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REVIEW 4 major objections 4 minor 59 references

Non-Hermitian effects on the quantum parameter estimation in pseudo-Hermitian systems

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A non-Hermitian system's changing norm contributes a positive term to the quantum Fisher information, and the formula reduces to the Hermitian one when the norm is fixed.

desk verdict The central formula is wrong: Eq. (19)'s norm-velocity term carries an artifact 16, and the paper's main quantitative claims do not hold as derived. read the letter →

arxiv 2505.19079 v1 pith:DNL423KY submitted 2025-05-25 quant-ph

L. H. Wei , H. J. Xing , L. B. Fu , H. D. Liu This is my paper · ORCID
classification quant-ph MSC 81P5081Q12 PACS 03.65.Ta03.65.-w
keywords quantumFisherinformationnon-Hermitiansystemspseudo-HermitianPTsymmetryparameterestimationNaimarkdilationprojectedHilbertspacemetrology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out an explicit formula for the quantum Fisher information (QFI) of states in non-Hermitian systems, where the state norm changes with the estimated parameter. Using a projected Hilbert space and an operator $L$ defined by $\partial_\theta \rho = \tfrac{1}{2}(L\rho + \rho L^{\dagger})$, it argues that the QFI is $F = \langle L^{\dagger} L \rangle$, which for pure states becomes $F = 16e^{2\alpha}(\partial_\theta \alpha)^2 + 4e^{2\alpha}[\langle\partial_\theta\psi|\partial_\theta\psi\rangle - |\langle\partial_\theta\psi|\psi\rangle|^2]$. The first term is new: it is the information carried by the time-dependent normalization $e^{2\alpha}$. When $\alpha=0$ the formula reduces to the standard Hermitian QFI, so the framework is an extension rather than a replacement. Concrete single-qubit pseudo-Hermitian and PT-symmetric examples show how the norm factor can enhance or modulate the achievable estimation precision.

What carries the argument

The load-bearing object is the projected-Hilbert-space decomposition $|\Psi_\theta\rangle = e^{\alpha + i\beta}|\psi_\theta\rangle$ with normalized $|\psi_\theta\rangle$, together with the non-Hermitian symmetric logarithmic derivative $L$ defined by $\partial_\theta \rho = \tfrac{1}{2}(L\rho + \rho L^{\dagger})$. The QFI is identified with $\langle L^{\dagger} L \rangle$, and the decomposition converts $\partial_\theta \rho$ into terms involving $\partial_\theta \alpha$ and $\partial_\theta|\psi\rangle$, producing the norm-velocity term. The same machinery gives mixed-state formulas through spectral decomposition, and Naimark dilation is used as a validation tool: embedding the pseudo-Hermitian qubit into an enlarged Hermitian system should preserve or increase the available parameter information.

What would settle it

Take the single-qubit pseudo-Hermitian Hamiltonian of Eq. (31), prepare the right eigenstate $|R_n\rangle$ with $n\neq 1/\sqrt{1+\delta_\lambda^2}$, and perform the optimal two-outcome measurement on the unnormalized state. If the minimal attainable variance does not saturate $1/F_x(n)$ from Eq. (41), the identification $F=\langle L^{\dagger}L\rangle$ is not the true Cramér-Rao bound for unnormalized states.

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Extended reading notes

Core claim

The central claim is that in a non-Hermitian system the single-parameter estimation precision is governed by $F = \langle L^{\dagger} L \rangle$ with $L$ fixed by $\partial_\theta \rho = \tfrac{1}{2}(L\rho + \rho L^{\dagger})$, and that this quantity separates into a conventional projective-state contribution plus a positive norm-velocity term $16e^{2\alpha}(\partial_\theta \alpha)^2$. The paper derives explicit pure- and mixed-state forms, Eqs. (19) and (27)-(30), shows they reduce to the Hermitian QFI for $\alpha=0$, and argues that the norm factor itself carries parameter information that can be exploited. In the pseudo-Hermitian qubit example, the dilated Hermitian system gives a higher QFI than the normalized projective state, and in the PT-symmetric example the unbroken region gives oscillatory QFI while the broken region gives exponential growth or decay; the optimal initial states are identified as $m=\pm 1$, $\phi=\pi$.

