  With the detection of gravitational waves from merging binary black holes,
over the last few years, there has been a considerable interest in the
literature to understand if these black holes could have originated in the
early universe. If the primordial scalar power over small scales is boosted
considerably when compared to the COBE normalized amplitude over large scales,
then, such an increased power can lead to a copious production of primordial
black holes that can constitute a significant fraction of the cold dark matter
density today. Recently, many models of inflation involving single or two
scalar fields have been constructed which lead to enhanced power on small
scales. In this work, we examine the evolution of the quantum state of the
curvature perturbations in such single and two field models of inflation using
measures of squeezing, entanglement entropy or quantum discord. We find that,
in the single as well as the two field models, the extent of squeezing of the
modes is enhanced to a certain extent (when compared to the scenarios involving
only slow roll) over modes which exhibit increased scalar power. Moreover, we
show that, in the two field models, the entanglement entropy associated with
the curvature perturbation, arrived at when the isocurvature perturbation has
been traced out, is also enhanced over the small scales. We also take the
opportunity to discuss the relation between entanglement entropy and quantum
discord in the case of the two field model. We conclude with a brief discussion
on the wider implications of the results.
