  We consider the fidelity of state transfer on an unweighted path on $n$
vertices, where a loop of weight $w$ has been appended at each of the end
vertices. It is known that if $w$ is transcendental, then there is pretty good
state transfer from one end vertex to the other; we prove a companion result to
that fact, namely that there is a dense subset of $[1,\infty)$ such that if $w$
is in that subset, pretty good state transfer between end vertices is
impossible. Under mild hypotheses on $w$ and $t$, we derive upper and lower
bounds on the fidelity of state transfer between end vertices at readout time
$t$. Using those bounds, we localise the readout times for which that fidelity
is close to $1$. We also provide expressions for, and bounds on, the
sensitivity of the fidelity of state transfer between end vertices, where the
sensitivity is with respect to either the readout time or the weight $w$.
Throughout, the results rely on detailed knowledge of the eigenvalues and
eigenvectors of the associated adjacency matrix.
