  Semiconductor quantum dots have emerged as an especially promising platform
for the generation of polarization-entangled photon pairs. However, it was
demonstrated recently that the two-photon excitation scheme employed in
state-of-the-art experiments limits the achievable degree of entanglement by
introducing which-path information. In this work, the combined impact of
two-photon excitation and longitudinal acoustic phonons on photon pairs emitted
by strongly-confining quantum dots is investigated. It is found that phonons
further reduce the achievable degree of entanglement even in the limit of
vanishing temperature due to phonon-induced pure dephasing and phonon-assisted
one-photon processes, which increase the reexcitation probability. In addition,
the degree of entanglement, as measured by the concurrence, decreases with
rising temperature and/or pulse duration, even if the excitonic fine-structure
splitting is absent and when higher electronic states are out of reach.
Furthermore, in the case of finite fine-structure splittings, phonons enlarge
the discrepancy in concurrence for different laser polarizations.
