  Exceptional points (EPs) correspond to degeneracies of open systems. These
are attracting much interest in optics, optoelectronics, plasmonics, and
condensed matter physics. In the classical and semiclassical approaches,
Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian
Hamiltonians such that at least two eigenfrequencies are identical and the
corresponding eigenstates coalesce. HEPs result from continuous, mostly slow,
nonunitary evolution without quantum jumps. Clearly, quantum jumps should be
included in a fully quantum approach to make it equivalent to, e.g., the
Lindblad master-equation approach. Thus, we suggest to define EPs via
degeneracies of a Liouvillian superoperator (including the full Lindbladian
term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two
main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must
have essentially different properties. Theorem 2 dictates a condition under
which, in the ``semiclassical'' limit, LEPs and HEPs recover the same
properties. In particular, we show the validity of Theorem 1 studying systems
which have (1) an LEP but no HEPs, and (2) both LEPs and HEPs but for shifted
parameters. As for Theorem 2, (3) we show that these two types of EPs become
essentially equivalent in the semiclassical limit. We introduce a series of
mathematical techniques to unveil analogies and differences between the HEPs
and LEPs. We analytically compare LEPs and HEPs for some quantum and
semiclassical prototype models with loss and gain.
