  Anderson localization, i.e. the suppression of diffusion in lattices with
random or incommensurate disorder, is a fragile interference phenomenon which
is spoiled out in the presence of dephasing effects or fluctuating disorder. As
a consequence, Anderson localization-delocalization phase transitions observed
in Hermitian systems, such as in one-dimensional quasicrystals when the
amplitude of the incommensurate potential is increased above a threshold, are
washed out when dephasing effects are included. Here we consider
localization-delocalization spectral phase transitions occurring in
non-Hermitian quasicrystals with local incommensurate gain and loss, and show
that, contrary to the Hermitian case, the non-Hermitian phase transition is
robust against dephasing effects. The results are illustrated by considering
synthetic quasicrystals in photonic mesh lattices.
