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APL Photonics

Bloch representation of quantum Fisher information in phase estimations with non-Hermitian Hamiltonians

Zi-Hao Li, Peng-Fei Wei, Zhuo Zhang, Zhucheng Zhang, Wei Zhong, Zhe Sun

Published Oct 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0293530

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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.
Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.
Quantum parameter estimation with Hermitian systems has been applied in various fields, but there are relatively few results concerning non-Hermitian systems. Here, we study the quantum parameter estimation for general non-Hermitian Hamiltonians and derive an intuitive expression of quantum Fisher information (QFI) for pure states. Furthermore, we propose the condition for optimal measurements, which is applicable to both Hermitian and non-Hermitian Hamiltonians. To illustrate these results, we calculate and study the QFI of a specific parity-time (PT)(\mathcal{PT})(PT) -symmetric non-Hermitian Hamiltonian and give the optimal measurement. Surprisingly, we find some interesting properties of this PT\mathcal{PT}PT-symmetric Hamiltonian QFI, such as the mutations in QFI at exceptional point. Moreover, we also compare the variance of estimation generated by the optimal measurement with the theoretical precision bound to verify the condition for optimal measurements we proposed.
Non-Hermitian quantum metrology, an emerging field at the intersection of quantum estimation and non-Hermitian physics, holds promise for revolutionizing precision measurement. Here, we present a comprehensive investigation of non-Hermitian quantum parameter estimation in the quantum regime, with a special focus on achieving Heisenberg scaling. We introduce a concise expression for the quantum Fisher information (QFI) that applies to general non-Hermitian Hamiltonians, enabling the analysis of estimation precision in these systems. Our findings unveil the remarkable potential of non-Hermitian systems to attain the Heisenberg scaling of 1/ t , where t represents time. Moreover, we derive optimal measurement conditions based on the proposed QFI expression, demonstrating the attainment of the quantum Cramér-Rao bound. By constructing non-unitary evolutions governed by two non-Hermitian Hamiltonians, one with parity-time symmetry and the other without specific symmetries, we experimentally validate our theoretical analysis. The experimental results affirm the realization of Heisenberg scaling in estimation precision, marking a substantial milestone in non-Hermitian quantum metrology.
Quantum Fisher information bounds the ultimate precision limit in the parameter estimation and has important applications in quantum metrology. In recent years, the theoretical and experimental studies of non-Hermitian Hamiltonians realized in quantum systems have attracted wide attention. Here, the parameter estimation based on eigenstates of non-Hermitian Hamiltonians is investigated, and the corresponding quantum Fisher information and quantum Cramér-Rao bound for the single-parameter and two-parameter estimations are given. In particular, the quantum Fisher information about estimating intrinsic momentum and external parameters in the non-reciprocal and gain-and-loss Su-Schrieffer-Heeger models, and non-Hermitian versions of the quantum Ising chain, Chern-insulator model and two-level system are calculated and analyzed. For these non-Hermitian models, the results show that in the case of single-parameter estimation in these non-Hermitian models, the quantum Fisher information increases significantly in the gapless regime and near the exceptional points, which can improve the accuracy limit of parameter estimation. For the two-parameter estimation, the determinant of the quantum Fisher information matrix also increases obviously near the gapless and exceptional points. In addition, a higher overall accuracy can be achieved in the topological regime than in the trivial regime, and the topological bound in two-parameter estimation can be determined by the Chern number.
Quantum Fisher Information (QFI) is a fundamental quantity in quantum parameter estimation theory, characterizing the ultimate precision bound of parameter estimation. In this work, we investigate QFI for quantum states in non-Hermitian systems. By employing the projected Hilbert space method and spectral decomposition, we derive an explicit expression for the QFI in terms of the density matrix and parameter generators. This formulation not only recovers the well-known results in the Hermitian case but also captures the non-Hermitian effects induced by the time-dependent norm of the state. To validate our theoretical framework, we analyze a single-qubit pseudo-Hermitian system and apply Naimark dilation theory to embed it into an equivalent Hermitian system. The comparison between the original and dilated systems demonstrates the consistency and applicability of the proposed QFI formula in non-Hermitian settings. In addition, we investigate a PT\mathcal{PT}PT-symmetric system to further explore the influence of non-Hermiticity on QFI. Our findings offer a new perspective for analyzing and enhancing QFI in non-Hermitian systems, paving the way for promising applications in non-Hermitian quantum metrology and sensing.
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.