Improved key-rate bounds for practical decoy-state quantum-key-distribution systems

The decoy-state scheme is the most widely implemented quantum-key-distribution protocol in practice. In order to account for the finite-size key effects on the achievable secret key generation rate, a rigorous statistical fluctuation analysis is required. Originally, a heuristic Gaussian-approximation technique was used for this purpose, which, despite its analytical convenience, was not sufficiently rigorous. The fluctuation analysis has recently been made rigorous by using the Chernoff bound. There is a considerable gap, however, between the key-rate bounds obtained from these techniques and that obtained from the Gaussian assumption. Here we develop a tighter bound for the decoy-state method, which yields a smaller failure probability. This improvement results in a higher key rate and increases the maximum distance over which secure key exchange is possible. By optimizing the system parameters, our simulation results show that our method almost closes the gap between the two previously proposed techniques and achieves a performance similar to that of conventional Gaussian approximations.

Figure 1

Comparison of the width of the confidence interval versus failure probability for three methods: the Gaussian analysis (solid line), the Chernoff-Hoeffding (CH) method [24] (dotted line), and our method (dash-dotted line). In each scheme, we find lower and upper bounds for the expectation value $\mathbb{E}\left[\chi \right]$ from an observed value $\chi$, at a given failure probability and at $\chi \to \infty$. The vertical axis then represents $({\mathbb{E}}^U\left[\chi \right]-{\mathbb{E}}^L\left[\chi \right])/2\sigma$, for $\sigma =\sqrt{\chi }$.

Figure 2

Lower and upper bounds of the expectation value versus observed values of $\chi$ for the Gaussian analysis (dotted line) and our method (solid line). In both cases, the failure probability is fixed at $\varepsilon ={10}^{-10}$.

Figure 3

Total failure probability $\varepsilon$ versus the observed value $\chi$ when we fix the deviation from the mean value given by $n_{\alpha }\sigma$, for $n_{\alpha }=3,5,7,9$ from top to bottom.

Figure 4

Comparison of the key rates obtained by the three methods: the Gaussian analysis, the Chernoff-Hoeffding method [24], and our method. The infinite-key-length case is also shown.

Figure 5

Maximum secure transmission distance versus the number of pluses sent by Alice $N$. The simulation parameters are listed in Table 3. No secure keys can be generated for $N\le {10}^7$. The asymptotic limit for the maximum secure transmission distance is 142 km when $N\ge {10}^{14}$.