  Determining the ultimate classical information carrying capacity of
electromagnetic waves requires quantum-mechanical analysis to properly account
for the bosonic nature of these waves. Recent work has established capacity
theorems for bosonic single-user and broadcast channels, under the presumption
of two minimum output entropy conjectures. Despite considerable accumulated
evidence that supports the validity of these conjectures, they have yet to be
proven. In this paper, it is shown that the second conjecture suffices to prove
the classical capacity of the bosonic wiretap channel, which in turn would also
prove the quantum capacity of the lossy bosonic channel. The preceding minimum
output entropy conjectures are then shown to be simple consequences of an
Entropy Photon-Number Inequality (EPnI), which is a conjectured
quantum-mechanical analog of the Entropy Power Inequality (EPI) form classical
information theory.