Load-bearing premise

The whole derivation rests on identifying the non-Hermitian QFI with $\langle L^{\dagger}L\rangle$ for unnormalized states; if the true precision limit for such states is different, the norm-velocity term $16e^{2\alpha}(\partial_\theta\alpha)^2$ would not be part of the ultimate bound.

Editorial extensions

If this is right

  • For pure states the QFI is the sum of the usual Hermitian term and $16e^{2\alpha}(\partial_\theta \alpha)^2$, so a parameter-dependent norm always adds nonnegative information.
  • The mixed-state formulas (27)-(30) generalize the standard QFI and reduce to it at $\alpha=0$.
  • In the single-qubit pseudo-Hermitian example, the QFI depends on the arbitrary normalization $n$ of the right eigenstate, and the Naimark-dilated Hermitian system yields a higher QFI than the normalized projective state.
  • In PT-symmetric two-level systems the optimal initial state is $m=\pm 1$, $\phi=\pi$; QFI oscillates in the unbroken region and grows or decays exponentially in the broken region, favouring the unbroken region for estimation.
  • At an exceptional point the formula gives $F=0$ for an eigenstate, but the authors note that the degenerate eigenstates do not form a complete basis, so the zero value may not be physically meaningful.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If $F=\langle L^{\dagger}L\rangle$ is the operative precision bound, then gain and loss rates for a parameter are themselves metrological resources: a small parameter change can be encoded into the exponent $\alpha$ and read out through the state norm, a route the paper only sketches.
  • The comparison with Naimark dilation suggests a general recipe: for any pseudo-Hermitian sensor, the dilated Hermitian system is the fair benchmark, and the non-Hermitian QFI should be compared with the dilated QFI rather than with a naive Hermitian formula.
  • The derivation does not settle the operational meaning of measurements on unnormalized states; connecting Eq. (19) to a concrete measurement protocol with realistic postselection would make the norm-velocity term directly testable.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a quantum Fisher information (QFI) for non-Hermitian systems by defining F=⟨L†L⟩ through an SLD-type equation ∂θρ=1/2(Lρ+ρL†), and uses the projective Hilbert space decomposition |Ψ⟩=e^{α+iβ}|ψ⟩ to separate the norm factor e^{2α} from the normalized state |ψ⟩. It derives explicit pure- and mixed-state formulas, Eq. (19) and Eqs. (25)-(30), and applies them to a single-qubit pseudo-Hermitian system with Naimark dilation as well as to a PT-symmetric Hamiltonian. The paper claims that the norm-velocity term 16e^{2α}(∂θα)^2 represents a genuine non-Hermitian enhancement of precision, while reducing to the standard Hermitian QFI when α=0.

Significance. The topic is timely, and the paper contains concrete examples, including a comparison with a Naimark-dilated Hermitian system, that could be valuable if the formalism were correct. However, the central derivation is internally inconsistent: Eq. (18a) is not a consequence of the defining SLD equation, and the same problem reappears in the mixed-state formulas. Because all later quantitative claims inherit this error, the claimed norm-dependent enhancement is not established. The paper therefore cannot be recommended for publication in its current form.

