Nonconvexity of private capacity and classical environment-assisted capacity of a quantum channel

The capacity of classical channels is convex. This is not the case for the quantum capacity of a channel: The capacity of a mixture of different quantum channels exceeds the mixture of the individual capacities and thus is nonconvex. Here we show that this effect goes beyond the quantum capacity and holds for the private and classical environment-assisted capacities of quantum channels.

Figure 1

Operational interpretation of nonconvexity. Above, Alice has full control over which channel is applied in the transmission, but she has to apply each channel with some probability. Below, a black box chooses the channel for Alice (with the same probabilities). Nonconvexity implies that Alice might communicate at a strictly higher rate in the scenario below.

Figure 2

The figure shows the difference between (21) and (16) normalized by $logd$ when $d$ goes to infinity. A value larger than zero implies nonconvexity of $\mathcal{P}$.