On Simultaneous Information and Energy Transmission through Quantum Channels

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.