Quantum Channel Capacities Per Unit Cost

Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for the additive white Gaussian noise channel, the cost associated with a real number input symbol is the square of its magnitude. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel; in particular, we consider classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special case in which there is a zero-cost quantum state, we obtain expressions for the various capacities per unit cost in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various quantum Gaussian channels and introduce the notion of a blocklength-constrained capacity per unit cost.