major comments (4)
  1. [§II, Eqs. (18a)-(19)] Equation (18a) is inconsistent with the defining equation (2) when k lies in the support of ρ. For ρ=e^{2α}|ψ⟩⟨ψ| with ⟨ψ|ψ⟩=1, the matrix element of Eq. (2) gives Re⟨ψ|L|ψ⟩=2∂θα, whereas Eq. (18a) evaluated at k=|ψ⟩ would require ⟨ψ|L|ψ⟩=4∂θα+2⟨ψ|∂θψ⟩. Since ⟨ψ|∂θψ⟩ is purely imaginary, the real parts disagree unless ∂θα=0. Thus Eq. (18a) is valid only for k orthogonal to |ψ⟩, and the completeness sum leading to Eq. (19) misweights the k=|ψ⟩ term. Moreover, Eq. (2) leaves Im⟨ψ|L|ψ⟩ unconstrained, so F=⟨L†L⟩ is not uniquely fixed by the SLD equation. For example, imposing L=L† gives F=4e^{2α}[(∂θα)^2+⟨∂θψ|∂θψ⟩−|⟨∂θψ|ψ⟩|^2], not Eq. (19).
  2. [§II, Eq. (15)] The identification of the QFI with F=⟨L†L⟩ is assumed rather than derived from an optimal-measurement bound for unnormalized states. The standard Braunstein-Caves argument applies to normalized density matrices; when trρ≠1 the Cramér-Rao bound and the optimization over measurements require separate justification. Since every later formula rests on this identification, the absence of such a derivation is a load-bearing gap.
  3. [§II, Eqs. (24a)-(27)] Equation (24a) is also inconsistent with Eq. (2). For ρ=e^{2α}∑_i p_i|i⟩⟨i|, the correct matrix element is ⟨l|∂θρ|k⟩=(e^{2α}/2)(p_k⟨l|L|k⟩+p_l⟨l|L†|k⟩). Equation (24a) omits the second term, and as a result Eqs. (25)-(27) are not derived from the SLD equation; the rank and summation structure of those formulas is therefore ambiguous and unsupported.
  4. [§III, Eq. (41) and Fig. 1] The dependence of the QFI on the arbitrary parameter n in Eq. (41) is a normalization artifact. Since |R_n⟩=n√(1+δ_λ^2)|ψ1⟩, changing n only rescales the density matrix by a constant factor. For a physical state, the QFI should be invariant under this rescaling (or should be computed after normalizing trρ=1). The claimed enhancement with increasing n in Fig. 1 is therefore a consequence of the scale dependence of F=⟨L†L⟩ for unnormalized states, not a physical resource.
minor comments (4)
  1. [§II, Eq. (3)] Equation (3) does not reproduce the standard Hermitian pure-state QFI: the correct expression is 4(⟨∂θψ|∂θψ⟩−|⟨∂θψ|ψ⟩|^2), with a minus sign and an absolute value. The plus sign and omitted absolute value are likely typos, but they are confusing in a paper whose central object is the QFI.
  2. [§IV, Eqs. (51)-(53)] The notation in the PT-symmetric section is inconsistent: the eigenstate parameter x is defined by sin x=(r/s)sinω, while Eq. (51) introduces a combined phase φ, and Eq. (53) then contains terms such as sin2α and cos2α that are not defined in that context. This makes the PT-symmetric formulas very hard to verify.
  3. [§III, Eq. (43)] The sentence defining |ψ1⟩ as 1/√2(|R⟩_n|0⟩) is dimensionally inconsistent: |ψ1⟩ is a two-component vector, while the right-hand side uses a tensor product with an ancilla state. This should be clarified or corrected.
  4. [Throughout] There are several typographical issues, including 'Schwartz inequality' for 'Schwarz inequality' in §II and 'accouting' for 'accounting' in §IV. These should be corrected in a revision.

Circularity Check

1 steps flagged · score 7.0 of 10

The norm-velocity term in Eq. (19) is not derived from the SLD equation; it is built into the definition F=⟨L†L⟩ and the asymmetric SLD choice in Eq. (18a).

  1. self definitional [Section II, Eqs. (15), (18a), (19)]
    "Consequently, the QFI can be expressed as Fθ=⟨L†L⟩. ... ⟨k|∂θρθ|ψθ⟩= e2α/2 ⟨kθ|L|ψθ⟩ ... F= ... =16e2α(∂θα)2+4e2α[⟨∂θψθ|∂θψθ⟩−|⟨∂θψθ|ψθ⟩|2]."

    The central non-Hermitian effect is obtained from the definition F=⟨L†L⟩ together with a chosen L, not from Eq. (2). For ρ=e^{2α}|ψ⟩⟨ψ|, Eq. (2) only fixes Re⟨ψ|L|ψ⟩=2∂θα; the imaginary part, and hence ⟨L†L⟩, is free. Eq. (18a) is not a corollary of Eq. (2): at k=|ψ⟩ it demands ⟨ψ|L|ψ⟩=4∂θα, whereas Eq. (2) gives Re⟨ψ|L|ψ⟩=2∂θα. Accepting Eq. (18a) as the effective definition of L is exactly the statement ∂θρ=(1/2)Lρ on the support, and then F=e^{2α}∥L|ψ⟩∥² evaluates to 16e^{2α}(∂θα)²+4e^{2α}[⟨∂θψ|∂θψ⟩−|⟨∂θψ|ψ⟩|²]. Thus Eq. (19) is the definition of L and F rewritten, not a prediction forced by the SLD equation; the later enhancement claims (Eqs. (41), (53), (56) and Figs. 1-4) inherit this built-in norm term.

