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Photonics Research
Vol. 11, Issue 3, 463 (2023)

Continuous variable quantum key distribution with a shared partially characterized entangled source

Shanna Du^1、2、†, Pu Wang^{1、2、3、†}, Jianqiang Liu^1、2, Yan Tian^1、2, and Yongmin Li^1、2、*

Author Affiliations

¹State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China

²Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

³School of Information, Shanxi University of Finance and Economics, Taiyuan 030006, China

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DOI: 10.1364/PRJ.475943 Cite this Article Set citation alerts

Shanna Du, Pu Wang, Jianqiang Liu, Yan Tian, Yongmin Li. Continuous variable quantum key distribution with a shared partially characterized entangled source[J]. Photonics Research, 2023, 11(3): 463 Copy Citation Text

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Abstract

Locking the sophisticated and expensive entanglement sources at the shared relay node is a promising choice for building a star-type quantum network with efficient use of quantum resources, where the involved parties only need to equip low-cost and simple homodyne detectors. Here, to our best knowledge, we demonstrate the first experimental continuous variable quantum key distribution with an entanglement source between the two users. We consider a practical partially characterized entangled source and establish the security analysis model of the protocol under realistic conditions. By applying a biased base technology, the higher key rate than that of the original protocol is achieved. The experimental results demonstrate that the distance between two users can reach up to 60 km over telecom single-mode fiber, implying the feasibility for high-rate and secure communication with a shared entangled source at metropolitan distances.

K_{R R}^{\infty} = (1 - P_{A}) (1 - P_{B}) (β I_{A B}^{x} - χ_{B E}^{x}) + P_{A} P_{B} (β I_{A B}^{p} - χ_{B E}^{p}),

(1)

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χ_{B E}^{x (p)} = S (ρ_{E}) - S (ρ_{E / B}^{x (p)}) = S (ρ_{A_{1} B_{1}}) - S (ρ_{A_{1} F G / B}^{x (p)}),

(2)

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K_{R R}^{\infty} = P_{A} P_{B} (β I_{A B}^{x} - χ_{B E}^{x}) + (1 - P_{A}) (1 - P_{B}) (β I_{A B}^{p} - χ_{B E}^{p}) .

(A1)

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I_{A B}^{x} = \frac{1}{2} \log_{2} \frac{V_{A}^{x}}{V_{A / B}^{x}} = \frac{1}{2} \log_{2} \frac{V_{A}^{x}}{V_{A}^{x} - {(C_{A B}^{x})}^{2} / V_{B}^{x}}, I_{A B}^{p} = \frac{1}{2} \log_{2} \frac{V_{A}^{p}}{V_{A / B}^{p}} = \frac{1}{2} \log_{2} \frac{V_{A}^{p}}{V_{A}^{p} - {(C_{A B}^{p})}^{2} / V_{B}^{p}},

(A2)

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χ_{B E}^{x (p)} = S (ρ_{E}) - S (ρ_{E / B}^{x (p)}) .

(A3)

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χ_{B E}^{x (p)} = S (ρ_{A_{1} B_{1}}) - S (ρ_{A_{1} F G / B}^{x (p)}),

(A4)

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Ω = \oplus_{j = 1}^{k} [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}],

(A5)

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χ_{B E}^{x (p)} = \sum_{i = 1}^{2} f (λ_{i}) - \sum_{i = 3}^{5} f (λ_{i}^{x (p)}),

(A6)

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γ_{A_{0} B_{0}} = [\begin{matrix} V_{A_{0}} I & C_{A_{0} B_{0}} σ_{z} \\ C_{A_{0} B_{0}} σ_{z} & V_{B_{0}} I \end{matrix}],

(A7)

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V_{A_{0}} = V_{B_{0}} = \frac{1}{2} (1 / s + s + Δ V_{0}), C_{A_{0} B_{0}} = \frac{1}{2} (1 / s - s + Δ V_{0}), ⟨ {({\hat{x}}_{A_{0}} - {\hat{x}}_{B_{0}})}^{2} ⟩ = ⟨ {({\hat{p}}_{A_{0}} + {\hat{p}}_{B_{0}})}^{2} ⟩ = 2 s,

