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Journal of Physics Condensed Matter

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

Elnaz Alizadeh, Hossein Rangani Jahromi, M. Amniat-Talab

Published Dec 9, 2025• 2 citations
Publisher

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.

DOI
10.1088/1361-648x/ae2a8f

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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.
Non-Hermitian systems with parity-time (PT\mathcal{PT}PT) symmetry and anti-PT\mathcal{PT}PT symmetry give rise to exceptional points (EPs) with important properties due to the coalescence of eigenvectors. Because of the limitations on the Hermiticity of the Hamiltonian, these kinds of non-Hermitian systems have recently attracted extensive attention and emerged as a new way to engineer the response of open quantum systems. We propose an efficient and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to characterize non-Markovian effects in (anti-)PT ⁣−\mathcal{PT}\!-PT−symmetric systems. Our proposed witness of non-Markovianity can also detect exactly the EPs and hence the critical behavior of the non-Hermitian systems, especially high-dimensional qudits. Our theoretical predictions are completely consistent with results extracted from recent experiments studying the non-Markovianity and criticality in (anti-)PT ⁣−\mathcal{PT}\!-PT−symmetric systems.
Exceptional points (EPs), indicative of parity-time (PT) symmetry breaking, play a central role in non-Hermitian physics, yet most studies begin from deliberately engineered effective Hamiltonians whose parameters are tuned to exhibit exceptional behavior. In realistic open quantum systems, however, dynamics are governed by Lindblad superoperators whose spectral structure is high-dimensional, symmetry-constrained, and not obviously reducible to minimal non-Hermitian models. A general framework for discovering exceptional dynamics directly from microscopic dissipative models has been lacking. Here we introduce a symmetry-resolved approach for identifying and characterizing exceptional points directly from the full Liouvillian generator. Correlated dissipation induces graph symmetries that decompose Liouville space into low-dimensional invariant sectors, within which minimal non-Hermitian blocks govern the onset of EPs and PT-breaking behavior. We further introduce a numerical diagnostic - the exceptional-point strength E\mathcal{E}E - based on eigenvector conditioning, which quantifies proximity to defective dynamics without requiring analytic reduction. Applied to tight-binding models with correlated dephasing and relaxation, the method reproduces analytically predicted exceptional seams and reveals universal scaling of E\mathcal{E}E near EP manifolds. More broadly, the framework enables systematic discovery of hidden exceptional structure in complex or high-dimensional open systems and is naturally compatible with matrix-free and tensor-network implementations for scalable many-body applications.
Exceptional points (EPs) of a non-Hermitian Hamiltonian with parity-time-reversal (PT) symmetry have the potential to drastically enhance the capabilities of metrology and sensing through their power-law growing sensitivity to external perturbation. With the ability of generating and tuning dissipation in a single trapped ion system, we observe rich dynamics and detailed quantum phase transitions from the PT-symmetric phase to the symmetry-breaking phase. In this single qubit full quantum system, we develop a method to precisely determine the location of EP without any fitting parameter, and extract the eigenvalues in a unified way through all parameter regions. We can also obtain the full density matrix by quantum state tomography. Finally, we suggest from theoretical analysis that the periodically driving PT-symmetric non-Hermitian system can be used to measure the magnitude, frequency, and phase of time-dependent perturbation with EP enhancement.
We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, can be exploited as a powerful and easily computable tool for quantum phase estimation in a nnn-qubit system. We find that, when both the HSS and quantum Fisher information (QFI) are computed with respect to the phase parameter encoded into the initial state of the nnn-qubit register, the zeros of the HSS dynamics are essentially the same as those of the QFI dynamics. Moreover, the positivity (negativity) of the time-derivative of the HSS exactly coincides with the positivity (negativity) of the time-derivative of the QFI. Our results also provide strong evidence for contractivity of the HSS under completely positive and trace preserving maps in high-dimensional systems, as predicted in previous studies.
Exceptional points (EPs), the degeneracy points of non-Hermitian systems, have recently attracted great attention because of their potential of enhancing the sensitivity of quantum sensors. Unlike the usual degeneracies in Hermitian systems, at EPs, both the eigenenergies and eigenvectors coalesce. Although EPs have been widely explored, the range of EPs studied is largely limited by the underlying systems, for instance, higher-order EPs are hard to achieve. Here we propose an extendable method to simulate non-Hermitian systems and study EPs with quantum circuits. The system is inherently parity-time (PT) broken due to the non-symmetric controlling effects of the circuit. Inspired by the quantum Zeno effect, the circuit structure guarantees the success rate of the post-selection. A sample circuit is implemented in a quantum programming framework, and the phase transition at EP is demonstrated. Considering the scalable and flexible nature of quantum circuits, our model is capable of simulating large-scale systems with higher-order EPs.
ABSTRACT Exceptional points (EPs), singularities in non‐Hermitian systems where eigenvalues and eigenstates coalesce, exhibit unique spectral responses and enhanced sensitivity. However, in quantum‐interference settings such as two‐photon measurements, the interference transition generally does not occur at the EPs, limiting their applications for quantum sensing. To address this issue, we develop an anti‐parity‐time‐symmetric model based on indirectly coupled waveguides and introduce a cascaded‐waveguide configuration that produces an effective rotation of the system's eigenbasis. This modification shifts the two‐photon bunching‐antibunching transition to coincide with the EP. Near the EP, a small refractive‐index perturbation leads to a sharp change in visibility, which becomes more pronounced by increasing waveguide length. This passive, gain‐free design may minimize noise and offer a feasible route toward EP‐enhanced quantum phase sensing on integrated platforms.