full rationale

The paper is not circular in the usual self-citation sense: Ref. [51] is a coauthor citation for the projective-space decomposition, but it is paired with an independent reference [52], and no load-bearing argument reduces to a self-citation chain. The Hermitian limit α=0 and the Naimark-dilation comparison are independent checks that give the paper some external anchor. However, the central derivation is self-definitional. Eq. (15) simply asserts Fθ=⟨L†L⟩ for non-Hermitian systems, and Eq. (18a) then selects an L that is inconsistent with the stated SLD equation (2): for k=|ψ⟩, Eq. (2) fixes Re⟨ψ|L|ψ⟩=2∂θα, while Eq. (18a) gives ⟨ψ|L|ψ⟩=4∂θα. Substituting that chosen L into the definition F=⟨L†L⟩ produces exactly the claimed norm-velocity term 16e^{2α}(∂θα)^2. The apparent non-Hermitian enhancement is therefore not an independently derived consequence of the dynamics or of an optimal-measurement bound; it is the definition of F and the choice of L restated as a result. The later optimization over the arbitrary normalization n in Eq. (41) and Fig. 1 is a visible symptom of the same built-in effect, since a physical QFI should be independent of renormalizing the state. These considerations justify a score of 7: the central claim reduces by construction, even though the paper also contains independent limiting checks and is not driven by self-citation.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivation relies mainly on a new definition of QFI for non-Hermitian systems (an ad hoc assumption), the projective Hilbert space decomposition, and the assumption that probabilities are conserved. The only hand-chosen numbers are the normalization n and the PT initial-state parameters m and φ, which are optimized rather than predicted.

free parameters (3)
  • n = 1/2, √2/2, 1 in Fig. 1
    QFI F_x(n) in Eq. (41) depends on the arbitrary normalization of the right eigenstate |R⟩_n; no principle fixes n, so it acts as a free parameter.
  • m = m = ±1
    Superposition amplitude in the PT initial state |ψ0⟩ = N(|ε+⟩ + m e^{iφ}|ε−⟩); optimized to maximize QFI.
  • φ = φ = π
    Relative phase in the PT initial state; optimized to maximize QFI.
assumptions (4)
  • ad hoc to paper The non-Hermitian QFI is defined as Fθ = ⟨L†L⟩, with L solving ∂θρ = 1/2(Lρ + ρL†).
    Introduced in Sec. II (Eqs. 13-15) and used throughout; not derived from an optimal measurement bound for unnormalized states.
  • domain assumption Projective decomposition |Ψ⟩ = e^{α+iβ}|ψ⟩ with ⟨ψ|ψ⟩=1, and the evolution equations for α and β in Eq. (17).
    The pure-state QFI formula depends on this decomposition; the evolution equations are quoted without derivation.
  • domain assumption The probabilities p_i in the spectral decomposition are conserved under non-Hermitian evolution.
    Stated after Eq. (23) and used to obtain Eq. (28), dropping the ∂p_k terms.
  • domain assumption The Naimark dilation Hamiltonian H_tot maps to the pseudo-Hermitian H_s via M(H_tot)=H_s(δλ), with the specific coefficients b and c in Sec. III.
    The before-and-after dilation comparison relies on this mapping being physically equivalent.

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Cite this review

Pith. "Pith review of Non-Hermitian effects on the quantum parameter estimation in pseudo-Hermitian systems." pith.science (2026). https://pith.science/paper/DNL423KY

@misc{pith2026250519079,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian effects on the quantum parameter estimation in pseudo-Hermitian systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNL423KY}},
  note         = {Machine review of arXiv:2505.19079}
}
abstract

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 $\mathcal{PT}$-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.

Figures

Figures reproduced from arXiv: 2505.19079 by the authors.

