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Photonics Research
Vol. 8, Issue 9, 1522 (2020)

Experimental free-space quantum secure direct communication and its security analysis

Dong Pan^1、2、†, Zaisheng Lin^{2、3、4、5、†}, Jiawei Wu^1、2, Haoran Zhang^1、2, Zhen Sun^2、3, Dong Ruan^1、2, Liuguo Yin^{2、3、4、5、6、*}, and Gui Lu Long^{1、2、3、4、5、7、*}

Author Affiliations

¹State Key Laboratory of Low-dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China

²Frontier Science Center for Quantum Information, Beijing 100084, China

³School of Information Science and Technology, Tsinghua University, Beijing 100084, China

⁴Beijing National Research Center for Information Science and Technology, Beijing 100084, China

⁵Beijing Academy of Quantum Information Sciences, Beijing 100193, China

⁶e-mail: yinlg@tsinghua.edu.cn

⁷e-mail: gllong@tsinghua.edu.cn

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DOI: 10.1364/PRJ.388790 Cite this Article Set citation alerts

Dong Pan, Zaisheng Lin, Jiawei Wu, Haoran Zhang, Zhen Sun, Dong Ruan, Liuguo Yin, Gui Lu Long. Experimental free-space quantum secure direct communication and its security analysis[J]. Photonics Research, 2020, 8(9): 1522 Copy Citation Text

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Abstract

We report an experimental implementation of free-space quantum secure direct communication based on single photons. The quantum communication scheme uses phase encoding, and the asymmetric Mach–Zehnder interferometer is optimized so as to automatically compensate phase drift of the photons during their transitions over the free-space medium. At a 16 MHz pulse repetition frequency, an information transmission rate of 500 bps over a 10 m free space with a mean quantum bit error rate of

0.49 % \pm 0.27 %

is achieved. The security is analyzed under the scenario that Eve performs the collective attack for single-photon state and the photon number splitting attack for multi-photon state in the depolarizing channel. Our results show that quantum secure direct communication is feasible in free space.

C_{s} = \max_{{p_{0}}} {I (A : B) - I (A : E)},

(1)

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U {| 0 ⟩}_{B} {| 0 ⟩}_{B} | E ⟩ = {| 0 ⟩}_{B} {| 0 ⟩}_{B} | E_{0000} ⟩ + {| 0 ⟩}_{B} {| 1 ⟩}_{B} | E_{0001} ⟩ = | φ_{1} ⟩, U {| 1 ⟩}_{B} {| 1 ⟩}_{B} | E ⟩ = {| 1 ⟩}_{B} {| 0 ⟩}_{B} | E_{1110} ⟩ + {| 1 ⟩}_{B} {| 1 ⟩}_{B} | E_{1111} ⟩ = | φ_{2} ⟩, U {| + ⟩}_{B} {| + ⟩}_{B} | E ⟩ = | φ_{3} ⟩, U {| - ⟩}_{B} {| - ⟩}_{B} | E ⟩ = | φ_{4} ⟩,

(2)

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Y U {| 0 ⟩}_{B} {| 0 ⟩}_{B} | E ⟩ = - {| 0 ⟩}_{B} {| 1 ⟩}_{B} | E_{0000} ⟩ + {| 0 ⟩}_{B} {| 0 ⟩}_{B} | E_{0001} ⟩ = | φ_{5} ⟩, Y U {| 1 ⟩}_{B} {| 1 ⟩}_{B} | E ⟩ = - {| 1 ⟩}_{B} {| 1 ⟩}_{B} | E_{1110} ⟩ + {| 1 ⟩}_{B} {| 0 ⟩}_{B} | E_{1111} ⟩ = | φ_{6} ⟩, Y U {| + ⟩}_{B} {| + ⟩}_{B} | E ⟩ = | φ_{7} ⟩, Y U {| - ⟩}_{B} {| - ⟩}_{B} | E ⟩ = | φ_{8} ⟩ .

