  Surpassing the repeaterless bound is a crucial task on the way towards
realizing long-distance quantum key distribution. In this paper, we focus on
the protocol proposed by Azuma et al. in [Nature Communications 6, 10171
(2015)], which can beat this bound with idealized devices. We investigate the
robustness of this protocol against imperfections in realistic setups,
particularly the multiple-photon pair components emitted by practical
entanglement sources. In doing so, we derive necessary conditions on the
photon-number statistics of the sources in order to beat the repeaterless
bound. We show, for instance, that parametric down-conversion sources do not
satisfy the required conditions and thus cannot be used to outperform this
bound.
