  This work considers a Poisson noise channel with an amplitude constraint. It
is well-known that the capacity-achieving input distribution for this channel
is discrete with finitely many points. We sharpen this result by introducing
upper and lower bounds on the number of mass points. Concretely, an upper bound
of order $\mathsf{A} \log^2(\mathsf{A})$ and a lower bound of order
$\sqrt{\mathsf{A}}$ are established where $\mathsf{A}$ is the constraint on the
input amplitude. In addition, along the way, we show several other properties
of the capacity and capacity-achieving distribution. For example, it is shown
that the capacity is equal to $ - \log P_{Y^\star}(0)$ where $P_{Y^\star}$ is
the optimal output distribution. Moreover, an upper bound on the values of the
probability masses of the capacity-achieving distribution and a lower bound on
the probability of the largest mass point are established. Furthermore, on the
per-symbol basis, a nonvanishing lower bound on the probability of error for
detecting the capacity-achieving distribution is established under the maximum
a posteriori rule.
