The statistical fluctuation study of quantum key distribution in means of uncertainty principle

Laser defects in emitting single photon, photon signal attenuation and propagation of error cause our serious headaches in practical long-distance quantum key distribution (QKD) experiment for a long time. In this paper, we study the uncertainty principle in metrology and use this tool to analyze the statistical fluctuation of the number of received single photons, the yield of single photons and quantum bit error rate (QBER). After that we calculate the error between measured value and real value of every parameter, and concern the propagation error among all the measure values. We paraphrase the Gottesman–Lo–Lutkenhaus–Preskill (GLLP) formula in consideration of those parameters and generate the QKD simulation result. In this study, with the increase in coding photon length, the safe distribution distance is longer and longer. When the coding photon’s length is \(N = 10^{11}\), the safe distribution distance can be almost 118 km. It gives a lower bound of safe transmission distance than without uncertainty principle’s 127 km. So our study is in line with established theory, but we make it more realistic.