  While measurement-device-independent (MDI) quantum key distribution (QKD)
allows two trusted parties to establish a shared secret key from a distance
without needing to trust a central detection node, their quantum sources must
be well-characterized, with side-channels at the source posing the greatest
loophole to the protocol's security. In this paper, we identify a
time-dependent side-channel in a common polarization-based QKD source that
employs a Faraday mirror for phase stabilization. We apply the recently
developed numerical proof technique from [Phys. Rev. A 99, 062332 (2019)] to
quantify the sensitivity of the secret key rate to the quantum optical model
for the side-channel, and to develop strategies to mitigate the information
leakage. In particular, we find that the MDI three-state and BB84 protocols,
while yielding the same key rate under ideal conditions, have diverging results
in the presence of a side-channel, with BB84 proving more advantageous. While
we consider only a representative case example, we expect the strategies
developed and key rate analysis method to be broadly applicable to other leaky
sources.
