Searching for exceptional points and inspecting non-contractivity of trace distance in (anti-) PT -symmetric systems

Non-Hermitian systems with parity-time (PT) symmetry and anti-PT symmetry lead to exceptional points (EPs) with intriguing properties related to, e.g., chiral transport and enhanced sensitivity, due to the coalescence of eigenvectors. In this paper, we propose an easily computable tool, based on the Hilbert–Schmidt speed (HSS), not requiring the diagonalization of the evolved density matrix, to detect exactly the EPs, especially in high-dimensional systems. Our theoretical predictions, made without the need for modification of the Hilbert space, are completely consistent with results extracted from recent experiments studying the criticality in (anti-)PT-symmetric systems. Moreover, not modifying the Hilbert space of the non-Hermitian system, we find that the trace distance whose dynamics is known as a faithful witness of non-Markovianity, may be non-contractive under the non-Hermitian evolution of the system. Therefore, it losses one of the most important characteristics which must be met by any standard witness of non-Markovianity. We also address the non-contractivity of quantum Fisher information in non-Hermitian systems.