  The non-Hermitian paradigm of quantum systems displays salient features
drastically different from Hermitian counterparts. In this work, we focus on
one such aspect, the difference of evolving quantum ensembles under
$H_{\mathrm{nh}}$ (right ensemble) versus its Hermitian conjugate,
$H_{\mathrm{nh}}^{\dagger}$ (left ensemble). We propose a formalism that
quantifies the (dis-)similarity of these right and left ensembles, for single-
as well as many-particle quantum properties. Such a comparison gives us a scope
to measure the extent to which non-Hermiticity gets translated from the
Hamiltonian into physically observable properties. We test the formalism in two
cases: First, we construct a non-Hermitian Hamiltonian using a set of imperfect
Bell states, showing that the non-Hermiticity of the Hamiltonian does not
automatically comply with the non-Hermiticity at the level of observables.
Second, we study the interacting Hatano--Nelson model with asymmetric hopping
as a paradigmatic quantum many-body Hamiltonian. Interestingly, we identify
situations where the measures of non-Hermiticity computed for the Hamiltonian,
for single-, and for many-particle quantum properties behave distinctly from
each other. Thus, different notions of non-Hermiticity can become useful in
different physical scenarios. Furthermore, we demonstrate that the measures can
mark the model's Parity--Time (PT) symmetry-breaking transition. Our findings
can be instrumental in unveiling new exotic quantum phases of non-Hermitian
quantum many-body systems as well as in preparing resourceful states for
quantum technologies.
