Hermitian vs non-Hermitian quantum thermometry

Abdallah Slaoui; Anass Hminat; Mourad Telmini; Rachid Ahl Laamara

REVIEW 3 major objections 5 minor 1 cited by

Hermitian vs non-Hermitian quantum thermometry

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that anti-PT-symmetric qubits are the most precise dephasing-based thermometers in Ohmic environments, retaining coherence longest and maximizing the quantum Fisher information for temperature.

desk verdict The APT-superiority claim in this thermometry paper rests on a decoherence factor that is never derived, and the QSNR section contradicts the QFI section; this version should not go to referees. read the letter →

arxiv 2509.10840 v1 pith:RSX7OBY4 submitted 2025-09-13 quant-ph math-phmath.MP

Anass Hminat , Abdallah Slaoui , Rachid Ahl Laamara , Mourad Telmini This is my paper · ORCID

classification quant-phmath-phmath.MP

keywords quantumthermometrydephasingFisherinformationanti-PTsymmetryPTOhmicspectraldensityqubitprobeCramér-Raobound

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

This paper sets out to show that qubit dephasing can be used as a resource for thermometry and that the symmetry of the qubit determines how good the thermometer is. For a qubit coupled to a bosonic bath with an Ohmic spectral density, the paper claims that anti-PT-symmetric qubits—those whose Hamiltonian anticommutes with the combined parity-time operation—keep their coherence longest and yield the highest quantum Fisher information for the bath temperature, followed by PT-symmetric qubits, with Hermitian qubits last. The reason to care is that this turns decoherence, usually the enemy of quantum devices, into a precise probe of temperature without requiring the probe to reach thermal equilibrium with the sample, and it suggests that non-Hermitian engineering can protect quantum sensors in noisy low-temperature environments. The paper also identifies the optimal probe preparation and the optimal measurement, and shows that the best estimate happens at a finite interaction time rather than at the steady state.

What carries the argument

The central object is the dephasing dynamical map, whose off-diagonal entries decay as $e^{-\Gamma(T,t)}$, and the quantum Fisher information it induces, $H(T,t)=\sin^2\theta\,(\partial_T\Gamma)^2/(e^{2\Gamma}-1)$. The bath is Ohmic with spectral density $J(\omega,\omega_c)=J_0(\omega^s/\omega_c^{s-1})e^{-\omega/\omega_c}$, and the symmetry enters through the phase function $\Omega(\tilde t)$ in Eq. (24): Hermitian uses $\Omega(t)$, PT uses $-\Omega(t)$, and anti-PT uses $\Omega_2(t)-\Omega_1(t)$, with the anti-PT reduced density matrix obtained through a time-dependent Dyson map to an equivalent Hermitian system. Optimizing $H$ over the preparation angle, the interaction time, and the temperature yields the paper's rankings and identifies the optimal measurement that saturates the quantum Cramér-Rao bound.

What would settle it

Deriving $\Gamma_{\mathrm{APT}}(T,t)$ explicitly from the time-dependent Dyson map for Hamiltonian (18) and comparing it with $\Omega_2(t)-\Omega_1(t)$ would settle the calculation; a discrepancy at the optimal times changes $H(T,t)$ and the APT ranking. Experimentally, measuring the coherence decay of anti-PT, PT, and Hermitian qubits in one Ohmic bath and checking the fitted ordering $a\approx 0.685<1.688<3.141$ (Hermitian, PT, APT) would settle the claim directly.

Watch

Extended reading notes

Core claim

The paper's central claim is a ranking: under pure dephasing in an Ohmic environment, an anti-PT-symmetric qubit has the slowest decoherence and gives the largest quantum Fisher information $H(T,t)$ for temperature, while a Hermitian qubit gives the smallest and a PT-symmetric qubit sits between them. Alongside this, the paper claims that the optimal initial state is the equatorial superposition $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ for all temperatures, times, and Ohmicity parameters; that the maximum of $H(T,t)$ occurs at a finite interaction time, before the qubit reaches its stationary state or complete dephasing, except in the super-Ohmic low-temperature case where the optimum coincides with thermalization; and that the optimal measurement saturates the quantum Cramér-Rao bound. The paper further reports that the quantum signal-to-noise ratio is lowest for anti-PT symmetry and highest for Hermitian symmetry, saturating to a universal value at high temperature.

