  Losses are one of the main bottlenecks for the distribution of entanglement
in quantum networks, which can be overcome by the implementation of quantum
repeaters. The most basic form of a quantum repeater chain is the swap ASAP
repeater chain. In such a repeater chain, elementary links are
probabilistically generated and deterministically swapped as soon as two
adjacent links have been generated. As each entangled state is waiting to be
swapped, decoherence is experienced, turning the fidelity of the entangled
state between the end nodes of the chain into a random variable. Fully
characterizing the (average) fidelity as the repeater chain grows is still an
open problem. Here, we analytically investigate the case of equally-spaced
repeaters, where we find exact analytic formulae for all moments of the
fidelity up to 25 segments. We obtain these formulae by providing a general
solution in terms of a generating function; a function whose n'th term in its
Maclaurin series yields the moments of the fidelity for n segments. We
generalize this approaches as well to a global cut-off policy -- a method for
increasing fidelity at the cost of longer entanglement delivery times --
allowing for fast optimization of the cut-off parameter by eliminating the need
for Monte Carlo simulation. We furthermore find simple approximations of the
average fidelity that are exponentially tight, and, for up to 10 segments, the
full distribution of the delivered fidelity. We use this to analytically
calculate the secret-key rate when the distributed entanglement is used for
quantum-key distribution, both with and without binning methods. In follow-up
work we exploit a connection to a model in statistical physics to numerically
calculate quantities of interest for the inhomogeneous multipartite case.
