Researching | Measurement-device-independent quantum key distribution for nonstandalone networks

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Photonics Research
Vol. 9, Issue 10, 1881 (2021)

Measurement-device-independent quantum key distribution for nonstandalone networks | Editors' Pick

Guan-Jie Fan-Yuan^1、2、3, Feng-Yu Lu^1、2、3, Shuang Wang^{1、2、3、*}, Zhen-Qiang Yin^1、2、3, De-Yong He^1、2、3, Zheng Zhou^1、2、3, Jun Teng^1、2、3, Wei Chen^1、2、3, Guang-Can Guo^1、2、3, and Zheng-Fu Han^1、2、3

Author Affiliations

¹CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China

²CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China

³State Key Laboratory of Cryptology, Beijing 100878, China

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DOI: 10.1364/PRJ.428309 Cite this Article Set citation alerts

Guan-Jie Fan-Yuan, Feng-Yu Lu, Shuang Wang, Zhen-Qiang Yin, De-Yong He, Zheng Zhou, Jun Teng, Wei Chen, Guang-Can Guo, Zheng-Fu Han. Measurement-device-independent quantum key distribution for nonstandalone networks[J]. Photonics Research, 2021, 9(10): 1881 Copy Citation Text

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Abstract

Untrusted node networks initially implemented by measurement-device-independent quantum key distribution (MDI-QKD) protocol are a crucial step on the roadmap of the quantum Internet. Considering extensive QKD implementations of trusted node networks, a workable upgrading tactic of existing networks toward MDI networks needs to be explicit. Here, referring to the nonstandalone (NSA) network of 5G, we propose an NSA-MDI scheme as an evolutionary selection for existing phase-encoding BB84 networks. Our solution can upgrade the BB84 networks and terminals that employ various phase-encoding schemes to immediately support MDI without hardware changes. This cost-effective upgrade effectively promotes the deployment of MDI networks as a step of untrusted node networks while taking full advantage of existing networks. In addition, the diversified demands on security and bandwidth are satisfied, and network survivability is improved.

Q_{ι_{a} ι_{b}}^{X} |_{θ_{a} = θ_{b}} = \frac{ι_{a} ι_{b}}{2}, Q_{ι_{a} ι_{b}}^{X} |_{θ_{a} \neq θ_{b}} = 0, E_{ι_{a} ι_{b}}^{X} = \frac{Q^{X} |_{θ_{a} \neq θ_{b}}}{Q^{X} |_{θ_{a} = θ_{b}} + Q^{X} |_{θ_{a} \neq θ_{b}}} = 0, Q_{ι_{a} ι_{b}}^{Y} |_{θ_{a} = θ_{b}} = \frac{{(ι_{a} + ι_{b})}^{2} - 2 ι_{a} ι_{b}}{8}, Q_{ι_{a} ι_{b}}^{Y} |_{θ_{a} \neq θ_{b}} = \frac{{(ι_{a} + ι_{b})}^{2} + 2 ι_{a} ι_{b}}{8}, E_{ι_{a} ι_{b}}^{Y} = \frac{Q^{Y} |_{θ_{a} = θ_{b}}}{Q^{Y} |_{θ_{a} = θ_{b}} + Q^{Y} |_{θ_{a} \neq θ_{b}}} = \frac{ι_{a}^{2} + ι_{b}^{2}}{2 {(ι_{a} + ι_{b})}^{2}} \overset{ι_{a} = ι_{b}}{=} \frac{1}{4} .

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R = P_{μ}^{X^{2}} (μ^{2} e^{- 2 μ} Y_{11}^{X, L} (1 - H_{2} (e_{11}^{Y, U})) - Q_{μ_{a} μ_{b}}^{X} f_{e} H_{2} ((E_{μ_{a} μ_{b}}^{X}))),

(2)

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R = \frac{1}{N} (s_{0}^{X} + s_{1}^{X} (1 - H_{2} (e_{1, p}^{X})) - λ_{EC} - 6 \log_{2} \frac{21}{ε_{\sec}} - \log_{2} \frac{2}{ε_{cor}}),

(3)

