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Non-Hermitian systems with parity-time ($\mathcal{PT}$) symmetry andanti-$\mathcal{PT}$ symmetry give rise to exceptional points (EPs) withintriguing properties related to, e.g., chiral transport and enhancedsensitivity, due to the coalescence of eigenvectors. In this paper, we proposea powerful and easily computable tool, based on the Hilbert-Schmidt speed(HSS), which does not require the diagonalization of the evolved densitymatrix, to detect exactly the EPs and hence the critical behavior of the(anti-)$\mathcal{PT}\!-$symmetric systems, especially high-dimensional ones.Our theoretical predictions, made without the need for modification of theHilbert space, which is performed by diagonalizing one of the observables, arecompletely consistent with results extracted from recent experiments studyingthe criticality in (anti-)$\mathcal{PT}\!-$symmetric systems. Nevertheless, notmodifying the Hilbert space of the non-Hermitian system, we find that the tracedistance, a measure of distinguishability of two arbitrary quantum states,whose dynamics is known as a faithful witness of non-Markovianity in Hermitiansystems, may be non-contractive under the non-Hermitian evolution of thesystem. Therefore, it lacks one of the most important characteristics whichmust be met by any standard witness of non-Markovianity.

Searching for exceptional points and inspecting non-contractivity of trace distance in (anti-)$\mathcal{PT}\!-$symmetric systems