A Hermitian bypass to the non-Hermitian quantum theory

Abstract

Describing systems with non-Hermitian (NH) operators remains a challenge in quantum theory due to instabilities (e.g., exceptional points and decoherence) arising from interactions with the environment. We propose a framework to express the energy states of NH Hamiltonians using a well-defined basis (dub computational basis) derived from a related Hermitian operator. This suitably shifts the singularities from the basis states to the expansion coefficients, allowing for their easier mathematical treatment and parametric control. Furthermore, we introduce a `space-time' transformation on the computational basis that defines a generic dual space map for the energy states. Interestingly, this transformation leads to a symmetry for real/imaginary energy values, revealing the existence of weaker condition than hermiticity or the $\mathcal{PT}$ symmetry. This leads to clearer understanding and novel interpretations of key features like exceptional points, dual space, and weaker symmetry-enforced real eigenvalues. The applicability of our framework extends to various branches of physics where NH operators manifest as ladder operators, order parameters, self-energies, projectors, and other entities.