Figure 1
Figure 1. FIG. 1: QFI [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Evolution of the QFI [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Evolution of the QFI [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The green dashed line represents the variation of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Works this paper leans on

59 extracted references · 55 canonical work pages

  1. [53]

    Quantum parameter estima- tion of non-hermitian systems with optimal measurements,

    Xinglei Yu and Chengjie Zhang, “Quantum parameter estima- tion of non-hermitian systems with optimal measurements,” Phys. Rev. A108, 022215 (2023)

    work page 2023

  2. [1]

    Quantum imaging beyond the diffraction limit by optical centroid measurements,

    Mankei Tsang, “Quantum imaging beyond the diffraction limit by optical centroid measurements,” Phys. Rev. Lett.102, 253601 (2009)

    work page 2009

  3. [2]

    Quantum projection noise in an atomic fountain: A high stability cesium frequency standard,

    G. Santarelli, Ph. Laurent, P. Lemonde, A. Clairon, A. G. Mann, S. Chang, A. N. Luiten, and C. Salomon, “Quantum projection noise in an atomic fountain: A high stability cesium frequency standard,” Phys. Rev. Lett.82, 4619–4622 (1999)

    work page 1999

  4. [3]

    Gravitational radiation detection with laser interferometry,

    Rana X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys.86, 121–151 (2014)

    work page 2014

  5. [4]

    Quantifying non-markovianity via lo- cal quantum fisher information,

    Yassine Dakir, Abdallah Slaoui, Lalla Btissam Drissi, and 9 Rachid Ahl Laamara, “Quantifying non-markovianity via lo- cal quantum fisher information,” (2024), arXiv:2409.10163 [quant-ph]

    arXiv 2024

  6. [5]

    Quantum illumi- nation strategy for parameter estimation,

    L. Davidovich and R. L. de Matos Filho, “Quantum illumi- nation strategy for parameter estimation,” Phys. Rev. A108, 042612 (2023)

    work page 2023

  7. [6]

    Quantum parameter estimation with optimal control,

    Jing Liu and Haidong Yuan, “Quantum parameter estimation with optimal control,” Phys. Rev. A96, 012117 (2017)

    work page 2017

  8. [7]

    PT-symmetric quantum sensing: Advantages and restric- tions,

    Yan-Yi Wang, Chun-Wang Wu, Wei Wu, and Ping-Xing Chen, “PT-symmetric quantum sensing: Advantages and restric- tions,” Phys. Rev. A109, 062611 (2024)

    work page 2024

Show all 59 references
  1. [8]

    Quantum metrology in open systems: Dissipative cram ´er-rao bound,

    S. Alipour, M. Mehboudi, and A. T. Rezakhani, “Quantum metrology in open systems: Dissipative cram ´er-rao bound,” Phys. Rev. Lett.112, 120405 (2014)

    2014

  2. [9]

    Ultimate precision bound of quantum and subwavelength imaging,

    Cosmo Lupo and Stefano Pirandola, “Ultimate precision bound of quantum and subwavelength imaging,” Phys. Rev. Lett.117, 190802 (2016)

    2016

  3. [10]

    Quantum limits for pre- cisely estimating the orientation and wobble of dipole emitters,

    Oumeng Zhang and Matthew Lew, “Quantum limits for pre- cisely estimating the orientation and wobble of dipole emitters,” (2020)

    2020

  4. [11]

    Quantum illumination with gaussian states,

    Si-Hui Tan, Baris I. Erkmen, Vittorio Giovannetti, Saikat Guha, Seth Lloyd, Lorenzo Maccone, Stefano Pirandola, and Jef- frey H. Shapiro, “Quantum illumination with gaussian states,” Phys. Rev. Lett.101, 253601 (2008)

    2008

  5. [12]

    Quantum in- terferometric metrology with entangled photons,

    Yuanyuan Chen, Ling Hong, and Lixiang Chen, “Quantum in- terferometric metrology with entangled photons,” Frontiers in Physics10(2022), 10.3389/fphy.2022.892519

    2022

  6. [13]

    Parameter estimation with mixed-state quantum computation,

    Rolando Somma and Sergio Boixo, “Parameter estimation with mixed-state quantum computation,” Physical Review A77 (2007), 10.1103/PhysRevA.77.052320

    2007 doi

  7. [14]

    Quantum phase estimation algorithm with gaussian spin states,

    Luca Pezz `e and Augusto Smerzi, “Quantum phase estimation algorithm with gaussian spin states,” PRX Quantum2(2021), 10.1103/PRXQuantum.2.040301

    2021 doi

  8. [15]

