The non-triviality of Hilbert space geometries in non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein formalism which transforms the Hilbert spaces of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be transformed into a Hermitian one without altering the physics. Thus, we show how to find a reference frame (corresponding to Einsteins quantum elevator) in which a non-Hermitian system, described by a non-trivial Hilbert space, reduces to a Hermitian system within the standard formalism of quantum mechanics for a Hilbert space.
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