Exact minimum and maximum of yield with a finite number of decoy light intensities

In this paper, for the decoy state method using a finite number of decoy light intensities, we present an improved upper and lower bounds for the asymptotic yield y_n for n-photon states. In particular if all the light intensities are less than or equal to one, they are not only a lower or upper bound, but in fact are the exact minimum or maximum.