  Remote quantum entanglement can enable numerous applications including
distributed quantum computation, secure communication, and precision sensing.
In this paper, we consider how a quantum network-nodes equipped with limited
quantum processing capabilities connected via lossy optical links-can
distribute high-rate entanglement simultaneously between multiple pairs of
users (multiple flows). We develop protocols for such quantum "repeater" nodes,
which enable a pair of users to achieve large gains in entanglement rates over
using a linear chain of quantum repeaters, by exploiting the diversity of
multiple paths in the network. Additionally, we develop repeater protocols that
enable multiple user pairs to generate entanglement simultaneously at rates
that can far exceed what is possible with repeaters time sharing among
assisting individual entanglement flows. Our results suggest that the
early-stage development of quantum memories with short coherence times and
implementations of probabilistic Bell-state measurements can have a much more
profound impact on quantum networks than may be apparent from analyzing linear
repeater chains. This framework should spur the development of a general
quantum network theory, bringing together quantum memory physics, quantum
information theory, and computer network theory.
