Private-Capacity Bounds for Bosonic Wiretap Channels
  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. 
Ligong Wang, Jeffrey H. Shapiro, Nivedita Chandrasekaran, and Gregory   W. Wornell
quant-ph