  We consider quantum and private communications assisted by repeaters, from
the basic scenario of a single repeater chain to the general case of an
arbitrarily-complex quantum network, where systems may be routed through single
or multiple paths. In this context, we investigate the ultimate rates at which
two end-parties may transmit quantum information, distribute entanglement, or
generate secret keys. These end-to-end capacities are defined by optimizing
over the most general adaptive protocols that are allowed by quantum mechanics.
Combining techniques from quantum information and classical network theory, we
derive single-letter upper bounds for the end-to-end capacities in repeater
chains and quantum networks connected by arbitrary quantum channels,
establishing exact formulas under basic decoherence models, including bosonic
lossy channels, quantum-limited amplifiers, dephasing and erasure channels. For
the converse part, we adopt a teleportation-inspired simulation of a quantum
network which leads to upper bounds in terms of the relative entropy of
entanglement. For the lower bounds we combine point-to-point quantum protocols
with classical network algorithms. Depending on the type of routing (single or
multiple), optimal strategies corresponds to finding the widest path or the
maximum flow in the quantum network. Our theory can also be extended to
simultaneous quantum communication between multiple senders and receivers.