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{| e^{i ϕ_{a}} \sqrt{μ_{a}} ⟩}_{a} {| e^{i ϕ_{b}} \sqrt{μ_{b}} ⟩}_{a},

(A1)

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{| e^{i ϕ_{a}} \sqrt{\frac{μ_{a}}{2}} ⟩}_{a_{l}} {| e^{i (ϕ_{a} + θ_{a})} \sqrt{\frac{μ_{a}}{2}} ⟩}_{a_{s}} {| e^{i ϕ_{b}} \sqrt{\frac{μ_{b}}{2}} ⟩}_{b_{l}} | e^{i (ϕ_{b} + θ_{b})} {\sqrt{\frac{μ_{b}}{2}} ⟩}_{b_{s}},

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{| e^{i ϕ_{a}} \sqrt{\frac{μ_{a} η_{a}}{2}} ⟩}_{a_{l}} | e^{i (ϕ_{a} + θ_{a})} {\sqrt{\frac{μ_{a} η_{a}}{2}} ⟩}_{a_{s}} \otimes {| e^{i ϕ_{b}} \sqrt{\frac{μ_{b} η_{b}}{2}} ⟩}_{b_{l}} {| e^{i (ϕ_{b} + θ_{b})} \sqrt{\frac{μ_{b} η_{b}}{2}} ⟩}_{b_{s}},

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| e^{i ϕ_{a}} \sqrt{\frac{μ_{a} η_{a}}{2}} - e^{i ϕ_{b}} {\sqrt{\frac{μ_{b} η_{b}}{2}} ⟩}_{l l} \otimes | e^{i (ϕ_{a} + θ_{a} + θ_{c})} \sqrt{\frac{μ_{a} η_{a}}{2}} + {e^{i (ϕ_{b} + θ_{b} + θ_{c})} \sqrt{\frac{μ_{b} η_{b}}{2}} ⟩}_{s s} \otimes | e^{i (ϕ_{a} + θ_{c})} \sqrt{\frac{μ_{a} η_{a}}{2}} + e^{i (ϕ_{b} + θ_{c})} {\sqrt{\frac{μ_{b} η_{b}}{2}} ⟩}_{l s} \otimes | e^{i (ϕ_{a} + θ_{a})} \sqrt{\frac{μ_{a} η_{a}}{2}} - e^{i (ϕ_{b} + θ_{b})} {\sqrt{\frac{μ_{b} η_{b}}{2}} ⟩}_{s l},

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{| \frac{\sqrt{μ_{a} η_{a}}}{2 \sqrt{2}} (e^{i (ϕ_{a} + θ_{c})} + e^{i (ϕ_{a} + θ_{a})}) + \frac{\sqrt{μ_{b} η_{b}}}{2 \sqrt{2}} (e^{i (ϕ_{b} + θ_{c})} - e^{i (ϕ_{b} + θ_{b})}) ⟩}_{D_{1}} \otimes {| \frac{\sqrt{μ_{a} η_{a}}}{2 \sqrt{2}} (e^{i (ϕ_{a} + θ_{c})} - e^{i (ϕ_{a} + θ_{a})}) + \frac{\sqrt{μ_{b} η_{b}}}{2 \sqrt{2}} (e^{i (ϕ_{b} + θ_{c})} + e^{i (ϕ_{b} + θ_{b})}) ⟩}_{D_{2}} .

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{| ψ_{1} ⟩}_{D_{1}} \otimes {| ψ_{2} ⟩}_{D_{2}} .

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p_{μ}^{D_{1}} = 1 - (1 - Y_{0}) (1 - P_{ap}) \exp (- {| ψ_{1} |}^{2}), p_{μ}^{D_{2}} = 1 - (1 - Y_{0}) (1 - P_{ap}) \exp (- {| ψ_{2} |}^{2}),

