1Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical Science and Technology, Guangxi University, Nanning 530004, China
2Guangxi Key Laboratory of Multimedia Communications and Network Technology, School of Computer, Electronics, and Information,Guangxi University, Nanning 530004, China
Reference-frame-independent measurement-device-independent quantum key distribution is a promising candidate for building star-type quantum secure networks because it does not require reference alignment and removes all detector-side-channel attacks. However, prior works considered only a symmetric case in which the channels of both users have the same loss. In a realistic quantum secure network, the losses of various channels are likely to be different owing to their geographical locations. In this study, we present an asymmetric protocol for scalable reference-frame-independent measurement-device-independent quantum key distribution networks. By allowing independent adjustments of signal intensities of both users, our protocol provides a higher key rate than previous symmetric protocols in a realistic quantum secure network. The simulation results demonstrate that our protocol works well under realistic experimental conditions and obtains a key rate that is approximately one order of magnitude higher than that of previous methods. Our study paves the way for high-rate quantum secure communication network development.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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${p_d}$ denotes the dark count rate, $\epsilon$ is the failure probability of statistical analysis, ${f_e}$ is the error correction efficiency, $\alpha (\rm dB/km)$ is the channel loss coefficient, $N$ is the total sending pulses, ${\eta _d}$ is the detector efficiency, ${e_Z}$ represents the optical misalignment of the $Z$ basis, and ${e_{X,Y}}$ represents the optical misalignment of the $X$ and $Y$ bases.
Table 2.
Three Protocols with Different Distance Combinationsa
Protocol
x
Rate
Comparison with Four-Intensity Protocol
Four-intensity protocol
0.13
10 km
60 km
…
Four-intensity protocol fiber
1
60 km
60 km
Our protocol
0.13
10 km
60 km
Four-intensity protocol
0.26
12 km
45 km
…
Four-intensity protocol fiber
1
45 km
45 km
Our protocol
0.26
12 km
45 km
Four-intensity protocol
0.13
10 km
60 km
…
Four-intensity protocol fiber
1
60 km
60 km
Our protocol
0.13
10 km
60 km
Four-intensity protocol
0.26
12 km
45 km
…
Four-intensity protocol fiber
1
45 km
45 km
Our protocol
0.26
12 km
45 km
The first method uses the classical four-intensity protocol without additional fibers. The second method uses the four-intensity protocol and includes additional fibers to the short arm to match the long arm. The third method is our protocol, which produces different intensities and does not require additional fibers; $N= {10^{11}}$ and $x = {\eta _A}/{\eta _B}$.
Table 3.
Examples of Parameters Used for Our Asymmetric Protocola
SKR
60 km
10 km
0.13
0.939
0.664
0.141
0.494
0.194
0.193
0.151
0.155
0.955
0.084
0.018
0.494
0.189
0.188
0.147
0.151
60 km
60 km
1
0.822
0.377
0.082
0.303
0.263
0.262
0.158
0.159
0.822
0.377
0.082
0.289
0.270
0.273
0.169
0.158
45 km
12 km
0.26
0.965
0.566
0.118
0.607
0.150
0.150
0.143
0.144
0.973
0.144
0.030
0.615
0.144
0.143
0.140
0.142
45 km
45 km
1
0.932
0.357
0.076
0.494
0.192
0.191
0.147
0.155
0.932
0.357
0.076
0.495
0.191
0.191
0.149
0.147
SKR
60 km
10 km
0.13
0.870
0.637
0.127
0.409
0.226
0.229
0.150
0.142
0.903
0.080
0.016
0.411
0.222
0.220
0.138
0.146
60 km
60 km
1
0.612
0.363
0.075
0.190
0.306
0.307
0.156
0.153
0.612
0.363
0.075
0.188
0.306
0.309
0.159
0.149
45 km
12 km
0.26
0.787
0.363
0.030
0.408
0.223
0.223
0.153
0.153
0.942
0.139
0.027
0.552
0.167
0.167
0.138
0.135
45 km
45 km
1
0.858
0.343
0.069
0.408
0.225
0.226
0.148
0.142
0.858
0.343
0.069
0.420
0.218
0.219
0.147
0.143
${{\rm Pr}_{{Z_A}}}$ (${{\rm Pr}_{{Z_B}}}$) is the probability that Alice (Bob) chooses the $Z$ basis; ${{\rm Pr}_{X{{(Y)}_{A(B)}}}}$ and ${{\rm Pr}_{{Y_{A(B)}}}}$ are the independent probabilities of Alice (Bob) choosing the $X$ and $Y$ bases, respectively; ${{\rm Pr}_{{X_{A(B)}}|{\nu _{A(B)}}}}$ and ${{\rm Pr}_{{Y_{A(B)}}|{\nu _{A(B)}}}}$ are the independent probabilities of Alice (Bob) choosing the intensities ${\nu _{A(B)}}$ in $X$ and $Y$ bases, respectively.
