Improved statistical fluctuation analysis for measurement-device-independent quantum key distribution with three-intensity decoy-state method

We present an improved statistical fluctuation analysis for measurement device independent quantum key distribution with three-intensity decoy-state method. Taking the statistical fluctuations for different sources jointly, we present more tightened formulas for some key quantities used in calculating the secure final key. Numerical simulation shows that, given the total number of pulses $10^{12}$, our method improves the key rate by about 97\% for a distance of 50kms compared with the result given by Xu., et al. (Phys. Rev. A 89, 052333); and improves the key rate by $146\%$ for a distance of 100kms compared with the result from full optimization of all parameters but treating the statistical fluctuations traditionally, i.e., treating the fluctuations for different sources separately.