(A8)

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γ_{A_{1} B_{1}} = [\begin{matrix} V_{A_{1}}^{x} & 0 & C_{A_{1} B_{1}}^{x} & 0 \\ 0 & V_{A_{1}}^{p} & 0 & C_{A_{1} B_{1}}^{p} \\ C_{A_{1} B_{1}}^{x} & 0 & V_{B_{1}}^{x} & 0 \\ 0 & C_{A_{1} B_{1}}^{p} & 0 & V_{B_{1}}^{p} \end{matrix}],

(A9)

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V_{A_{1}}^{x} = V_{A_{1}}^{p} = V_{A_{1}} = T_{A} (V_{A_{0}} + ε_{A}) + 1 - T_{A}, V_{B_{1}}^{x} = V_{B_{1}}^{p} = V_{B_{1}} = T_{B} [V_{B_{0}} η_{S} + (1 - η_{S}) + ε_{B}] + 1 - T_{B}, C_{A_{1} B_{1}}^{x} = \sqrt{η_{S} T_{A} T_{B}} C_{A_{0} B_{0}} + g \sqrt{1 - T_{A}} \sqrt{1 - T_{B}}, C_{A_{1} B_{1}}^{p} = - \sqrt{η_{S} T_{A} T_{B}} C_{A_{0} B_{0}} + g^{'} \sqrt{1 - T_{A}} \sqrt{1 - T_{B}},

(A10)

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γ_{E_{1} E_{2}} = [\begin{matrix} ω_{A} I & G \\ G & ω_{B} I \end{matrix}], G = [\begin{matrix} g & 0 \\ 0 & g^{'} \end{matrix}],

(A11)

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γ_{A_{1} F G / B}^{x (p)} = γ_{A_{1} F G} - σ_{A_{1} F G; B} {(X γ_{B} X)}^{M P} σ_{A_{1} F G; B}^{T},

(A12)

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γ_{A_{1} F G B} = [\begin{matrix} γ_{A_{1} F G} & σ_{A_{1} F G; B} \\ σ_{A_{1} F G; B}^{T} & γ_{B} \end{matrix}] .

(A13)

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γ_{A B} = [\begin{matrix} V_{A}^{x} & 0 & C_{A B}^{x} & 0 \\ 0 & V_{A}^{p} & 0 & C_{A B}^{p} \\ C_{A B}^{x} & 0 & V_{B}^{x} & 0 \\ 0 & C_{A B}^{p} & 0 & V_{B}^{p} \end{matrix}],

(A14)

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V_{A}^{x} = V_{A}^{p} = η_{A} V_{A_{1}} + 1 - η_{A} + v_{e l A}, V_{B}^{x} = V_{B}^{p} = η_{B} V_{B_{1}} + 1 - η_{B} + v_{e l B}, C_{A B}^{x} = \sqrt{η_{A} η_{B}} C_{A_{1} B_{1}}^{x}, C_{A B}^{p} = \sqrt{η_{A} η_{B}} C_{A_{1} B_{1}}^{p} .

(A15)

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g^{'} = - g = ϕ, ϕ = \min {\sqrt{(ω_{A} - 1) (ω_{B} + 1)}, \sqrt{(ω_{A} + 1) (ω_{B} - 1)}} .

(B1)

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I_{A B}^{x} = \frac{1}{2} \log_{2} \frac{V_{A_{1}} V_{B_{1}}}{V_{A_{1}} V_{B_{1}} - {(C_{A_{1} B_{1}}^{x})}^{2}}, I_{A B}^{p} = \frac{1}{2} \log_{2} \frac{V_{A_{1}} V_{B_{1}}}{V_{A_{1}} V_{B_{1}} - {(C_{A_{1} B_{1}}^{p})}^{2}} .

