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Journals >Photonics Research >Volume 10 >Issue 7 >Page 1703 > Article

Photonics Research
Vol. 10, Issue 7, 1703 (2022)

Measurement-device-independent quantum key distribution protocol with phase post-selection

Cong Jiang^1、2, Xiao-Long Hu³, Zong-Wen Yu⁴, and Xiang-Bin Wang^{1、2、5、6、7、*}

Author Affiliations

¹Jinan Institute of Quantum Technology, Jinan 250101, China

²State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China

³School of Physics, State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China

⁴Data Communication Science and Technology Research Institute, Beijing 100191, China

⁵Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China

⁶Shenzhen Institute for Quantum Science and Engineering, and Physics Department, Southern University of Science and Technology, Shenzhen 518055, China

⁷Frontier Science Center for Quantum Information, Beijing, China

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DOI: 10.1364/PRJ.445617 Cite this Article Set citation alerts

Cong Jiang, Xiao-Long Hu, Zong-Wen Yu, Xiang-Bin Wang. Measurement-device-independent quantum key distribution protocol with phase post-selection[J]. Photonics Research, 2022, 10(7): 1703 Copy Citation Text

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Abstract

Measurement-device-independent quantum key distribution (MDI-QKD) protocol can remove all the loopholes of the detection devices and, thus, has attracted much attention. Based on the technique of single-photon interference, we propose a modified MDI-QKD protocol with phase post-selection. We prove the security of the announcement of the private phases in the

X

basis and show how to apply the phase post-selection method to the double-scanning four-intensity MDI-QKD protocol. The numerical results show that the phase post-selection method can significantly improve the key rates at all distances. In the double-scanning method, two parameters need to be scanned in the calculation of the final key rate, and the global parameter optimization is pretty time-consuming. We propose an accelerated method that can greatly reduce the running time of the global parameter optimization program. This makes the method practically useful in an unstable channel.

| e^{i θ} \sqrt{μ} ⟩ = \sum_{m = 0}^{\infty} \frac{e^{- μ / 2} e^{i m θ} {\sqrt{μ}}^{m}}{\sqrt{m!}} | m ⟩,

(1)

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1 - | \cos (θ_{a j} - θ_{b j}) | \leq λ .

(2)

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1 - | \cos (θ_{a j} - θ_{b j}) | \leq λ .

(3)

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ρ = \frac{1}{4 π} \int_{0}^{2 π} d θ_{a j} [Ω (δ_{i}) + Ω (δ_{i} + π)],

(4)

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Ω (δ_{i}) = | e^{i (θ_{a j} + γ_{a j})} \sqrt{μ_{x_{A}}} ⟩ ⟨ e^{i (θ_{a j} + γ_{a j})} \sqrt{μ_{x_{A}}} | \otimes | e^{i (θ_{a j} + γ_{b j} - δ_{i})} \sqrt{μ_{x_{B}}} ⟩ ⟨ e^{i (θ_{a j} + γ_{b j} - δ_{i})} \sqrt{μ_{x_{B}}} | .

(5)

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ρ = c_{11} | 11 ⟩ ⟨ 11 | + (1 - c_{11}) ρ^{'},

(6)

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ρ_{α β} = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} a_{m}^{α} b_{n}^{β} | m n ⟩ ⟨ m n |,

(7)

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a_{m}^{α} = \frac{μ_{α_{A}}^{m} e^{- μ_{α_{A}}}}{m!}, b_{n}^{β} = \frac{μ_{β_{B}}^{n} e^{- μ_{β_{B}}}}{n!},

(8)

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N_{α β} = p_{α_{A}} p_{β_{B}} N .

(9)

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ρ_{Q} = c_{11} | 11 ⟩ ⟨ 11 | + (1 - c_{11}) ρ^{'},

(10)

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{⟨ s_{11} ⟩}^{L} = \frac{{⟨ S_{+} ⟩}^{L} + \frac{a_{1}^{y} b_{2}^{y}}{N_{x x}} M - {⟨ S_{-} ⟩}^{U} - a_{1}^{y} b_{2}^{y} H}{a_{1}^{x} a_{1}^{y} (b_{1}^{x} b_{2}^{y} - b_{2}^{x} b_{1}^{y})},

(11)

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⟨ S_{+} ⟩ = \frac{a_{1}^{y} b_{2}^{y}}{N_{x x}} ⟨ {\bar{m}}_{x x} ⟩ + \frac{a_{1}^{x} b_{2}^{x} a_{0}^{y}}{N_{o y}} ⟨ n_{o y} ⟩ + \frac{a_{1}^{x} b_{2}^{x} b_{0}^{y}}{N_{y o}} ⟨ n_{y o} ⟩,