    Quantum spectral clustering algorithm for unsuper- vised learning,

    Qingyu Li, Yuhan Huang, Shan Jin, Xiaokai Hou, and Xiaot- ing Wang, “Quantum spectral clustering algorithm for unsuper- vised learning,” (2022)

    2022

  9. [16]

    Quantum bayesian optimization,

    Zhongxiang Dai, Gregory Kang Ruey Lau, Arun Verma, Yao Shu, Bryan Kian Hsiang Low, and Patrick Jaillet, “Quantum bayesian optimization,” (2023), arXiv:2310.05373 [cs.LG]

    2023 arXiv

  10. [17]

    HARALD CRAM ´ER,Mathematical Methods of Statistics (PMS-9)(Princeton University Press, 1999)

    1999

  11. [18]

    Information and the accuracy attainable in the estimation of statistical parameters,

    C. Radhakrishna Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” inBreakthroughs in Statistics: F oundations and Basic Theory, edited by Samuel Kotz and Norman L. Johnson (Springer New York, New York, NY , 1992) pp. 235–247

    1992

  12. [19]

    Quantum detection and estimation theory,

    Carl W. Helstrom, “Quantum detection and estimation theory,” Journal of Statistical Physics1, 231–252 (1969)

    1969

  13. [20]

    Statistical dis- tance and the geometry of quantum states,

    Samuel L. Braunstein and Carlton M. Caves, “Statistical dis- tance and the geometry of quantum states,” Phys. Rev. Lett.72, 3439–3443 (1994)

    1994

  14. [21]

    Quantum fisher information for density matrices with arbitrary ranks,

    Jing Liu, Xiao-Xing Jing, Wei Zhong, and Xiao-Guang Wang, “Quantum fisher information for density matrices with arbitrary ranks,” Communications in Theoretical Physics61, 45 (2014)

    2014

  15. [22]

    Classical fisher information in non- linear interferometry,

    Russell J. Stevensonet al., “Classical fisher information in non- linear interferometry,” Physical Review A98, 053832 (2018)

    2018

  16. [23]

    Prethermalization in the hubbard model under interaction imbalance,

    Takahiro Mori et al., “Prethermalization in the hubbard model under interaction imbalance,” Phys. Rev. B100, 245127 (2019)

    2019

  17. [24]

    Dynamical quantum phase transitions: A classi- fication scheme,

    Markus Heyl, “Dynamical quantum phase transitions: A classi- fication scheme,” Phys. Rev. B103, L180302 (2021)

    2021

  18. [25]

    Nonequilibrium quantum criticality in open systems,

    F. Dalla Torre et al., “Nonequilibrium quantum criticality in open systems,” Phys. Rev. B101, 085114 (2020)

    2020

  19. [26]

    Exploring nonequilibrium dynamics of bose-hubbard lattices with machine learning,

    Giuseppe Carleo et al., “Exploring nonequilibrium dynamics of bose-hubbard lattices with machine learning,” Phys. Rev. A 102, 033619 (2020)

    2020

  20. [27]

    Reconstructing nonequilibrium regimes of quantum many-body systems,

    Gabriele Perfetto et al., “Reconstructing nonequilibrium regimes of quantum many-body systems,” Phys. Rev. X9, 041008 (2019)

    2019

  21. [28]

    Designing nonequilibrium states of quantum matter through stochastic resetting,

    Igor Lesanovsky et al., “Designing nonequilibrium states of quantum matter through stochastic resetting,” Phys. Rev. B104, L180302 (2021)

    2021

  22. [29]

    Nonequilibrium spin dynamics in disordered many-body systems,

    S. Rout et al., “Nonequilibrium spin dynamics in disordered many-body systems,” Phys. Rev. B102, 045112 (2020)

    2020

  23. [30]

    Observation of parity- time symmetry breaking transitions in a dissipative floquet sys- tem of ultracold atoms,

    Jiaming Li, Andrew K. Harter, Ji Liu, Leonardo de Melo, Yogesh N. Joglekar, and Le Luo, “Observation of parity- time symmetry breaking transitions in a dissipative floquet sys- tem of ultracold atoms,” Nature Communications10(2019), 10.1038/s41467-019-08596-1

    2019 doi

  24. [31]

    Observa- tion of parity-time symmetry breaking in a single-spin system,

    Yang Wu, Wenquan Liu, Jianpei Geng, Xingrui Song, Xiangyu Ye, Chang-Kui Duan, Xing Rong, and Jiangfeng Du, “Observa- tion of parity-time symmetry breaking in a single-spin system,” Science364, 878–880 (2019)

