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arXiv:2110.07805 · Physical Review Research

Quantum Fisher information perspective on sensing in anti-PT symmetric systems

Jiaxuan Wang, Debsuvra Mukhopadhyay, G. S. Agarwal

Published Feb 18, 2022• 19 citations
arXivHTMLPDF

Abstract

The efficient sensing of weak environmental perturbations via special degeneracies called exceptional points in non-Hermitian systems has attracted enormous attention in the last few decades. However, in contrast to the extensive literature on parity-time (PT) symmetric systems, the exotic hallmarks of anti-PT symmetric systems are only beginning to be realized now. Very recently, a characteristic resonance of vanishing linewidth in anti-PT symmetric systems was shown to exhibit tremendous sensitivity to intrinsic nonlinearities. Given the primacy of sensing in non-Hermitian systems, in general, and the immense topicality of anti-PT symmetry, we investigate the statistical bound to the measurement sensitivity for any arbitrary perturbation in a dissipatively coupled, anti-PT symmetric system. Using the framework of quantum Fisher information and the long-time solution to the full master equation, we analytically compute the Cram\'er-Rao bound for the system properties such as the detunings and the couplings. As an illustrative example of this formulation, we inspect and reaffirm the role of a long-lived resonance in dissipatively interacting systems for sensing applications.

DOI
10.1103/physrevresearch.4.013131

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The efficient sensing of weak environmental perturbations via special degeneracies called exceptional points in non-Hermitian systems has attracted enormous attention in the last few decades. However, in contrast to the extensive literature on parity-time (PT) symmetric systems, the exotic hallmarks of anti-PT symmetric systems are only beginning to be realized now. Very recently, a characteristic resonance of vanishing linewidth in anti-PT symmetric systems was shown to exhibit tremendous sensitivity to intrinsic nonlinearities. Given the primacy of sensing in non-Hermitian systems, in general, and the immense topicality of anti-PT symmetry, we investigate the statistical bound to the measurement sensitivity for any arbitrary perturbation in a dissipatively coupled, anti-PT symmetric system. Using the framework of quantum Fisher information and the long-time solution to the full master equation, we analytically compute the Cram'er-Rao bound for the system properties such as the detunings and the couplings. As an illustrative example of this formulation, we inspect and reaffirm the role of a long-lived resonance in dissipatively interacting systems for sensing applications.
In this paper we study exceptional-point (EP) effects and quantum sensing in a parity-time (PT)-symmetric two-qubit system with the Ising-type interaction. We explore EP properties of the system by analyzing degeneracy of energy eigenvalues or entanglement of eigenstates. We investigate entanglement dynamics of the two qubits in detail. In particular, we demonstrate that the system can create the steady-state entanglement in the PT-broken phase and collapse-revival phenomenon of entanglement in the PT-symmetric phase during the long-time evolution. We show that entanglement can be generated more quickly than the corresponding Hermitian system. Finally, we prove that the sensitivity of eigenstate quantum sensing for the parameters exhibits the remarkable enharncement at EPs, and propose a quantum-coherence measurement to witness the existence of EPs.
We investigate the dynamics of a general two-level based anti-parity-time (anti-PT\mathcal{PT}PT)-symmetric qubit and study its decoherence as well as entanglement entropy properties. We compare our findings with that of the corresponding parity-time (PT\mathcal{PT}PT)-symmetric and Hermitian qubits. To begin, we consider the time-dependent Dyson map to find the exact analytical dynamics for a general non-Hermitian qubit system weakly coupled with a thermal bath for pure dephasing, before specializing it to the case of a general anti-PT\mathcal{PT}PT-symmetric qubit. Basing the comparison under the same coupling strength or increasing the non-Hermiticity, we observe that the decoherence function and entanglement entropy of the anti-PT\mathcal{PT}PT-symmetric qubit decays and grows more slowly, respectively, compared to the PT\mathcal{PT}PT-symmetric and Hermitian qubits. Similarly, the corresponding variance and area of Fisher information are much higher compared to the PT\mathcal{PT}PT-symmetric and Hermitian qubits. These results demonstrate that anti-PT\mathcal{PT}PT-symmetric qubits may be better suited for quantum computing and quantum information processing applications than conventional Hermitian or even PT\mathcal{PT}PT-symmetric qubits.
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.
Non-Hermitian systems exhibit great potential for quantum-enhanced sensing, yet a unified information-theoretic framework to quantify their sensing capabilities remains elusive. Here, we investigate quantum Fisher information (QFI) in phase estimation processes governed by non-Hermitian Hamiltonians. In two-dimensional systems, we derive a closed-form formula for QFI using the Bloch representation, which applies to both pure and mixed states, as well as Hermitian and non-Hermitian Hamiltonians. Such a general formula allows us to rigorously establish upper bounds on QFI and to identify the conditions under which these bounds are achievable. By scrutinizing a specific non-Hermitian Hamiltonian with parity-time (PT) symmetry, we demonstrate that the upper bound of QFI is exactly attained in the PT-symmetry regime, at the exceptional point, and in the PT-symmetry-breaking regime. Finally, we extend our analysis to mixed states in higher dimensions, establishing a practical inequality that sets the lower and upper bounds of QFI.
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 the dephasing dynamics of a qubit as an effective mechanism for estimating the temperature of its surrounding environment for different symmetries. Our approach is fundamentally quantum, leveraging the qubit’s susceptibility to decoherence without necessitating thermal equilibrium with the system under study. We also examine how symmetry properties affect the accuracy of information retrieval and the robustness of quantum information storage in such systems, highlighting their potential advantages in mitigating decoherence effects. The optimization of quantum Fisher information is performed with respect to both the interaction duration and the environmental temperature, focusing on Ohmic-like spectral density environments. Furthermore, we explicitly identify the optimal qubit measurement that attains the quantum Cramér–Rao bound for precision. Our findings reveal that optimal estimation arises from a complex interplay between the qubit’s dephasing dynamics and the Ohmic characteristics of the environment, with a particular focus on non-Hermitian systems that exhibit enhanced resilience to decoherence. We examine the decoherence dynamics of a two-level spin system governed by an anti-parity-time-symmetric Hamiltonian. The obtained results are systematically compared with those corresponding to both PT-symmetric and conventional Hermitian qubit models. To this end, we first employ a time-dependent Dyson transformation to derive the exact analytical evolution of a generic non-Hermitian qubit interacting with an external environment. The formalism is then applied to the particular case of an anti-PT-symmetric qubit. Our analysis reveals that the coherence of the non-Hermitian-symmetric qubit is preserved over significantly longer times, as evidenced by the slower decay of its decoherence function relative to its Hermitian counterparts. Furthermore, the generation of entanglement entropy is found to proceed at a reduced rate, indicating a more gradual buildup of system–environment correlations. In addition, the anti-PT-symmetric configuration exhibits substantially enhanced quantum Fisher information together with a smaller variance than those observed in the corresponding PT-symmetric and Hermitian systems. These findings highlight the remarkable robustness of anti-PT-symmetric qubits against environmental decoherence and suggest that they constitute a promising platform for the implementation of quantum information processing and quantum computing protocols.