} - x_{2}}{\sqrt[]{2 η}}〉}_{E_{2}}\} \end{array}

。（15）

\{\begin{array}{l} ρ_{B}^{0} = (1 - ε) |e_{00}〉 〈e_{00}| + ε |e_{10}〉 〈e_{10}| \\ ρ_{B}^{1} = (1 - ε) |e_{11}〉 〈e_{11}| + ε |e_{01}〉 〈e_{01}| \end{array}

，（16）

χ = S (ρ) - S (ρ_{B}^{0}) / 2 - S (ρ_{B}^{1}) / 2

，（17）

G = [\begin{matrix} 1 - ε & δ t \\ δ t & ε \end{matrix}]

，（18）

t = 〈e_{00}| e_{10}〉 = 〈e_{11}| e_{01}〉 = e x p (- 2 \frac{1 + ξ - η}{1 + ξ} α^{2})

，（19）

δ = \sqrt[]{ε (1 - ε)},

（20）

\frac{1}{2} [1 \pm \sqrt[]{1 - 4 δ^{2} {(1 - t)}^{2}}] 。

（21）

S (ρ_{B}^{0}) = S (ρ_{B}^{1})

。（22）

G = [\begin{matrix} 1 - ε & δ s & δ t & (1 - ε) s t u \\ δ s & ε & ε s t / u & δ t \\ δ t & ε s t / u & ε & δ s \\ (1 - ε) s t u & δ t & δ s & 1 - ε \end{matrix}]

，（23）

s = 〈e_{00}| e_{01}〉 = 〈e_{11}| e_{10}〉 = e x p [- 2 \frac{ξ (2 + ξ)}{1 + ξ} m^{2}]

，（24）

u = e x p (4 \frac{ξ \sqrt[]{η}}{1 + ξ} α |m|)

。（25）

\{\begin{array}{l} \frac{1}{4 u} (v_{+} \pm \sqrt[]{v_{+} - w_{+}}) \\ \frac{1}{4 u} (v_{-} \pm \sqrt[]{v_{-} + w_{-}}) \end{array}

，（26）

v_{\pm} = u \pm s t [ε + (1 - ε) u^{2}],

（27）

w_{\pm} = 4 δ^{2} u [s t {(1 - u)}^{2} \pm (1 - s^{2}) (1 - t^{2}) u]

。（28）

K = η_{p r} (I_{A B} - χ)

，（29）

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Youdong Sun, Yongxin Xing, Tianyi Wang. Security Analysis of Three-State Continuous Variable Quantum Key Distribution Protocol Based on Discrete Modulation[J]. Laser & Optoelectronics Progress, 2021, 58(7): 0727004

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