${p_d}$ denotes the dark count rate, $\epsilon$ is the failure probability of statistical analysis, ${f_e}$ is the error correction efficiency, $\alpha (\rm dB/km)$ is the channel loss coefficient, $N$ is the total sending pulses, ${\eta _d}$ is the detector efficiency, ${e_Z}$ represents the optical misalignment of the $Z$ basis, and ${e_{X,Y}}$ represents the optical misalignment of the $X$ and $Y$ bases.
Table 2.
Three Protocols with Different Distance Combinationsa
Protocol
x
Rate
Comparison with Four-Intensity Protocol
Four-intensity protocol
0.13
10 km
60 km
…
Four-intensity protocol fiber
1
60 km
60 km
Our protocol
0.13
10 km
60 km
Four-intensity protocol
0.26
12 km
45 km
…
Four-intensity protocol fiber
1
45 km
45 km
Our protocol
0.26
12 km
45 km
Four-intensity protocol
0.13
10 km
60 km
…
Four-intensity protocol fiber
1
60 km
60 km
Our protocol
0.13
10 km
60 km
Four-intensity protocol
0.26
12 km
45 km
…
Four-intensity protocol fiber
1
45 km
45 km
Our protocol
0.26
12 km
45 km
The first method uses the classical four-intensity protocol without additional fibers. The second method uses the four-intensity protocol and includes additional fibers to the short arm to match the long arm. The third method is our protocol, which produces different intensities and does not require additional fibers; $N= {10^{11}}$ and $x = {\eta _A}/{\eta _B}$.
Table 3.
Examples of Parameters Used for Our Asymmetric Protocola
SKR
60 km
10 km
0.13
0.939
0.664
0.141
0.494
0.194
0.193
0.151
0.155
0.955
0.084
0.018
0.494
0.189
0.188
0.147
0.151
60 km
60 km
1
0.822
0.377
0.082
0.303
0.263
0.262
0.158
0.159
0.822
0.377
0.082
0.289
0.270
0.273
0.169
0.158
45 km
12 km
0.26
0.965
0.566
0.118
0.607
0.150
0.150
0.143
0.144
0.973
0.144
0.030
0.615
0.144
0.143
0.140
0.142
45 km
45 km
1
0.932
0.357
0.076
0.494
0.192
0.191
0.147
0.155
0.932
0.357
0.076
0.495
0.191
0.191
0.149
0.147
SKR
60 km
10 km
0.13
0.870
0.637
0.127
0.409
0.226
0.229
0.150
0.142
0.903
0.080
0.016
0.411
0.222
0.220
0.138
0.146
60 km
60 km
1
0.612
0.363
0.075
0.190
0.306
0.307
0.156
0.153
0.612
0.363
0.075
0.188
0.306
0.309
0.159
0.149
45 km
12 km
0.26
0.787
0.363
0.030
0.408
0.223
0.223
0.153
0.153
0.942
0.139
0.027
0.552
0.167
0.167
0.138
0.135
45 km
45 km
1
0.858
0.343
0.069
0.408
0.225
0.226
0.148
0.142
0.858
0.343
0.069
0.420
0.218
0.219
0.147
0.143
${{\rm Pr}_{{Z_A}}}$ (${{\rm Pr}_{{Z_B}}}$) is the probability that Alice (Bob) chooses the $Z$ basis; ${{\rm Pr}_{X{{(Y)}_{A(B)}}}}$ and ${{\rm Pr}_{{Y_{A(B)}}}}$ are the independent probabilities of Alice (Bob) choosing the $X$ and $Y$ bases, respectively; ${{\rm Pr}_{{X_{A(B)}}|{\nu _{A(B)}}}}$ and ${{\rm Pr}_{{Y_{A(B)}}|{\nu _{A(B)}}}}$ are the independent probabilities of Alice (Bob) choosing the intensities ${\nu _{A(B)}}$ in $X$ and $Y$ bases, respectively.
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