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arXiv:1507.01222 · Physical Review A

Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions

Hongxin Li, Haodong Jiang, Ming Gao, Zhi Ma, Chuangui Ma, Wei Wang

Published Dec 30, 2015• 13 citations
arXivHTMLPDF

Abstract

The statistical fluctuation problem is a critical factor in all quantum key distribution (QKD) protocols under finite-key conditions. The current statistical fluctuation analysis is mainly based on independent random samples, however, the precondition cannot always be satisfied because of different choices of samples and actual parameters. As a result, proper statistical fluctuation methods are required to solve this problem. Taking the after-pulse contributions into consideration, this paper gives the expression for the secure key rate and the mathematical model for statistical fluctuations, focusing on a decoy-state QKD protocol [Z.-C. Wei et al., Sci. Rep. 3, 2453 (2013)] with a biased basis choice. On this basis, a classified analysis of statistical fluctuation is represented according to the mutual relationship between random samples. First, for independent identical relations, a deviation comparison is made between the law of large numbers and standard error analysis. Second, a sufficient condition is given that the Chernoff bound achieves a better result than Hoeffding's inequality based on only independent relations. Third, by constructing the proper martingale, a stringent way is proposed to deal issues based on dependent random samples through making use of Azuma's inequality. In numerical optimization, the impact on the secure key rate, the comparison of secure key rates, and the respective deviations under various kinds of statistical fluctuation analyses are depicted.

DOI
10.1103/physreva.92.062344

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The statistical fluctuation problem is a critical factor in all quantum key distribution (QKD) protocols under finite-key conditions. The current statistical fluctuation analysis is mainly based on independent random samples, however, the precondition cannot always be satisfied because of different choices of samples and actual parameters. As a result, proper statistical fluctuation methods are required to solve this problem. Taking the after-pulse contributions into consideration, this paper gives the expression for the secure key rate and the mathematical model for statistical fluctuations, focusing on a decoy-state QKD protocol [Z.-C. Wei et al., Sci. Rep. 3, 2453 (2013)] with a biased basis choice. On this basis, a classified analysis of statistical fluctuation is represented according to the mutual relationship between random samples. First, for independent identical relations, a deviation comparison is made between the law of large numbers and standard error analysis. Second, a sufficient condition is given that the Chernoff bound achieves a better result than Hoeffding's inequality based on only independent relations. Third, by constructing the proper martingale, a stringent way is proposed to deal issues based on dependent random samples through making use of Azuma's inequality. In numerical optimization, the impact on the secure key rate, the comparison of secure key rates, and the respective deviations under various kinds of statistical fluctuation analyses are depicted.
The BB84 quantum key distribution (QKD) combined with decoy-state method is currently the most practical protocol, which has been proved secure against general attacks in the finite-key regime. Thereinto, statistical fluctuation analysis methods are very important in dealing with finite-key effects, which directly affect secret key rate, secure transmission distance and most importantly, the security. There are two tasks of statistical fluctuation in decoy-state BB84 QKD. One is the deviation between expected value and observed value for a given expected value or observed value. The other is the deviation between phase error rate of computational basis and bit error rate of dual basis. Here, we provide the rigorous and optimal analytic formula to solve the above tasks, resulting to higher secret key rate and longer secure transmission distance. Our results can be widely applied to deal with statistical fluctuation in quantum cryptography protocols.
Abstract One-decoy-state quantum key distribution with infinite length pulses can even approach the asymptotic case, i.e., infinite decoy states and infinite length pulses. Due to the limitation of experimental time, we must consider the statistical fluctuation caused by the finite number of pulses. The central limit theorem analysis method is an effective method, which can easily give the upper and lower bounds of experimental observations. Combining the central limit theorem method with the theorem that the ratio of two independent normal random variables can be approximated as a normal random variable under certain conditions, we propose a more efficient method to estimate the secure key rate. Compared with using central limit theorem method directly, our method has fewer bounds, smaller estimation deviation and higher estimation accuracy. The simulation results show that our method can significantly improve the secure key rate and transmission distance when considering statistical fluctuation and keeping the pulse number and security parameters unchanged, especially in the case of small pulse number (e.g. 107).
Phase-matching quantum key distribution is a promising scheme for remote quantum key distribution, breaking through the traditional linear key-rate bound. In practical applications, finite data size can cause significant system performance to deteriorate when data size is below 1010. In this work, an improved statistical fluctuation analysis method is applied for the first time to two decoy-states phase-matching quantum key distribution, offering a new insight and potential solutions for improving the key generation rate and the maximum transmission distance while maintaining security. Moreover, we also compare the influence of the proposed improved statistical fluctuation analysis method on system performance with those of the Gaussian approximation and Chernoff–Hoeffding boundary methods on system performance. The simulation results show that the proposed scheme significantly improves the key generation rate and maximum transmission distance in comparison with the Chernoff–Hoeffding approach, and approach the results obtained when the Gaussian approximation is employed. At the same time, the proposed scheme retains the same security level as the Chernoff–Hoeffding method, and is even more secure than the Gaussian approximation.
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.
In order to solve the statistical fluctuation problem caused by the finite data length in the practical quantum key distribution system, four commonly used statistical methods, DeMoivre-Laplace theorem, Chebyshev inequality, Chernoff boundary and Hoeffding boundary, are used to analyze. The application conditions of each method are discussed, and the effects of data length and confidence level on quantum key distribution security performance are simulated and analyzed. The simulation results show that the applicable conditions of Chernoff boundary are most consistent with the reality of the practical quantum key distribution system with finite-length data. Under the same experimental conditions, the secure key generation rate and secure transmission distance obtained by Chernoff boundary are better than those of the other three methods. When the data length and confidence level change, the stability of the security performance obtained by the Chernoff boundary is the best.
Measurement-device-independent quantum key distribution (MDI-QKD) is immune to detector side channel attacks, which is a notorious security loophole problem in QKD. Importantly, both finite-key effects and intensity fluctuations of the practical photon sources are nonnegligible in a full key rate analysis of practical QKD. In this paper, for the case with finite-key effects and without intensity fluctuations, we present a way to estimate the phase error rate for MDI-QKD and compare performances of the parameter estimation based on Hoeffding’s inequality and the Chernoff bound. By using Azuma’s inequality, we also present the finite-key analysis with composable security against general attacks for a biased decoy-state MDI-QKD protocol with intensity fluctuations. Our simulation results show that the effect of our phase error rate estimation is almost the same as the results in Phys. Rev. A91, 022313 (2015)PLRAAN1050-294710.1103/PhysRevA.91.022313, and the parameter estimation using the Chernoff bound is tighter than that using Hoeffding’s inequality. In addition, the influence of intensity fluctuations is more obvious when the data size of total transmitting signals is small. Our results highlight that it is important to keep the stability of the intensity modulator and estimate accurately the emitted pulse’s intensity for practical implementations.