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{| ψ_{1} |}^{2} = \frac{μ_{a} η_{a}}{4} (1 + \cos (θ_{c} - θ_{a})) + \frac{μ_{b} η_{b}}{4} (1 - \cos (θ_{c} - θ_{b})) + \frac{\sqrt{μ_{a} η_{a} μ_{b} η_{b}}}{4} (\cos (ϕ_{a} - ϕ_{b}) + \cos (ϕ_{a} - ϕ_{b} + θ_{a} - θ_{c}) - \cos (ϕ_{b} - ϕ_{a} + θ_{b} - θ_{c}) - \cos (ϕ_{a} - ϕ_{b} + θ_{a} - θ_{b})) {| ψ_{2} |}^{2} = \frac{μ_{a} η_{a}}{4} (1 - \cos (θ_{c} - θ_{a})) + \frac{μ_{b} η_{b}}{4} (1 + \cos (θ_{c} - θ_{b})) + \frac{\sqrt{μ_{a} η_{a} μ_{b} η_{b}}}{4} (\cos (ϕ_{a} - ϕ_{b}) + \cos (ϕ_{a} - ϕ_{b} + θ_{c} - θ_{b}) - \cos (ϕ_{b} - ϕ_{a} + θ_{c} - θ_{a}) - \cos (ϕ_{a} - ϕ_{b} + θ_{a} - θ_{b})) .

(A8)

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A = \frac{\sqrt{μ_{a} η_{a}}}{2}, B = \frac{\sqrt{μ_{b} η_{b}}}{2} .

(A9)

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Q_{μ}^{X} |_{θ_{a} = θ_{b}} = \sum_{θ_{a} = θ_{b} \in {0, π}} \frac{1}{4 π^{2}} \int_{0}^{2 π} \int_{0}^{2 π} p_{μ}^{D_{1}} p_{μ}^{D_{2}} d ϕ_{a} d ϕ_{b} = 2 (1 - (1 - Y_{0}) (1 - P_{ap}) (2 - 2 A^{2} - 2 B^{2}) + {(1 - Y_{0})}^{2} {(1 - P_{ap})}^{2} (1 - 2 A^{2}) (1 - 2 B^{2})), Q_{μ}^{X} |_{θ_{a} \neq θ_{b}} = \sum_{θ_{a} \neq θ_{b} \in {0, π}} \frac{1}{4 π^{2}} \int_{0}^{2 π} \int_{0}^{2 π} p_{μ}^{D_{1}} p_{μ}^{D_{2}} d ϕ_{a} d ϕ_{b} = 2 (1 - (1 - Y_{0}) (1 - P_{ap}) (2 - 2 A^{2} - 2 B^{2}) + {(1 - Y_{0})}^{2} {(1 - P_{ap})}^{2} (1 - 2 A^{2} - 2 B^{2})) .

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Q_{μ}^{X} = Q_{μ}^{X} |_{θ_{a} = θ_{b}} + Q_{μ}^{X} |_{θ_{a} \neq θ_{b}}, E_{μ}^{X} Q_{μ}^{X} = e_{d} Q_{μ}^{X} |_{θ_{a} = θ_{b}} + (1 - e_{d}) Q_{μ}^{X} |_{θ_{a} \neq θ_{b}},

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Q_{μ}^{Y} |_{θ_{a} = θ_{b}} = \sum_{θ_{a} = θ_{b} \in {\frac{π}{2}, \frac{3 π}{2}}} \frac{1}{4 π^{2}} \int_{0}^{2 π} \int_{0}^{2 π} p_{μ}^{D_{1}} p_{μ}^{D_{2}} d ϕ_{a} d ϕ_{b} = 2 (1 - (1 - Y_{0}) (1 - P_{ap}) (2 - 2 A^{2} - 2 B^{2}) + {(1 - Y_{0})}^{2} {(1 - P_{ap})}^{2} ({(1 - A^{2} - B^{2})}^{2} - 2 A^{2} B^{2})), Q_{μ}^{Y} |_{θ_{a} \neq θ_{b}} = \sum_{θ_{a} \neq θ_{b} \in {\frac{π}{2}, \frac{3 π}{2}}} \frac{1}{4 π^{2}} \int_{0}^{2 π} \int_{0}^{2 π} p_{μ}^{D_{1}} p_{μ}^{D_{2}} d ϕ_{a} d ϕ_{b} = 2 (1 - (1 - Y_{0}) (1 - P_{ap}) (2 - 2 A^{2} - 2 B^{2}) + {(1 - Y_{0})}^{2} {(1 - P_{ap})}^{2} ({(1 - A^{2} - B^{2})}^{2} + 2 A^{2} B^{2})),

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Q_{μ}^{Y} = Q_{μ}^{Y} |_{θ_{a} = θ_{b}} + Q_{μ}^{Y} |_{θ_{a} \neq θ_{b}}, E_{μ}^{Y} Q_{μ}^{Y} = e_{d} Q_{μ}^{Y} |_{θ_{a} = θ_{b}} + (1 - e_{d}) Q_{μ}^{Y} |_{θ_{a} \neq θ_{b}} .