(B2)

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χ_{B E}^{x} = f (λ_{1}) + f (λ_{2}) - f (λ_{3}^{x}), χ_{B E}^{p} = f (λ_{1}) + f (λ_{2}) - f (λ_{3}^{p}),

(B3)

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λ_{1} = \sqrt{\frac{ξ_{1} + ξ_{2}}{2}}, λ_{2} = \sqrt{\frac{ξ_{1} - ξ_{2}}{2}}, ξ_{1} = V_{A_{1}}^{2} + V_{B_{1}}^{2} + 2 C_{A_{1} B_{1}}^{x} C_{A_{1} B_{1}}^{p}, ξ_{2} = \sqrt{{(V_{A_{1}}^{2} - V_{B_{1}}^{2})}^{2} + 4 (V_{A_{1}}^{2} + V_{B_{1}}^{2}) C_{A_{1} B_{1}}^{x} C_{A_{1} B_{1}}^{p} + 4 V_{A_{1}} V_{B_{1}} [{(C_{A_{1} B_{1}}^{x})}^{2} + {(C_{A_{1} B_{1}}^{p})}^{2}]},

(B4)

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λ_{3}^{x} = \sqrt{V_{A_{1}}^{2} - \frac{V_{A_{1}} {(C_{A_{1} B_{1}}^{x})}^{2}}{V_{B_{1}}}}, λ_{3}^{p} = \sqrt{V_{A_{1}}^{2} - \frac{V_{A_{1}} {(C_{A_{1} B_{1}}^{p})}^{2}}{V_{B_{1}}}} .

(B5)

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k_{tot} = P_{A} P_{B} k_{A B}^{x} + (1 - P_{A}) (1 - P_{B}) k_{A B}^{p} = P_{A} P_{B} β I_{A B}^{x} + (1 - P_{A}) (1 - P_{B}) β I_{A B}^{p} - P_{A} P_{B} [f (λ_{1}) + f (λ_{2}) - f (λ_{3}^{x})] - (1 - P_{A}) (1 - P_{B}) [f (λ_{1}) + f (λ_{2}) - f (λ_{3}^{p})] .

(B6)

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k_{tot} = \frac{β (I_{A B}^{x} + I_{A B}^{p})}{4} - \frac{f (λ_{1}) + f (λ_{2})}{2} + \frac{f (λ_{3}^{x}) + f (λ_{3}^{p})}{4} .

(B7)

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γ_{A B} = [\begin{matrix} V_{A}^{x} & 0 & C_{A B}^{x} & 0 \\ 0 & V_{A}^{p} & 0 & C_{A B}^{p} \\ C_{A B}^{x} & 0 & V_{B}^{x} & 0 \\ 0 & C_{A B}^{p} & 0 & V_{B}^{p} \end{matrix}] .

(C1)

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{\hat{V}}_{A}^{x} = \frac{1}{m_{x}} \sum_{i = 1}^{m_{x}} x_{A i}^{2}, {\hat{V}}_{B}^{x} = \frac{1}{m_{x}} \sum_{i = 1}^{m_{x}} x_{B i}^{2}, {\hat{C}}_{A B}^{x} = \frac{1}{m_{x}} \sum_{i = 1}^{m_{x}} x_{A i} x_{B i}, {\hat{V}}_{A}^{p} = \frac{1}{m_{p}} \sum_{i = 1}^{m_{p}} p_{A i}^{2}, {\hat{V}}_{B}^{p} = \frac{1}{m_{p}} \sum_{i = 1}^{m_{p}} p_{B i}^{2}, {\hat{C}}_{A B}^{p} = \frac{1}{m_{p}} \sum_{i = 1}^{m_{p}} p_{A i} p_{B i} .

(C2)

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I_{A B}^{x} = \frac{1}{2} \log_{2} \frac{{\hat{V}}_{A}^{x}}{{\hat{V}}_{A | B}^{x}} = \frac{1}{2} \log_{2} \frac{{\hat{V}}_{A}^{x}}{{\hat{V}}_{A}^{x} - {({\hat{C}}_{A B}^{x})}^{2} / {\hat{V}}_{B}^{x}}, I_{A B}^{p} = \frac{1}{2} \log_{2} \frac{{\hat{V}}_{A}^{p}}{{\hat{V}}_{A | B}^{p}} = \frac{1}{2} \log_{2} \frac{{\hat{V}}_{A}^{p}}{{\hat{V}}_{A}^{p} - {({\hat{C}}_{A B}^{p})}^{2} / {\hat{V}}_{B}^{p}} .

