  We prove an upper bound on the private capacity of the single-mode noiseless
bosonic wiretap channel. Combined with a previous lower bound, we obtain the
low photon-number asymptotic expression for the private capacity. We then show
that the multiple-mode noiseless bosonic wiretap channel is equivalent to
parallel single-mode channels, hence the single-mode bounds can be applied.
Finally, we consider multiple-spatial-mode propagation through atmospheric
turbulence, and derive a private-capacity lower bound that only requires second
moments of the channel matrix.