(12)

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⟨ S_{-} ⟩ = \frac{a_{1}^{x} b_{2}^{x}}{N_{y y}} ⟨ n_{y y} ⟩ + \frac{a_{1}^{x} b_{2}^{x} a_{0}^{y} b_{0}^{y}}{N_{o o}} ⟨ n_{o o} ⟩,

(13)

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H = \frac{a_{0}^{x}}{N_{o x}} ⟨ n_{o x} ⟩ + \frac{b_{0}^{x}}{N_{x o}} ⟨ n_{x o} ⟩ - \frac{a_{0}^{x} b_{0}^{x}}{N_{o o}} ⟨ n_{o o} ⟩,

(14)

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{⟨ e_{11} ⟩}^{U} = \frac{M / N_{x x} - H / 2}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L}} .

(15)

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\cos \frac{Δ}{2} = 1 - λ,

(16)

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{⟨ e_{11} ⟩}^{' U} = \frac{\frac{⟨ m_{Q} ⟩}{Δ / π N_{x x}}}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L}} .

(17)

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{⟨ e_{11}^{ph} ⟩}^{U} = \min (\frac{M / N_{x x} - H / 2}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L}}, \frac{\frac{⟨ m_{Q} ⟩}{Δ / π N_{x x}}}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L}}) .

(18)

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s_{11, Z}^{L} = \frac{O^{L} (N_{z z} a_{1}^{z} b_{1}^{z} {⟨ s_{11} ⟩}^{L})}{N_{z z} a_{1}^{z} b_{1}^{z}},

(19)

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e_{11}^{ph, U} = \frac{O^{U} (N_{z z} a_{1}^{z} b_{1}^{z} s_{11, Z}^{L} {⟨ e_{11}^{ph} ⟩}^{U})}{N_{z z} a_{1}^{z} b_{1}^{z} s_{11, Z}^{L}},

(20)

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R (H, M) = p_{z_{A}} p_{z_{B}} {a_{1}^{z} b_{1}^{z} s_{11, Z}^{L} [1 - h (e_{11}^{ph, U})] - f S_{z z} h (E_{z z})} - \frac{1}{N} (\log_{2} \frac{8}{ε_{cor}} + 2 \log_{2} \frac{2}{ε^{'} \hat{ε}} + 2 \log_{2} \frac{1}{2 ε_{PA}}),

(21)

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R = \min_{\frac{H \in [H^{L}, H^{U}],}{M \in [M^{L}, M^{U}]}} R (H, M) .

(22)

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H (t_{11}) = {\begin{cases} H^{U}, & t_{11} + H^{U} / 2 \leq M^{U} / N_{x x}, \\ 2 (M^{U} / N_{x x} - t_{11}), & t_{11} + H^{U} / 2 > M^{U} / N_{x x} . \end{cases}

(23)

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{⟨ s_{11} ⟩}^{L} = \frac{{⟨ S_{+} ⟩}^{L} + a_{1}^{y} b_{2}^{y} [t_{11} - H (t_{11}) / 2] - {⟨ S_{-} ⟩}^{U}}{a_{1}^{x} a_{1}^{y} (b_{1}^{x} b_{2}^{y} - b_{2}^{x} b_{1}^{y})},

(24)

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{⟨ e_{11}^{ph} ⟩}^{U} = \frac{t_{11}}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L}} .

(25)

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R = \min_{t_{11} \in [T^{L}, T^{U'}]} R (t_{11}) .

(26)

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{⟨ e_{11}^{ph} ⟩}^{U} (t_{11}^{w}) = \frac{T^{Q}}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L} (t_{11}^{w})} > {⟨ e_{11}^{ph} ⟩}^{U} (t_{11}^{v}) = \frac{T^{Q}}{a_{1}^{x} b_{1}^{x} {⟨ s_{11} ⟩}^{L} (t_{11}^{v})} .

(27)

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E^{L} (X) = \frac{X}{1 + δ_{1} (X)},

(A1)

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E^{U} (X) = \frac{X}{1 - δ_{2} (X)},

(A2)

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{[\frac{e^{δ_{1}}}{{(1 + δ_{1})}^{1 + δ_{1}}}]}^{\frac{X}{1 + δ_{1}}} = ξ,

(A3)

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{[\frac{e^{- δ_{2}}}{{(1 - δ_{2})}^{1 - δ_{2}}}]}^{\frac{X}{1 - δ_{2}}} = ξ,

(A4)

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O^{U} (Y) = [1 + δ_{1}^{'} (Y)] Y,

(A5)

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O^{L} (Y) = [1 - δ_{2}^{'} (Y)] Y,

(A6)

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{[\frac{e^{δ_{1}^{'}}}{{(1 + δ_{1}^{'})}^{1 + δ_{1}^{'}}}]}^{Y} = ξ,

(A7)

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{[\frac{e^{- δ_{2}^{'}}}{{(1 - δ_{2}^{'})}^{1 - δ_{2}^{'}}}]}^{Y} = ξ .