(3)

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ρ_{BEA} = p_{0} \cdot ρ_{BE}^{0} + p_{1} \cdot ρ_{BE}^{1} = \frac{1}{4} (p_{0} | φ_{1} ⟩ ⟨ φ_{1} | + p_{0} | φ_{2} ⟩ ⟨ φ_{2} | + p_{0} | φ_{3} ⟩ ⟨ φ_{3} | + p_{0} | φ_{4} ⟩ ⟨ φ_{4} | + p_{1} | φ_{5} ⟩ ⟨ φ_{5} | + p_{1} | φ_{6} ⟩ ⟨ φ_{6} | + p_{1} | φ_{7} ⟩ ⟨ φ_{7} | + p_{1} | φ_{8} ⟩ ⟨ φ_{8} |),

(4)

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I (A : E) \leq χ = \max_{{U}} {S (ρ_{BEA}) - p_{0} \cdot S (ρ_{BE}^{0}) - p_{1} \cdot S (ρ_{BE}^{1})},

(5)

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G = \frac{1}{4} [\begin{matrix} p_{0} ⟨ φ_{1} | φ_{1} ⟩ & p_{0} ⟨ φ_{1} | φ_{2} ⟩ & \dots & \sqrt{p_{0} p_{1}} ⟨ φ_{1} | φ_{8} ⟩ \\ p_{0} ⟨ φ_{2} | φ_{1} ⟩ & p_{0} ⟨ φ_{2} | φ_{2} ⟩ & \dots & \sqrt{p_{0} p_{1}} ⟨ φ_{2} | φ_{8} ⟩ \\ ⋮ & ⋮ & □ & ⋮ \\ \sqrt{p_{0} p_{1}} ⟨ φ_{8} | φ_{1} ⟩ & \sqrt{p_{0} p_{1}} ⟨ φ_{8} | φ_{2} ⟩ & \dots & p_{1} ⟨ φ_{8} | φ_{8} ⟩ \end{matrix}] .

(6)

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⟨ E_{0000} | E_{1110} ⟩ = ⟨ E_{0001} | E_{1111} ⟩ = 0, ⟨ E_{0000} | E_{0001} ⟩ = ⟨ E_{1110} | E_{1111} ⟩ = 0, ⟨ E_{0001} | E_{1110} ⟩ = 0, ⟨ E_{0000} | E_{1111} ⟩ = 1 - 2 e_{2}^{BA},

(7)

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I {(A : E)}_{n = 2} = \frac{1}{2} h (2 e_{2}^{BA}) + \frac{1}{2} .

(8)

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t^{BA} = 10^{- (\frac{α^{BA}}{10})}, t^{BAB} = 10^{- (\frac{α^{BAB}}{10})},

(9)

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η^{BA} = t^{BA} η_{opt}^{BA} η_{D}^{A}, η^{BAB} = t^{BAB} η_{opt}^{BAB} η_{D}^{B},

(10)

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Q_{μ}^{BA} = \sum_{n = 0}^{\infty} Q_{μ, n}^{BA} = \sum_{n = 0}^{\infty} p (n, μ) Y_{n}^{A} = Y_{0}^{A} + 1 - e^{- η^{BA} μ}, Q_{μ}^{BAE} = \sum_{n = 0}^{\infty} Q_{μ, n}^{BAE} \leq \sum_{n = 0}^{\infty} [Q_{μ, n}^{BA} - p (n, μ) Y_{0}^{A}] \max {1, \frac{γ^{E}}{γ^{A}}}, Q_{μ}^{BAB} = \sum_{n = 0}^{\infty} Q_{μ, n}^{BAB} = \sum_{n = 0}^{\infty} p (n, μ) Y_{n}^{B} = Y_{0}^{B} + 1 - e^{- η^{BAB} μ},

(11)

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E_{μ}^{BA} = \frac{e_{0} Y_{0}^{A} + e_{\det}^{BA} (1 - e^{- η^{BA} μ})}{Q_{μ}^{BA}}, E_{μ}^{BAB} = \frac{e_{0} Y_{0}^{B} + e_{\det}^{BAB} (1 - e^{- η^{BAB} μ})}{Q_{μ}^{BAB}},

(12)

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I (A : B) = Q_{μ}^{BAB} [1 - h (E_{μ}^{BAB})],

(13)

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I {(A : E)}_{n = 1} = Q_{μ, n = 1}^{BAE} h (2 e_{1}^{BA}),

(14)

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C_{s} = Q_{μ}^{BAB} [1 - h (E_{μ}^{BAB})] - Q_{μ, n = 1}^{BAE} h (2 e_{1}^{BA}) - Q_{μ, n = 2}^{BAE} [\frac{1}{2} h (2 e_{2}^{BA}) + \frac{1}{2}] - Q_{μ, n \geq 3}^{BAE} \cdot 1 .