(A13)

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R = P_{μ}^{X^{2}} (μ^{2} e^{- 2 μ} Y_{11}^{X, L} (1 - H_{2} (e_{11}^{Y, U})) - Q_{μ_{a} μ_{b}}^{X} f_{e} H_{2} (E_{μ_{a} μ_{b}}^{X}),

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Y_{11}^{X, L} = \frac{1}{(μ_{a} - ω_{a}) (μ_{b} - ω_{b}) (ν_{a} - ω_{a}) (μ_{b} - ω_{b}) (μ_{a} - ω_{a})} \times ((μ_{a}^{2} - ω_{a}^{2}) (μ_{b} - ω_{b}) (Q_{ν_{a} ν_{b}}^{X, L} e^{(ν_{a} + ν_{b})} + Q_{ω_{a} ω_{b}}^{X, L} e^{(ω_{a} + ω_{b})} - Q_{ν_{a} ω_{b}}^{X, U} e^{(ν_{a} + ω_{b})} - Q_{ω_{a} ν_{b}}^{X, U} e^{(ω_{a} + ν_{b})}) - (ν_{a}^{2} - ω_{a}^{2}) (ν_{b} - ω_{b}) (Q_{μ_{a} μ_{b}}^{X, U} e^{(μ_{a} + μ_{b})} + Q_{ω_{a} ω_{b}}^{X, U} e^{(ω_{a} + ω_{b})} - Q_{μ_{a} ω_{b}}^{X, L} e^{(μ_{a} + ω_{b})} - Q_{ω_{a} μ_{b}}^{X, L} e^{(ω_{a} + μ_{b})})),

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e_{11}^{Y, U} = \frac{1}{(ν_{a} - ω_{a}) (ν_{b} - ω_{b}) Y_{11}^{Y, L}} \times (e^{(ν_{a} + ν_{b})} E Q_{ν_{a} ν_{b}}^{Y, U} + e^{(ω_{a} ω_{b})} E Q_{ω_{a} ω_{b}}^{Y, U} - e^{(ν_{a} + ω_{b})} E Q_{ν_{a} ω_{b}}^{Y, L} - e^{(ω_{a} + ν_{b})} E Q_{ω_{a} ν_{b}}^{Y, L}),

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Y_{11}^{Y, L} = \frac{1}{(μ_{a} - ω_{a}) (μ_{b} - ω_{b}) (ν_{a} - ω_{a}) (μ_{b} - ω_{b}) (μ_{a} - ω_{a})} \times ((μ_{a}^{2} - ω_{a}^{2}) (μ_{b} - ω_{b}) (Q_{ν_{a} ν_{b}}^{Y, L} e^{(ν_{a} + ν_{b})} + Q_{ω_{a} ω_{b}}^{Y, L} e^{(ω_{a} + ω_{b})} - Q_{ν_{a} ω_{b}}^{Y, U} e^{(ν_{a} + ω_{b})} - Q_{ω_{a} ν_{b}}^{Y, U} e^{(ω_{a} + ν_{b})}) - (ν_{a}^{2} - ω_{a}^{2}) (ν_{b} - ω_{b}) (Q_{μ_{a} μ_{b}}^{Y, U} e^{(μ_{a} + μ_{b})} + Q_{ω_{a} ω_{b}}^{Y, U} e^{(ω_{a} + ω_{b})} - Q_{μ_{a} ω_{b}}^{Y, L} e^{(μ_{a} + ω_{b})} - Q_{ω_{a} μ_{b}}^{Y, L} e^{(ω_{a} + μ_{b})})),

