  Network Utility Maximization (NUM) is a mathematical framework that has
endowed researchers with powerful methods for designing and analyzing classical
communication protocols. NUM has also enabled the development of distributed
algorithms for solving the resource allocation problem, while at the same time
providing certain guarantees, e.g., that of fair treatment, to the users of a
network. We extend here the notion of NUM to quantum networks, and propose
three quantum utility functions -- each incorporating a different entanglement
measure. We aim both to gain an understanding of some of the ways in which
quantum users may perceive utility, as well as to explore structured and
theoretically-motivated methods of simultaneously servicing multiple users in
distributed quantum systems. Using our quantum NUM constructions, we develop an
optimization framework for networks that use the single-photon scheme for
entanglement generation, which enables us to solve the resource allocation
problem while exploring rate-fidelity tradeoffs within the network topologies
that we consider. We learn that two of our utility functions, which are based
on distillable entanglement and secret key fraction, are in close agreement
with each other and produce similar solutions to the optimization problems we
study. Our third utility, based on entanglement negativity, has more favorable
mathematical properties, and tends to place a higher value on the rate at which
users receive entangled resources, compared to the two previous utilities,
which put a higher emphasis on end-to-end fidelity. These contrasting behaviors
thus provide ideas regarding the suitability of quantum network utility
definitions to different quantum applications.
