Public and private resource trade-offs for a quantum channel

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.