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We show how to calculate the secure final key rate in the four-intensity decoy-state measurement-device-independent quantum key distribution protocol with both source errors and statistical fluctuations with a certain failure probability. Our results rely only on the range of only a few parameters in the source state. All imperfections in this protocol have been taken into consideration without assuming any specific error patterns of the source. [ABSTRACT FROM AUTHOR]

The existing decoy-state measurement-device-independent quantum key distribution (MDI-QKD) theory assumes perfect control of the source states, which is an impossible task for any real setup. In this paper, we study the decoy-state MDI-QKD method with source errors without presuming any unverifiable error patterns. In our method we need the range of only a few parameters in the source state. [ABSTRACT FROM AUTHOR]