Optimal decoy intensity for decoy quantum key distribution

In the decoy quantum key distribution, we show that a smaller decoy intensity gives a better key generation rate in the asymptotic setting when we employ only one decoy intensity and the vacuum pulse. In particular, the counting rate of single photon can be perfectly estimated when the decoy intensity is infinitesimal. The same property holds even when the intensities cannot be perfectly identified. Further, we propose a protocol to improve the key generation rate over the existing protocol under the same decoy intensity.