  We show how generic non-Hermitian tight-binding lattice models can be
realized in an unconditional, quantum-mechanically consistent manner by
constructing an appropriate open quantum system. We focus on the quantum steady
states of such models for both fermionic and bosonic systems. Surprisingly, key
features and spatial structures in the steady state cannot be simply understood
from the non-Hermitian Hamiltonian alone. Using the 1D Hatano-Nelson model as a
paradigmatic example, we show that the steady state has a marked sensitivity to
boundary conditions. This dependence however is qualitatively and
quantitatively distinct from the non-Hermitian skin effect, and has no simple
connection to non-Hermitian topology. Further, particle statistics play an
unexpected role: the steady-state density profile is dramatically different for
fermions versus bosons. Our work highlights the key role of fluctuations in
quantum realizations of non-Hermitian dynamics, and provides a starting point
for future work on engineered steady states of open quantum systems.