Load-bearing premise

The anti-PT advantage rests entirely on a decoherence factor $\Gamma_{\mathrm{APT}}(T,t)$ that the paper imports from reference [62] rather than deriving from its own Dyson-map construction; if that imported factor is wrong or misapplied to this setup, the central ranking collapses.

Editorial extensions

If this is right

A thermometer using the anti-PT qubit needs fewer repetitions to reach a target variance, since the quantum Fisher information sets the per-measurement precision bound.
The optimal measurement is explicitly identified, so a protocol can be implemented that actually reaches the quantum Cramér-Rao bound rather than only quoting it.
Timing matters: the probe should be measured at the optimal interaction time rather than after it has equilibrated, except in super-Ohmic low-temperature environments where the optimum coincides with stationarity.
At high temperatures the environmental spectrum stops mattering: the quantum signal-to-noise ratio becomes universal, which simplifies calibration of hot-sample thermometers.
Non-Markovian memory effects do not improve the estimate, so a memoryless treatment is sufficient for optimizing the protocol.

Reading between the lines

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

The paper's two metrics point in opposite directions—anti-PT maximizes QFI while Hermitian maximizes QSNR—so the practical winner depends on whether one is limited by the number of measurement repetitions (QFI) or by the dynamic range of normalized sensitivity (QSNR); the paper itself does not resolve that choice.
Equation (37) suggests a general design principle: a good dephasing thermometer should maximize the temperature derivative of the decoherence factor while keeping $\Gamma$ near unity, so the denominator $e^{2\Gamma}-1$ does not erase the signal; this principle could be applied to other non-Hermitian or multi-qubit probes.
The exponential decay constants reported from the fits ($a\approx 0.685$ for Hermitian, $1.688$ for PT, $3.141$ for APT) give a concrete experimental signature: in existing anti-PT platforms (coupled atomic beams, electrical-circuit resonators, microcavities), measuring the pure-dephasing decay ordering in the same Ohmic bath would directly test the central ranking.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The right thing to say up front: the paper has a reasonable research question, but the central claim doesn't hold together as written. The two things you need to know are (1) the claimed APT advantage in QFI depends entirely on an APT decoherence factor that is never derived or even stated, and (2) the QSNR results in Sec. V.B directly contradict the QFI results in Sec. IV.B, because QSNR = T²H. You cannot have APT with the highest H at the optimal time and the lowest QSNR at the optimal time in the same model.

What is genuinely there: comparing Hermitian, PT-symmetric, and anti-PT-symmetric qubits for dephasing-based thermometry is a sensible extension of existing work. The paper correctly borrows the standard dephasing-thermometry machinery from Razavian et al. and the APT dynamics from Cen and Saxena, and it runs a fairly thorough numerical scan over Ohmicity parameter, interaction time, and temperature. If the APT model were correctly applied, the conclusion that APT qubits suppress decoherence and therefore improve thermometric precision would be a useful, if incremental, design rule for quantum sensing.

The soft spots are serious. Equation (37) shows QFI depends only on the decoherence factor Gamma, but Gamma for the APT case is never given; the paper gestures at Omega_2 − Omega_1 with a citation to Ref. [62] and moves on. A reader cannot reproduce the APT curves or check whether the comparison is fair. The QSNR contradiction is worse: Sec. V.B first says the high-T QSNR saturates to a universal value, then lists symmetry-dependent plateaus of 0.25, 0.16, and 0.04—and those plateaus, combined with QSNR = T²H, imply the opposite QFI ordering from what the figures show. This is an internal inconsistency in the paper's own equations, not a matter of interpretation. There are also smaller issues: the abstract promises identification of the optimal measurement, but none is specified, and the text has typos, swapped s-labels in figure captions, and duplicated sentences. Those would be minor on their own, but here they sit on top of a load-bearing gap.

I would not send this to peer review in its current form. The missing APT decoherence factor and the QSNR contradiction are not fixable by tweaking language; the authors need to derive Gamma_APT explicitly, redo the QSNR calculation, and reconcile the ordering claims before the paper is coherent. The topic is fine, and a corrected version might be worth a look, but this version does not give a serious referee enough to work with.

Referee Report

3 major / 5 minor

Summary. The manuscript studies temperature estimation of an Ohmic thermal bath via dephasing of a single qubit, comparing Hermitian, PT-symmetric, and anti-PT-symmetric (APT) qubit Hamiltonians. It derives a QFI formula for the dephasing channel (Eq. 37), identifies the optimal initial state (θ=π/2), and reports numerical optimization of the QFI over interaction time and temperature for sub-Ohmic, Ohmic, and super-Ohmic environments. It also reports the QSNR, decoherence dynamics, von Neumann entropy, and exponential decay fits. The paper's central claim is that APT symmetry yields the highest QFI and the strongest decoherence resilience, with Hermitian dynamics giving the lowest QFI. However, the QSNR section reports the opposite ordering, which is logically incompatible with the QFI results.

