Protocol choice and parameter optimization in decoy-state measurement-device-independent quantum key distribution (2024)

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  Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber

  Figure 1

  Key rate comparison with the infinite data set. The dotted black curve is the perfect key rate with infinite decoy states. The blue solid curve is our optimized key rate using the numerical approach with two decoy states, where the intensities are $\omega =0.0005$, $\nu =0.01$, and optimized $\mu$. For comparison purposes, we present the nonoptimized and partially optimized key rates using the methods and parameters of Refs.[27, 29, 30]: the black dashed curve is from using[30] with $\omega =0$, $\nu =0.01$, and optimized $\mu$; the red (dark gray) dashed curve is from using[29] with $\omega =0.01$, $\nu =0.1$, and $\mu =0.3$; the green (light gray) dashed curve is from using[27] with $\omega =0$, $\nu =0.1$, and $\mu =0.5$. Notice that if the parameter optimization is also applied to Refs[27, 29], all the key rates are almost the same. In the asymptotic case, parameter optimization is simple, as only the intensities are required to be optimized and a smaller value of decoy-state intensity can, in principle, result in a better estimation. Parameter optimization can still increase the key rate and extend the secure distance.

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  Figure 2

  Practical key rate comparison (with statistical fluctuations). The optimal parameters and key rate for a distance of 50 km (standard fiber) are shown in Table3. All the key rates are simulated with $N={10}^{12}$. The blue solid and red dashed-dotted curves (almost overlapping) are, respectively, our optimized key rates (after a full optimization) using the numerical (Appendixpp1-s1) and analytical (Appendixpp1-s2) methods with two decoy states. The black dashed curve is from using the method of Ref.[30], where only partial parameters (i.e., the intensities) are optimized. The green (gray) dashed curve is from using the method of Ref.[27], where some typical parameters are assumed without optimization. Without full parameter optimization, the key rates in Refs[27, 30] are around one order of magnitude lower than ours across different distances. Our method can enable secure MDI-QKD over distances 25 km longer than those in[27, 30]. These results highlight the importance of parameter optimization in practical decoy-state MDI-QKD.

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  Figure 3

  Asymptotic key rates with different numbers of decoy states. The solid curve is the one with infinite decoy states. The dashed, dash-dotted, and dotted curves are, respectively, the one-, two-, and three-decoy-state results using numerical methods (see Appendixpp1-s1). The signal state $\mu$ is optimized in all cases, while some reasonable values of decoy states are adopted: for one decoy state, $\nu =0.0005$; for two decoy states, $\nu =0.01$ and $\omega =0.0005$; for three decoy states, ${\nu }_1=0.1$, ${\nu }_2=0.01$, and $\omega =0.0005$. We emphasize that the key rates with analytical methods of Appendixpp1-s2 almost overlap with the ones presented here, which shows that the analytical approaches provide a very good estimation. The estimation using two decoy states gives a key rate that is nearly the same as the one using three decoy states and is higher than that for the one-decoy-state case. Therefore, two decoy states can already result in a near-optimal estimation, and more decoy states cannot improve the key rate.

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  Figure 4

  Secret key rate in logarithmic scale as a function of the distance under different numbers of decoy states. The main figure is for data set $N={10}^{12}$, and the insert is for $N={10}^{14}$. The key rates are obtained using numerical methods with one (dashed curve), two (solid curve), and three (dash-dotted curve) decoy states. The key rates with two and three decoy states almost overlap. In simulation, we perform a full parameter optimization for all cases. Our results show that after a full parameter optimization, the two-decoy-state method can give an almost optimal key rate, which is higher than the one for one decoy state. Three decoy states cannot help increase the key rate.

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  Figure 5

  Coordinate descent (CD). The CD algorithm searches along one coordinate direction in each iteration, and it uses a different coordinate direction cyclically. For instance, on the equal-error contour of two-dimensional subspace, CD starts at point A (arbitrarily) and descends vertically along direction $e_1$ to B, then horizontally along direction $e_2$ toward C. After cyclic iterations of vertical and horizontal descent, the algorithm stops at D, where it is very close to the optimal. This simplified two-dimensional example illustrates how a generalized search in any dimensional space can be done analogously.

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  Figure 6

  Plot of the intensity $\mu$ as a function of the transmission distance for the decoy-state MDI-QKD with infinite decoy states.

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  Figure 7

  (a) Convexity of the key rate function. The key rate is simulated by sweeping $\mu$ and $\nu$ and optimizing other parameters. (b) Key rate as a function of the decoy state $\omega$. As long as the intensity of $\omega$ is below $1\times {10}^{-3}$, the key rate is very close to the optimum. A perfect vacuum ($\omega =0$) is not essentially required in practical decoy-state QKD experiments.

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