Reference-frame-independent measurement-device-independent quantum key distribution with mismatched-basis statistics

The source flaw associated with the basis vector in the reference-frame-independent measurement-device-independent quantum key distribution (RFI-MDI-QKD) has not been systematically studied. As a result, it is often assumed that bit error is equal to phase error, which is not theoretically rigorous. Here, we propose a postprocessing method to estimate the phase error rate from the discarded mismatched-basis statistics, where the qubit source does not need to be characterized in detail. The source flaw in the basis vector of the RFI-MDI-QKD protocol can thus be corrected using this method. The numerical simulation results clearly demonstrate that the RFI-MDI-QKD protocol with uncharacterized sources is also insensitive to the misalignment of the reference frame.