Boosting up quantum key distribution by learning statistics of practical single photon sources

We propose a simple quantum-key-distribution (QKD) scheme for practical single photon sources (SPSs), which works even with a moderate suppression of the second-order correlation $g^{(2)}$ of the source. The scheme utilizes a passive preparation of a decoy state by monitoring a fraction of the signal via an additional beam splitter and a detector at the sender's side to monitor photon number splitting attacks. We show that the achievable distance increases with the precision with which the sub-Poissonian tendency is confirmed in higher photon number distribution of the source, rather than with actual suppression of the multi-photon emission events. We present an example of the secure key generation rate in the case of a poor SPS with $g^{(2)} = 0.19$, in which no secure key is produced with the conventional QKD scheme, and show that learning the photon-number distribution up to several numbers is sufficient for achieving almost the same achievable distance as that of an ideal SPS.