(C3)

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{\hat{V}}_{A_{1}}^{x} = 1 + \frac{{\hat{V}}_{A}^{x} - 1 - v_{e l A}}{η_{A}}, {\hat{V}}_{A_{1}}^{p} = 1 + \frac{{\hat{V}}_{A}^{p} - 1 - v_{e l A}}{η_{A}}, {\hat{V}}_{B_{1}}^{x} = 1 + \frac{{\hat{V}}_{B}^{x} - 1 - v_{e l B}}{η_{B}}, {\hat{V}}_{B_{1}}^{p} = 1 + \frac{{\hat{V}}_{B}^{p} - 1 - v_{e l B}}{η_{B}}, {\hat{C}}_{A_{1} B_{1}}^{x} = {\hat{C}}_{A B}^{x} / \sqrt{η_{A} η_{B}}, {\hat{C}}_{A_{1} B_{1}}^{p} = {\hat{C}}_{A B}^{p} / \sqrt{η_{A} η_{B}} .

(C4)

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χ_{B E}^{x (p)} = S (ρ_{A_{1} B_{1}}) - S (ρ_{A_{1} F G / B}^{x (p)}) .

(C5)

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⟨ {({\hat{x}}_{d A} - {\hat{x}}_{d B})}^{2} ⟩ = ⟨ {({\hat{p}}_{d A} + {\hat{p}}_{d B})}^{2} ⟩ = 2 s, ⟨ {({\hat{x}}_{d A} + {\hat{x}}_{d B})}^{2} ⟩ = ⟨ {({\hat{p}}_{d A} - {\hat{p}}_{d B})}^{2} ⟩ = 2 s_{anti},

(D1)

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s = \frac{1}{2} (V_{d A} + V_{d B} - 2 C_{d A d B}), s_{anti} = \frac{1}{2} (V_{d A} + V_{d B} + 2 C_{d A d B}),

(D2)

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s = \frac{1}{2} [\frac{V_{d A} - ν_{e l d A} - (1 - η_{d A})}{η_{d A}} + \frac{V_{d B} - ν_{e l d B} - (1 - η_{d B})}{η_{d B}} - \frac{2 C_{d A d B}}{\sqrt{η_{d A} η_{d B}}}], s_{anti} = \frac{1}{2} [\frac{V_{d A} - ν_{e l d A} - (1 - η_{d A})}{η_{d A}} + \frac{V_{d B} - ν_{e l d B} - (1 - η_{d B})}{η_{d B}} + \frac{2 C_{d A d B}}{\sqrt{η_{d A} η_{d B}}}],

(D3)

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K^{\infty} = P_{A} P_{B} (β I_{A B}^{x} - χ_{B E}^{x}) + (1 - P_{A}) (1 - P_{B}) (β I_{A B}^{p} - χ_{B E}^{p}) \approx [P_{A} P_{B} + (1 - P_{A}) (1 - P_{B})] (β I_{A B} - χ_{B E}) .

(E1)

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K^{x} = (1 - P_{A}) (1 - P_{B}) \frac{n_{x}}{N_{x}} [β I_{A B}^{x (δ_{P E})} - χ_{B E}^{x (δ_{P E})} - Δ (n_{x})], K^{p} = P_{A} P_{B} \frac{n_{p}}{N_{p}} [β I_{A B}^{p (δ_{P E})} - χ_{B E}^{p (δ_{P E})} - Δ (n_{p})], K^{finite} = \max {K^{x}, 0} + \max {K^{p}, 0},

(E2)

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χ_{B E}^{x (p)} = S (ρ_{E}) - S (ρ_{E / B}^{x (p)}) = S (ρ_{A_{1} B_{1} C D}) - S (ρ_{A_{1} C D F G / B}^{x (p)}) .

(F1)

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Shanna Du, Pu Wang, Jianqiang Liu, Yan Tian, Yongmin Li. Continuous variable quantum key distribution with a shared partially characterized entangled source[J]. Photonics Research, 2023, 11(3): 463

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