(15)

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e_{1}^{BA} = \frac{E_{μ}^{BA}}{1 - \frac{p (n \geq 2, μ)}{Q_{μ}^{BA}}},

(16)

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e_{1}^{BA, U} = \frac{E_{ν_{3}}^{BA} Q_{ν_{3}}^{BA} e^{ν_{3}} - e_{0} Y_{0}^{A}}{Y_{1}^{A, L} ν_{3}},

(17)

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e_{2}^{BA, U} = \frac{2 (E_{ν_{2}}^{BA} Q_{ν_{2}}^{BA} e^{ν_{2}} - \frac{ν_{2}}{ν_{3}} E_{ν_{3}}^{BA} Q_{ν_{3}}^{BA} e^{ν_{3}} + \frac{ν_{2} - ν_{3}}{ν_{3}} e_{0} Y_{0}^{A})}{Y_{2}^{A, L} ν_{2} (ν_{2} - ν_{3})},

(18)

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Y_{1}^{A, L} = \frac{μ^{2} (Q_{ν_{2}}^{BA} e^{ν_{2}} - Q_{ν_{3}}^{BA} e^{ν_{3}}) - (ν_{2}^{2} - ν_{3}^{2}) (Q_{μ}^{BA} e^{μ} - Y_{0}^{A})}{μ (ν_{2} - ν_{3}) (μ - ν_{2} - ν_{3})},

(19)

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Y_{2}^{A, L} = \frac{2 μ (Q_{ν_{1}}^{BA} e^{ν_{1}} - Q_{ν_{2}}^{BA} e^{ν_{2}}) - 2 (ν_{1} - ν_{2}) (Q_{μ}^{BA} e^{μ} - Y_{0}^{A})}{μ (ν_{1} - ν_{2}) (ν_{1} + ν_{2} - μ)} .

(20)

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0 < ν_{3} < ν_{2} \leq \frac{2}{3} μ < ν_{1} \leq \frac{3}{4} μ, ν_{1} + ν_{2} > μ, ν_{2} + ν_{3} < μ, ν_{1} - ν_{2} - \frac{ν_{1}^{3} - ν_{2}^{3}}{μ^{2}} = 0 .

(21)

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Y_{n}^{A} - Y_{0}^{A} = \sum_{m = 0}^{\infty} f_{n} (m, μ) {1 - {(1 - γ^{A})}^{m} - [1 - {(1 - γ^{A})}^{m}] Y_{0}^{A}} \approx \sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{A})}^{m}]

(A1)

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Y_{n}^{E} = \sum_{m = 0}^{\infty} f_{n} (m, μ) {1 - {(1 - γ^{E})}^{m} - [1 - {(1 - γ^{E})}^{m}] Y_{0}^{E}} + Y_{0}^{E} \approx \sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{E})}^{m}],

(A2)

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Y_{n}^{E} = (Y_{n}^{A} - Y_{0}^{A}) \frac{\sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{E})}^{m}]}{\sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{A})}^{m}]} \leq (Y_{n}^{A} - Y_{0}^{A}) \max {1, \frac{γ^{E}}{γ^{A}}},

(A3)

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{\begin{matrix} \frac{\sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{E})}^{m}]}{\sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{A})}^{m}]} \leq 1 i f γ^{A} \geq γ^{E}, \\ \frac{\sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{E})}^{m}]}{\sum_{m = 0}^{\infty} f_{n} (m, μ) [1 - {(1 - γ^{A})}^{m}]} \leq \frac{γ^{E}}{γ^{A}} i f γ^{A} < γ^{E} . \end{matrix}

(A4)

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Q_{μ, n}^{B A E} = p (n, μ) Y_{n}^{E} \leq [Q_{μ, n}^{B A} - p (n, μ) Y_{n}^{A}] \max {1, \frac{γ^{E}}{γ^{A}}} .

(A5)

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Dong Pan, Zaisheng Lin, Jiawei Wu, Haoran Zhang, Zhen Sun, Dong Ruan, Liuguo Yin, Gui Lu Long. Experimental free-space quantum secure direct communication and its security analysis[J]. Photonics Research, 2020, 8(9): 1522

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