Detecting critical behavior in systems with PT and anti-PT symmetry using trace speed

Abstract

In this paper, we demonstrate that the trace speed (TS), a statistical measure based on the trace norm of the derivative of the density matrix, serves as a powerful and precise tool for locating exceptional points (EPs) in quantum systems with parity-time (PT) and anti-PT symmetry. This detection of EPs is performed by benchmarking TS against the Hilbert-Schmidt speed (HSS), another efficient statistical measure. Analytical derivations and numerical simulations reveal that the dynamics of TS undergo characteristic changes when traversing EPs, enabling their exact positions to be determined unambiguously. Furthermore, TS can be more readily accessible experimentally than HSS, since it relates directly to trace-norm changes that can be estimated from feasible measurement schemes. Owing to its high sensitivity to small parameter variations and straightforward laboratory implementation, TS is a promising figure of merit for ultra-sensitive quantum sensing and for the optimization of EP-based photonic architectures. These results provide a foundation for exploring more complex non-Hermitian systems and their technological applications.

Keywords: Hilbert–Schmidt speed (HSS); PT-symmetric systems; exceptional points; trace speed (TS).