(A8)

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| ψ_{1} ⟩ = | e^{i θ_{a}} \sqrt{μ_{a}} ⟩ \otimes | e^{i θ_{b}} \sqrt{μ_{b}} ⟩,

(B1)

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| ψ_{1} ⟩ = e^{- μ_{a} / 2 - μ_{b} / 2} e^{\sqrt{μ_{a}} e^{i θ_{a}} a_{+}^{†} \sqrt{μ_{b}} e^{i θ_{b}} b_{+}^{†}} | 00 ⟩ .

(B2)

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| ψ_{2} ⟩ = e^{- μ_{a} / 2 - μ_{b} / 2} e^{(\frac{\sqrt{μ_{a}}}{2} e^{i θ_{a}} + \frac{\sqrt{μ_{b}}}{2} e^{i θ_{b}}) (a_{H}^{†} + a_{V}^{†})} \times e^{(\frac{\sqrt{μ_{a}}}{2} e^{i θ_{a}} - \frac{\sqrt{μ_{b}}}{2} e^{i θ_{b}}) (b_{H}^{†} + b_{V}^{†})} | 00 ⟩,

(B3)

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\hat{M} = [I_{a_{H}} - (1 - p_{d} {) | 0}_{a_{H}} ⟩ ⟨ 0_{a_{H}} |] \otimes [I_{a_{V}} - (1 - p_{d} {) | 0}_{a_{V}} ⟩ ⟨ 0_{a_{V}} |] \otimes (1 - p_{d} {) | 0}_{b_{H}} ⟩ ⟨ 0_{b_{H}} | \otimes (1 - p_{d} {) | 0}_{b_{V}} ⟩ ⟨ 0_{b_{V}} | .

(B4)

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p_{1 H,1 V} = tr (\hat{M} | ψ_{2} ⟩ ⟨ ψ_{2} |) = e^{- μ_{a} - μ_{b}} {[e^{\frac{μ_{a}}{4} + \frac{μ_{b}}{4} + \frac{\sqrt{μ_{a} μ_{b}}}{2} \cos δ} - (1 - p_{d})]}^{2} {(1 - p_{d})}^{2},

(B5)

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p_{1 H,2 V} = p_{2 H,1 V} = e^{- μ_{a} - μ_{b}} [e^{\frac{μ_{a}}{4} + \frac{μ_{b}}{4} + \frac{\sqrt{μ_{a} μ_{b}}}{2} \cos δ} - (1 - p_{d})] \times [e^{\frac{μ_{a}}{4} + \frac{μ_{b}}{4} - \frac{\sqrt{μ_{a} μ_{b}}}{2} \cos δ} - (1 - p_{d})] {(1 - p_{d})}^{2},

(B6)

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p_{2 H,2 V} = e^{- μ_{a} - μ_{b}} {[e^{\frac{μ_{a}}{4} + \frac{μ_{b}}{4} - \frac{\sqrt{μ_{a} μ_{b}}}{2} \cos δ} - (1 - p_{d})]}^{2} {(1 - p_{d})}^{2} .

(B7)

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P_{W} = p_{1 H,2 V} + p_{2 H,1 V}, P_{R} = p_{1 H,1 V} + p_{2 H,2 V} .

(B8)

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m_{Q} = \frac{1}{2 π} N p_{x_{A}} p_{x_{B}} (\int_{- \frac{Δ}{2}}^{\frac{Δ}{2}} P_{W} d δ + \int_{π - \frac{Δ}{2}}^{π + \frac{Δ}{2}} P_{W} d δ),

(B9)

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n_{Q} = \frac{1}{2 π} N p_{x_{A}} p_{x_{B}} (\int_{- \frac{Δ}{2}}^{\frac{Δ}{2}} P_{R} d δ + \int_{π - \frac{Δ}{2}}^{π + \frac{Δ}{2}} P_{R} d δ) + m_{Q},

(B10)

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Cong Jiang, Xiao-Long Hu, Zong-Wen Yu, Xiang-Bin Wang. Measurement-device-independent quantum key distribution protocol with phase post-selection[J]. Photonics Research, 2022, 10(7): 1703

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