    2019

  25. [32]

    Quantum state tomography across the exceptional point in a single dissipative qubit,

    M. Naghiloo, M. Abbasi, Yogesh N. Joglekar, and K. W. Murch, “Quantum state tomography across the exceptional point in a single dissipative qubit,” Nature Physics15, 1232–1236 (2019)

    2019

  26. [33]

    Observation of non-hermitian topological ander- son insulator in quantum dynamics,

    Quan Lin, Tianyu Li, Lei Xiao, Kunkun Wang, Wei Yi, and Peng Xue, “Observation of non-hermitian topological ander- son insulator in quantum dynamics,” Nature Communications 13(2022), 10.1038/s41467-022-30938-9

    2022 doi

  27. [34]

    Non-hermitian bulk–boundary correspondence in quantum dynamics,

    Lei Xiao, Tianshu Deng, Kunkun Wang, Gaoyan Zhu, Zhong Wang, Wei Yi, and Peng Xue, “Non-hermitian bulk–boundary correspondence in quantum dynamics,” Nature Physics16, 761–766 (2020)

    2020

  28. [35]

    Time-dependent theory of non- hermitian schr ¨odinger equation: application to multiphoton- induced ionisation decay of atoms,

    F H M Faisal and J V Moloney, “Time-dependent theory of non- hermitian schr ¨odinger equation: application to multiphoton- induced ionisation decay of atoms,” Journal of Physics B: Atomic and Molecular Physics16, 3109 (1983)

    1983

  29. [36]

    Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian,

    Ali Mostafazadeh, “Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian,” Journal of Mathematical Physics 43, 205–214 (2002)

    2002

  30. [37]

    Physical aspects of pseudo-hermitian andpt-symmetric quantum mechanics,

    Ali Mostafazadeh and Ahmet Batal, “Physical aspects of pseudo-hermitian andpt-symmetric quantum mechanics,” Jour- nal of Physics A: Mathematical and General37, 11645–11679 (2004)

    2004

  31. [38]

    Pseudo-hermiticity, weak pseudo- hermiticity andη-orthogonality condition,

    B Bagchi and C Quesne, “Pseudo-hermiticity, weak pseudo- hermiticity andη-orthogonality condition,” Physics Letters A 301, 173–176 (2002)

    2002

  32. [39]

    Pseudo-hermiticity for a class of nondiag- onalizable hamiltonians,

    Ali Mostafazadeh, “Pseudo-hermiticity for a class of nondiag- onalizable hamiltonians,” Journal of Mathematical Physics43, 6343–6352 (2002)

    2002

  33. [40]

    Pseudounitary operators and pseudouni- tary quantum dynamics,

    Ali Mostafazadeh, “Pseudounitary operators and pseudouni- tary quantum dynamics,” Journal of Mathematical Physics45, 932–946 (2004)

    2004

  34. [41]

    Relations between quantum metrology and criticality in general su(1,1) systems,

    Rui Zhang, Wenkui Ding, Zhucheng Zhang, Lei Shao, Yuyu Zhang, and Xiaoguang Wang, “Relations between quantum metrology and criticality in general su(1,1) systems,” Phys. Rev. A110, 012413 (2024)

    2024

  35. [42]

    Quantum dynamic response-based nv- diamond magnetometry: Robustness to decoherence and appli- cations in motion detection of magnetic nanoparticles,

    Wenkui Ding, Xingyu Zhang, Jing Liu, Aixi Chen, and Xiaoguang Wang, “Quantum dynamic response-based nv- diamond magnetometry: Robustness to decoherence and appli- cations in motion detection of magnetic nanoparticles,” Phys. Rev. B110, 045202 (2024)

    2024

  36. [43]

    Fundamen- tal sensitivity limits for non-hermitian quantum sensors,

    Wenkui Ding, Xiaoguang Wang, and Shu Chen, “Fundamen- tal sensitivity limits for non-hermitian quantum sensors,” Phys. Rev. Lett.131, 160801 (2023)

    2023

  37. [44]

    Experimental estimation of the quantum 10 fisher information from randomized measurements,

