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arXiv:2303.05532 · Physical Review Letters

Multiparameter Estimation Perspective on Non-Hermitian Singularity-Enhanced Sensing

Javid Naikoo, Ravindra W. Chhajlany, Jan Kołodyński

Published Nov 29, 2023• 27 citations
arXivHTMLPDF

Abstract

Describing the evolution of quantum systems by means of non-Hermitian generators opens a new avenue to explore the dynamical properties naturally emerging in such a picture, e.g. operation at the so-called exceptional points, preservation of parity-time symmetry, or capitalizing on the singular behavior of the dynamics. In this Letter, we focus on the possibility of achieving unbounded sensitivity when using the system to sense linear perturbations away from a singular point. By combining multiparameter estimation theory of Gaussian quantum systems with the one of singular-matrix perturbations, we introduce the necessary tools to study the ultimate limits on the precision attained by such singularity-tuned sensors. We identify under what conditions and at what rate can the resulting sensitivity indeed diverge, in order to show that nuisance parameters should be generally included in the analysis, as their presence may alter the scaling of the error with the estimated parameter.

DOI
10.1103/physrevlett.131.220801

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Describing the evolution of quantum systems by means of non-Hermitian generators opens a new avenue to explore the dynamical properties naturally emerging in such a picture, e.g. operation at the so-called exceptional points, preservation of parity-time symmetry, or capitalizing on the singular behavior of the dynamics. In this Letter, we focus on the possibility of achieving unbounded sensitivity when using the system to sense linear perturbations away from a singular point. By combining multiparameter estimation theory of Gaussian quantum systems with the one of singular-matrix perturbations, we introduce the necessary tools to study the ultimate limits on the precision attained by such singularity-tuned sensors. We identify under what conditions and at what rate can the resulting sensitivity indeed diverge, in order to show that nuisance parameters should be generally included in the analysis, as their presence may alter the scaling of the error with the estimated parameter.
Open systems possess unique potentials in high-precision sensing, yet the majority of previous studies rely on the spectral singularities known as "exceptional points." Here, we theoretically propose and experimentally demonstrate universal non-Hermitian sensing in the absence of exceptional points. The scheme makes use of the intrinsic sensitivity of a non-Hermitian probe to weak external fields, which can be understood as the direct consequence of non-Hermiticity. We confirm the basic mechanism by simulating the sensor-field dynamics using photon interferometry, and, as a concrete example, demonstrate the enhanced sensing of signals encoded in the setting angle of a wave plate. While the sensitivity of the probe is ultimately limited by the measurement noise, we find the non-Hermitian sensor showing superior performance under background noises that cannot be suppressed through repetitive measurements. Our experiment opens the avenue of enhanced sensing without exceptional points, complementing existing efforts aimed at harnessing the unique features of open systems.
Critical quantum metrology relies on the extreme sensitivity of a system's eigenstates near the critical point of a quantum phase transition to Hamiltonian perturbations. This means that these eigenstates are extremely sensitive to all the parameters of the Hamiltonian. In realistic settings, there is always some degree of uncertainty in the control parameters used to tune the system to criticality. These uncertainties, while not the target of estimation, can significantly affect the attainable precision, effectively acting as nuisance parameters in the estimation process. Despite being a practically relevant source of noise, their impact on critical metrology has been largely overlooked. In this work we present a general framework that interpolates between single- and multiparameter estimation settings, enabling a systematic analysis of how such uncertainties influence sensitivity. We apply this framework to the paradigmatic transverse field Ising and Lipkin-Meshkov-Glick models, explicitly demonstrating how uncertainty in control parameters affects the metrological performance of critical sensors. For finite-size systems, we identify a fundamental trade-off between robustness to uncertainty and the ability to retain a quantum advantage at the critical point. Our results contribute to a deeper understanding of the practical limitations of critical quantum metrology and provide a route toward its more resilient implementation.
Non-Hermitian many-body systems can be spectrally unstable, so small perturbations may induce large eigenvalue shifts. The pseudospectrum quantifies this instability and provides a perturbation-robust diagnostic. For inverse-polynomially small εεε, we show that deciding whether a point z∈Cz\in\mathbb{C}z∈C is εεε-close to the spectrum is PSPACE-hard for 555-local operators, whereas deciding whether zzz lies in the εεε-pseudospectrum is QMA-complete for 444-local operators. This identifies pseudospectrum membership as a more realistic computational target. We then present a concrete quantum framework for deciding pseudospectrum membership, which combines a singular-value estimation step with a dissipative state preparation algorithm. Our Quantum Singular-value Gaussian-filtered Search (QSIGS) combines quantum singular value transformation (QSVT) with classical post-processing to achieve Heisenberg-limited query scaling for singular-value estimation. To prepare suitable input states, we introduce an algorithmic Lindbladian protocol for approximate ground right singular vectors and prove its effectiveness for the Hatano--Nelson model. Finally, we demonstrate the full pipeline on a trapped-ion quantum computer and distinguish points inside and outside the target pseudospectrum near the exceptional point of a minimal non-Hermitian qubit model.
Abstract The exceptional points (EPs) of non-Hermitian systems, where n different energy eigenstates merge into an identical one, have many intriguing properties that have no counterparts in Hermitian systems. In particular, the ϵ 1 / n dependence of the energy level splitting on a perturbative parameter ϵ near an n th order EP stimulates the idea of metrology with arbitrarily high sensitivity, since the susceptibility d ϵ 1/ n /d ϵ diverges at the EP. Here we theoretically study the sensitivity of parameter estimation near the EPs, using the exact formalism of quantum Fisher information (QFI). The QFI formalism allows the highest sensitivity to be determined without specifying a specific measurement approach. We find that the EP bears no dramatic enhancement of the sensitivity. Instead, the coalescence of the eigenstates exactly counteracts the eigenvalue susceptibility divergence and makes the sensitivity a smooth function of the perturbative parameter.
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Quantum metrology leverages the unique properties of quantum systems to enhance parameter estimation beyond classical limits. In this short review, we introduce the core concepts of quantum metrology, with a particular focus on its extension to multi-parameter estimation. We begin by outlining the foundational principles of quantum metrology and explore how quantum criticality can enhance measurement precision — especially near quantum phase transitions, where systems exhibit heightened sensitivity to external perturbations. Next, we consider the framework of multi-parameter quantum critical metrology, in which the simultaneous estimation of multiple parameters near criticality can lead to further improvements in accuracy. Finally, we discuss measurement-induced phase transitions, a phenomenon emerging in the context of continuous quantum measurements that introduces non-Hermitian dynamics and fundamentally alters the system’s behavior. This review provides a concise overview of these key developments and highlights their significance for future progress in quantum sensing technologies.