Three-intensity decoy state method for device independent quantum key distribution

We study the measurement device independent quantum key distribution (MDI-QKD) in practice with limited resource, when there are only 3 different states in implementing the decoy-state method. We present a more tightened explicit formula to estimate the lower bound of the yield of two-single-photon pulses. Moreover, we show that the bounding of this yield and phase flip error of single photon pulse pairs can be further improved by using other constraints which can be solved by a simple and explicit program. Results of numerical simulation for key rates with both the improved explicit formula and the program are presented. It shows that the results obtained with our methods here can significantly improve the key rate and secure distance of MDI QKD with only three intensities.