  All phase-encoded BB84 implementations have signal states with unbalanced
amplitudes in practice. Thus, the original security analyses a priori do not
apply to them. Previous security proofs use signal tagging of multi-photon
pulses to recover the behaviour of regular BB84. This is overly conservative,
as for unbalanced signals, the photon-number splitting attack does not leak
full information to Eve. In this work, we exploit the flag-state squashing
model to preserve some parts of the multi-photon generated private information
in our analysis. Using a numerical proof technique, we obtain significantly
higher key rates compared with previously published results in the low-loss
regime. It turns out that the usual scenario of untrusted dark counts runs into
conceptual difficulties in some parameter regime. Thus, we discuss the trusted
dark count scenario in this paper as well. We also report a gain in key rates
when part of the total loss is known to be induced by a trusted device. We
highlight that all these key rate improvements can be achieved without
modification of the experimental setup.
