  Collins and Popescu realized a powerful analogy between several resources in
classical and quantum information theory. The Collins-Popescu analogy states
that public classical communication, private classical communication, and
secret key interact with one another somewhat similarly to the way that
classical communication, quantum communication, and entanglement interact. This
paper discusses the information-theoretic treatment of this analogy for the
case of noisy quantum channels. We determine a capacity region for a quantum
channel interacting with the noiseless resources of public classical
communication, private classical communication, and secret key. We then compare
this region with the classical-quantum-entanglement region from our prior
efforts and explicitly observe the information-theoretic consequences of the
strong correlations in entanglement and the lack of a super-dense coding
protocol in the public-private-secret-key setting. The region simplifies for
several realistic, physically-motivated channels such as entanglement-breaking
channels, Hadamard channels, and quantum erasure channels, and we are able to
compute and plot the region for several examples of these channels.
