Improving key rates of the unbalanced phase-encoded BB84 protocol using the flag-state squashing model

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 multiphoton pulses to recover the behavior 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 multiphoton-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 regimes. 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.