  Recently, apparent nonphysical implications of non-Hermitian quantum
mechanics (NHQM) have been discussed in the literature. In particular, the
apparent violation of the no-signaling theorem, discrimination of nonorthogonal
states, and the increase of quantum entanglement by local operations were
reported, and therefore NHQM was not considered as a fundamental theory. Here
we show that these and other no-go principles (including the no-cloning and
no-deleting theorems) of conventional quantum mechanics still hold in
finite-dimensional non-Hermitian quantum systems, including parity-time
symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We
have developed a modified formulation of NHQM based on the geometry of Hilbert
spaces which is consistent with the conventional quantum mechanics for
Hermitian systems. Using this formulation the validity of these principles can
be shown in a simple and uniform approach.
