  The optimal rate at which information can be sent through a quantum channel
when the transmitted signal must simultaneously carry some minimum amount of
energy is characterized. To do so, we introduce the quantum-classical analogue
of the capacity-power function and generalize results in classical information
theory for transmitting classical information through noisy channels. We show
that the capacity-power function for a classical-quantum channel, for both
unassisted and private protocol, is concave and also prove additivity for
unentangled and uncorrelated ensembles of input signals for such channels. This
implies we do not need regularized formulas for calculation. We show these
properties also hold for all noiseless channels when we restrict the set of
input states to be pure quantum states. For general channels, we find that the
capacity-power function is piece-wise concave. We give an elegant visual proof
for this supported by numerical simulations. We connect channel capacity and
properties of random quantum states. In particular, we obtain analytical
expressions for the capacity-power function for the case of noiseless channels
using properties of random quantum states under an energy constraint and
concentration phenomena in large Hilbert spaces.