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Q_{α_{a} α_{b}}^{β, U} = Q_{α_{a} α_{b}}^{β} (1 + \frac{f ((ε {/ 2)}^{4} / 16)}{\sqrt{N_{α_{a} α_{b}}^{β} Q_{α_{a} α_{b}}^{β}}}) Q_{α_{a} α_{b}}^{β, L} = Q_{α_{a} α_{b}}^{β} (1 - \frac{f ((ε {/ 2)}^{3 / 2})}{\sqrt{N_{α_{a} α_{b}}^{β} Q_{α_{a} α_{b}}^{β}}}) E Q_{α_{a} α_{b}}^{β, U} = E Q_{α_{a} α_{b}}^{β} (1 + \frac{f ((ε {/ 2)}^{4} / 16)}{\sqrt{N_{α_{a} α_{b}}^{β} E Q_{α_{a} α_{b}}^{β}}}) E Q_{α_{a} α_{b}}^{β, L} = E Q_{α_{a} α_{b}}^{β} (1 - \frac{f ((ε {/ 2)}^{3 / 2})}{\sqrt{N_{α_{a} α_{b}}^{β} E Q_{α_{a} α_{b}}^{β}}}),

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R = \frac{1}{N} (s_{0}^{X} + s_{1}^{X} (1 - H_{2} (e_{1, p}^{X})) - λ_{EC} - 6 \log_{2} \frac{21}{ε_{\sec}} - \log_{2} \frac{2}{ε_{cor}}),

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s_{0}^{ω} = \frac{τ_{0}}{ν_{1} - ν_{2}} (\frac{e^{ν_{2}} ν_{1} n_{ν_{2}}^{β, U}}{P_{ν_{2}}} - \frac{e^{ν_{1}} ν_{2} n_{ν_{1}}^{β, L}}{P_{ν_{1}}}),

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s_{1}^{ω} = \frac{μ τ_{1}}{μ ν_{1} - μ ν_{2} - ν_{1}^{2} + ν_{2}^{2}} (\frac{e^{ν 1} n_{ν 1}^{β, L}}{P_{ν 1}} - \frac{e^{ν 2} n_{ν 2}^{β, U}}{P_{ν 2}} - \frac{ν_{1}^{2} - ν_{2}^{2}}{μ^{2}} (\frac{e^{μ} n_{μ}^{ω, U}}{P_{μ}} - \frac{s_{0}^{β}}{τ_{0}})),

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e_{1, p}^{β} = \frac{v_{1}^{\bar{β}}}{s_{1}^{\bar{β}}} + γ (ε_{\sec}, \frac{v_{1}^{\bar{β}}}{s_{1}^{\bar{β}}}, s_{1}^{\bar{β}}, s_{1}^{β}),

(C4)

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γ (a, b, c, d) = \sqrt{\frac{(c + d) (1 - b) b \ln 2}{cd}} \sqrt{\log_{2} (\frac{c + d}{c d (1 - b) b} \frac{21^{2}}{a^{2}})},

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v_{1}^{β} = \frac{τ_{1}}{ν_{1} - ν_{2}} (\frac{e^{ν 1} m_{ν_{1}}^{β, U}}{P_{ν_{1}}} - \frac{e^{ν 2} m_{ν_{2}}^{β, L}}{P_{ν_{2}}}) .

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n_{α}^{β, U} = n_{α}^{β} + \sqrt{\frac{n^{β}}{2} \ln \frac{21}{ε_{\sec}}}, n_{α}^{β, L} = n_{α}^{β} - \sqrt{\frac{n^{β}}{2} \ln \frac{21}{ε_{\sec}},} m_{α}^{β, U} = m_{α}^{β} + \sqrt{\frac{m^{β}}{2} \ln \frac{21}{ε_{\sec}},} m_{α}^{β, L} = m_{α}^{β} - \sqrt{\frac{m^{β}}{2} \ln \frac{21}{ε_{\sec}}},

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Guan-Jie Fan-Yuan, Feng-Yu Lu, Shuang Wang, Zhen-Qiang Yin, De-Yong He, Zheng Zhou, Jun Teng, Wei Chen, Guang-Can Guo, Zheng-Fu Han. Measurement-device-independent quantum key distribution for nonstandalone networks[J]. Photonics Research, 2021, 9(10): 1881

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