  Non-Hermitian boundaries commonly take place in many open quantum systems
locally coupled to a surrounding environment. Here, we reveal a type of
non-Hermitian effect induced by non-Hermitian boundaries, the non-Hermitian
proximity effect (NHPE), which describes the penetration of non-Hermiticity
from the boundary into the bulk. For gapped quantum systems, the NHPE generates
in-gap states with imaginary eigenenergies, termed ``imaginary in-gap states".
The imaginary in-gap states are localized at the system boundary and decay into
the bulk, analogous to the behaviors of the conventional impurity states.
However, in contrast to impurity states, the imaginary in-gap states exhibit
distinct dynamical behaviors under time-evolution. Moreover, they are
physically manifested as corner modes under open boundaries, as a combined
result of the non-Hermitian skin effect (NHSE) and NHPE. These results not only
uncover implicit similarities between quantum systems with non-Hermitian
boundaries and impurity physics, but also point to intriguing non-Hermitian
phenomena broadly relevant to open quantum systems.
