  Entanglement is one of the most important concepts in quantum physics. We
review recent progress in understanding the quantum entanglement in many-body
systems using large-$N$ solvable models: the Sachdev-Ye-Kitaev (SYK) model and
its generalizations. We present the study of entanglement entropy in the
original SYK Model using three different approaches: the exact diagonalization,
the eigenstate thermalization hypothesis, and the path-integral representation.
For coupled SYK models, the entanglement entropy shows linear growth and
saturation at the thermal value. The saturation is related to replica wormholes
in gravity. Finally, we consider the steady-state entanglement entropy of
quantum many-body systems under repeated measurements. The traditional symmetry
breaking in the enlarged replica space leads to the measurement-induced
entanglement phase transition.
