  In this paper, we study the unconditional security of the so-called
measurement device independent quantum key distribution (MDIQKD) with the
basis-dependent flaw in the context of phase encoding schemes. We propose two
schemes for the phase encoding, the first one employs a phase locking technique
with the use of non-phase-randomized coherent pulses, and the second one uses
conversion of standard BB84 phase encoding pulses into polarization modes. We
prove the unconditional security of these schemes and we also simulate the key
generation rate based on simple device models that accommodate imperfections.
Our simulation results show the feasibility of these schemes with current
technologies and highlight the importance of the state preparation with good
fidelity between the density matrices in the two bases. Since the
basis-dependent flaw is a problem not only for MDIQKD but also for standard
QKD, our work highlights the importance of an accurate signal source in
practical QKD systems.
  Note: We include the erratum of this paper in Appendix C. The correction does
not affect the validity of the main conclusions reported in the paper, which is
the importance of the state preparation in MDIQKD and the fact that our schemes
can generate the key with the practical channel mode that we have assumed.
