# Quantum Channel Capacities Per Unit Cost

## TL;DR

This paper extends the concept of capacity per unit cost to various quantum communication tasks, providing formulas, coding schemes, and capacity calculations for bosonic channels, addressing infinite capacity issues with blocklength constraints.

## Contribution

It introduces a generalized framework for capacity per unit cost across multiple quantum communication tasks and derives explicit formulas and coding schemes for these capacities.

## Key findings

- Derived formulas for capacity per unit cost in quantum communication
- Constructed explicit pulse-position-modulation coding schemes
- Computed capacities for bosonic Gaussian channels

## Abstract

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 bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. 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 such as 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 and natural case in which there is a zero-cost state, we obtain expressions 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 bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality, on which we elaborate in depth.