  By employing the Lindblad equation, we derive the evolution of the two-point
correlator for a free-fermion chain of length $L$ subject to bulk dephasing and
boundary losses. We use the Bethe ansatz to diagonalize the Liouvillian
${\mathcal L}^{\scriptscriptstyle(2)}$ governing the dynamics of the
correlator. The majority of its energy levels are complex. Precisely,
$L(L-1)/2$ complex energies do not depend on dephasing, apart for a trivial
shift. The remaining complex levels are perturbatively related to the
dephasing-independent ones for large $L$. The long-time dynamics is governed by
a band of real energies, which contains an extensive number of levels. They
give rise to diffusive scaling at intermediate times, when boundaries can be
neglected. Moreover, they encode the breaking of diffusion at asymptotically
long times. Interestingly, for large loss rate two boundary modes appear in the
spectrum. The real energies correspond to string solutions of the Bethe
equations, and can be treated effectively for large chains. This allows us to
derive compact formulas for the dynamics of the fermionic density. We check our
results against exact diagonalization, finding perfect agreement.
