Practical decoy state for quantum key distribution

Decoy states have recently been proposed as a useful method for substantially improving the performance of quantum key distribution (QKD). Here, we present a general theory of the decoy state protocol based on only two decoy states and one signal state. We perform optimization on the choice of intensities of the two decoy states and the signal state. Our result shows that a decoy state protocol with only two types of decoy states—the vacuum and a weak decoy state—asymptotically approaches the theoretical limit of the most general type of decoy state protocol (with an infinite number of decoy states). We also present a one-decoy-state protocol. Moreover, we provide estimations on the effects of statistical fluctuations and suggest that, even for long-distance (larger than 100 km) QKD, our two-decoy-state protocol can be implemented with only a few hours of experimental data. In conclusion, decoy state quantum key distribution is highly practical.

Figure 1

(Color online) The solid lines show the relative deviations of $Y_1^{L,{\nu }_1,{\nu }_2}$ and $e_1^{U,{\nu }_1,{\nu }_2}$ from the asymptotic values (i.e., the case ${\nu }_1,{\nu }_2\to 0$) as functions of $\nu ∕\mu$ (where $\nu ={\nu }_1$) with the fiber length 40 km, and the dashed lines show the case of 140 km. The bounds $Y_1^{L,{\nu }_1,{\nu }_2}$ and $e_1^{U,{\nu }_1,{\nu }_2}$ are given by Eqs. (34, 37), and the true values are given by Eqs. (7, 9). We consider the $\text{vacuum}+\text{weak}$ protocol here (${\nu }_1=\nu$ and ${\nu }_2=0$). The expected photon number is $\mu =0.48$ as calculated from Eq. (12). The parameters used are from GYS [5] as listed in Table .

Figure 2

(Color online) The dashed line shows the asymptotic decoy state method (with infinite number of decoy states) with a maximal secure distance of 142.05 km, using Eq. (1). The solid line shows our $\text{vacuum}+\text{weak}$ decoy method, Eq. (42), with $\mu =0.48$, ${\nu }_1=0.05$, and ${\nu }_2=0$. It uses a strong version of the GLLP protocol and its maximal distance is 140.55 km. The dotted line shows the asymptotic case of Wang’s decoy method, Eq. (43) with $\mu =0.30$. It uses a weak version of the GLLP protocol and its maximal distance is about 128.55 km. This shows that our $\text{vacuum}+\text{weak}$ decoy protocol comes very close to the asymptotic limit and performs better than even the asymptotic case of Wang’s decoy method. The data are from GYS [5] as listed in Table .

Figure 3

(Color online) The solid line shows the simulation result of the $\text{vacuum}+\text{weak}$ protocol [Eqs. (34, 37)] with statistical fluctuations. The dashed line shows the result for the one-decoy-state method [Eq. (41)]. Here, we pick the data-set size (total number of pulses emitted by Alice) to be $N=6\times {10}^9$. We find the optimal $\nu$ for each fiber length by numerically solving Eqs. (44, 46, 47). The confidence interval for statistical fluctuation is ten standard deviations [i.e., $1\text{–}\left(1.5\right)\times {10}^{-23}$]. The data are from GYS [5] as listed in Table . The expected photon number of the signal state is calculated by Eq. (12), getting $\mu =0.48$. The second decoy state (vacuum decoy) becomes useful at 82 km.

Figure 4

(Color online) The dotted line shows the performance of the perfect decoy state method (with infinite number of decoy states and no statistical fluctuations). The maximal distance is about 142 km. The solid line shows the simulation result of the $\text{vacuum}+\text{weak}$ protocol [Eqs. (34, 37)] with statistical fluctuations. Its maximal distance is about 125 km. The dashed line shows the result for the one-decoy-state method [Eq. (41)] with maximal distance 122 km. We pick a data-set size (i.e., total number of pulses emitted by Alice) to be $N=6\times {10}^9$. Note that even with statistical fluctuations and a rather modest data-set size, our $\text{vacuum}+\text{weak}$ decoy protocol comes rather close to asymptotic limit, particularly at short distances. The second decoy state (vacuum decoy) becomes useful at 82 km. The data are from GYS [5] as listed in Table . The expected photon number of the signal state is calculated by Eq. (12), getting $\mu =0.48$.

Figure 5

(Color online) Here, we consider the data-set size (i.e., the number of pulses emitted by Alice) to be $N=8.4\times {10}^{10}$, following Wang [13]. The dashed line shows the performance of the perfect decoy state method. Its maximal distance is 142 km. The solid line shows the simulation result of the $\text{vacuum}+\text{weak}$ decoy state method with statistical fluctuations. Its maximal distance is 132 km. The dotted line shows the asymptotic case (i.e., an idealized version) of Wang’s method. Its maximal distance is 128.55 km. This figure shows clearly that with a data-set size $N=8.4\times {10}^{10}$, our protocol, which considers statistical fluctuations, performs better even than the idealized version of Wang’s protocol, where statistical fluctuations are neglected. For our asymptotic case and two decoy state with statistical fluctuation $\mu =0.48$, and for Wang’s asymptotic case $\mu =0.3$, which are optimized.

Figure 6

(Color online) Here, we compare various protocols using the parameters of Bourenanne et al. [19], listed in Table and [18]. The dashed line shows the performance of the perfect decoy state method. It has a maximal secure distance of about 68.6 km. The solid line shows the simulation result of the $\text{vacuum}+\text{weak}$ decoy state method with statistical fluctuations. The maximal distance is about 67.2 km. The dotted line shows the asymptotic case (i.e., neglecting statistical fluctuations) of Wang’s method, whose maximal distance is about 55.5 km. For our asymptotic case and two decoy states with statistical fluctuation $\mu =0.77$, and for Wang’s asymptotic case $\mu =0.43$, which are optimized.