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

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