    Dongxiao Liet al., “Experimental estimation of the quantum 10 fisher information from randomized measurements,” Physical Review Research3, 043122 (2021)

    2021

  38. [45]

    Exploring many-body interactions through quantum fisher information,

    Paweł Cie ´sli´nski, Paweł Kurzy ´nski, Tomasz Sowi´nski, Walde- mar Kłobus, and Wiesław Laskowski, “Exploring many-body interactions through quantum fisher information,” Physical Re- view A110, 012407 (2024)

    2024

  39. [46]

    Metrological advantage of quantum resources in complex systems,

    Various authors, “Metrological advantage of quantum resources in complex systems,” Physical Review Letters116, 120401 (2016)

    2016

  40. [47]

    Quantum fisher information for states in expo- nential form,

    Zhang Jiang, “Quantum fisher information for states in expo- nential form,” Physical Review A89, 032128 (2014)

    2014

  41. [48]

    Direct measurement of quantum fisher in- formation,

    Xingyu Zhang, Xiao-Ming Lu, Jing Liu, Wenkui Ding, and Xiaoguang Wang, “Direct measurement of quantum fisher in- formation,” Physical Review A107, 012414 (2023)

    2023

  42. [49]

    Sym- metric logarithmic derivative for entangled quantum states,

    Chien-Hsiu Chou, Chih-Hao Lin, Chia-Hsiu Hsu,et al., “Sym- metric logarithmic derivative for entangled quantum states,” Physical Review A101, 052114 (2020)

    2020

  43. [50]

    On the inequality of sums of squares,

    Hermann Amandus Schwarz, “On the inequality of sums of squares,” Mathematical Annals3, 249–281 (1865)

  44. [51]

    Dynamics of 2×2 matrix non-hermitian quan- tum systems on bloch sphere,

    Libin Fu, “Dynamics of 2×2 matrix non-hermitian quan- tum systems on bloch sphere,” Communications in Theoretical Physics76, 045101 (2024)

    2024

  45. [52]

    Coherence and joseph- son oscillations between two tunnel-coupled one-dimensional atomic quasicondensates at finite temperature,

    Pjotrs Gri ˇsins and Igor E. Mazets, “Coherence and joseph- son oscillations between two tunnel-coupled one-dimensional atomic quasicondensates at finite temperature,” Phys. Rev. A 87, 013629 (2013)

    2013

  46. [54]

    Quantum fisher information width in quantum metrol- ogy,

    Bo Liu, Guolong Li, Yanming Che, Jie Chen, and Xiaoguang Wang, “Quantum fisher information width in quantum metrol- ogy,” Science China Physics, Mechanics & Astronomy62, 1–9 (2018)

    2018

  47. [55]

    Quan- tum sensing with a single-qubit pseudo-hermitian system,

    Yaoming Chu, Yu Liu, Haibin Liu, and Jianming Cai, “Quan- tum sensing with a single-qubit pseudo-hermitian system,” Phys. Rev. Lett.124, 020501 (2020)

    2020

  48. [56]

    Naimark-dilatedPT- symmetric brachistochrone,

    Uwe G ¨unther and Boris F. Samsonov, “Naimark-dilatedPT- symmetric brachistochrone,” Phys. Rev. Lett.101, 230404 (2008)

    2008

  49. [57]

    Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system

    Jian Shun Tang, Yi Tao Wang, Yong Jian Han, Chuan Feng Li, and Guang Can Guo, “Experimental investigation of the no-signalling principle in parity-time symmetric theory using an open quantum system.” inAPS March Meeting 2017(2017)

    2017

  50. [58]

    Experimental simulation of anti-parity-time symmetric lorentz dynamics,

    Qiang Li, Cheng-Jie Zhang, Cheng Ze-di, Wen-Zheng Liu, Jun- Feng Wang, Fei-Fei Yan, Zhi-Hai Lin, Ya Xiao, Kai Sun, Yi- Tao Wang, Jian-Shun Tang, Jin-Shi Xu, Chuan-Feng Li, and Guang-Can Guo, “Experimental simulation of anti-parity-time symmetric lorentz dynamics,” Optica6, 67–...

    2019

  51. [59]

    Complex extension of quantum mechanics,

    Carl M. Bender, Dorje C. Brody, and Hugh F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett.89, 270401 (2002)

    2002

Pith tools

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