  As a hallmark of pure quantum effect, quantum entanglement has provided
unconventional routes to condensed matter systems. Here, from the perspective
of quantum entanglement, we disclose exotic quantum physics in non-Hermitian
quasicrystals. We present a class of experimentally realizable models for
non-Hermitian quasicrystal chains, in which asymmetric hopping and complex
potential coexist. We diagnose global phase diagram by means of entanglement
from both real-space and momentum-space partition. By measuring entanglement
entropy, we numerically determine the metal-insulator transition point. We
combine real-space and momentum-space entanglement spectra to complementarily
characterize the delocalization phase and the localization phase. Inspired by
entanglement spectrum, we further analytically prove that a duality exists
between the two phase regions. The transition point is self-dual and exact,
further validating the numerical result from diagonalizing non-Hermitian
matrices. Finally, we identify mobility edge by means of entanglement.
