A Hermitian bypass to the non-Hermitian quantum theory

With the emergence of phenomena explainable only through non-Hermiticity and their potential applications in quantum technologies, the demand for a systematic theory of such operators has become unavoidable. In this work, we study a class of non-Hermitian Hamiltonians (H) in which {H, H^†} can be made a dynamical symmetry. By constructing a bilinear Hermitian operator, F= H^†H, we build a computational basis bounded by exceptional points. Expressing the energy eigenstates in this computational basis, we discover that (a) the energy eigenstates meet at the exceptional points when they collapse to any one of the computational basis states, (b) the biorthogonal dual space can be mapped via a discrete “space-time” transformation and (c) the energy eigenstates display an intrinsic entanglement in the computational space. Extending these results to odd-dimension Hilbert spaces, the computational basis leads to an irreducible representation where one eigenstate exhibits parameter-free energy, whereas the others follow the same theory as for even-dimensions.