Significance. If the APT-superiority claim were supported, the paper would offer a concrete and useful result for quantum thermometry: anti-PT-symmetric qubits would provide both higher precision and slower decoherence than Hermitian or PT-symmetric qubits in Ohmic environments. The manuscript also has positive features: it clearly formulates the QFI optimization, identifies the optimal state preparation, and reports explicit exponential decoherence fits with numerical parameters and uncertainties. The study builds on established single-qubit dephasing thermometry (Ref. [74]) and the APT-qubit decoherence model of Cen and Saxena [62], so the cross-symmetry comparison is a plausible extension. However, because the APT decoherence factor is never explicitly given and the QFI and QSNR orderings contradict each other, the central claim is not supported by the manuscript as written.

major comments (3)

[§IV.B and §V.B] The central claim that APT symmetry yields the highest QFI across all regimes is logically incompatible with the QSNR results reported in Section V.B. Since Eq. (10) defines Q_T = T²H(T), and both Section IV.C (Fig. 5) and Section V.B (Fig. 8) evaluate quantities at the optimal interaction time t_opt, the ordering of the QSNR plateaus (Hermitian 0.25, PT 0.16, APT 0.04) must match the ordering of H at t_opt. The paper instead reports APT as the highest QFI and the lowest QSNR, which cannot both be true for the same H(T,t_opt). One of the two sets of curves must be erroneous, and this undermines the main conclusion.
[§III (Eqs. 23-27) and §IV.A (Eq. 37)] Equation (37) expresses the QFI solely in terms of the decoherence factor Γ(T,t), yet the manuscript never provides Γ(T,t) for the APT case. The APT case is introduced only through Ω2(t)−Ω1(t) in Eq. (24), with a reference to Cen and Saxena [62], and the thermal decoherence formulas in Eqs. (38)-(39) are the standard Hermitian dephasing expressions. Because no time-dependent Dyson map or resulting reduced density matrix is shown for the APT qubit, the APT QFI curves in Figs. 4-5 cannot be reproduced from the manuscript as written. The APT-superiority claim therefore rests on an unstated imported model, and the claimed derivation is not actually provided.
[§V.B] Section V.B contains a direct self-contradiction: it first states that the QSNR 'saturates to a universal value, independent of the symmetry and the nature of the spectral density,' and then immediately reports symmetry-dependent plateaus of 0.04 (APT), 0.16 (PT), and 0.25 (Hermitian). The Introduction similarly promises a universal high-temperature saturation. If the universal value is meant to be independent only of the spectral density s but not of the symmetry, this needs to be stated; as written, the two sentences are incompatible.

minor comments (5)

[Abstract and §I] The word 'symmetrizes' is used where 'symmetries' is intended; this typo should be corrected.
[§I] The sentence 'whereas the anti-PT operator anticommutes with it, satisfying {H,PT}=0' appears twice in succession; one duplicate should be removed.
[§IV.A] The text says 'Inserting Γ(t,ω_c) as given in Eq. (6)'; Eq. (6) is the Cramér-Rao bound, not the decoherence factor. The cross-reference should point to the spectral density or decoherence equations.
[Fig. 6 caption] The caption assigns 'sub-Ohmic (s=1.0)' and 'Ohmic (s=0.5)', which is inconsistent with the rest of the paper where s=0.5 is sub-Ohmic and s=1.0 is Ohmic; these labels should be corrected.
[§V.A / Fig. 6 caption] The exponential fit model D(t)=exp(−t/a) and the fitting region are described only in the caption; the fitting methodology should be stated in the main text for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QFI and dephasing formulas are standard or imported from independent external sources (Razavian et al., Cen and Saxena), and no fitted parameter is dressed up as a prediction.

full rationale

The derivation chain is not circular. Equation (37) is the standard QFI expression for a dephased qubit, and Eqs. (38)-(39) are the standard Hermitian dephasing factor taken from Razavian et al. (Ref. [74]), an independent external source. The APT branch is imported from Cen and Saxena (Ref. [62]) through the Omega functions in Eqs. (25)-(27); Ref. [62] is not authored by the present authors, so this is an external-model transfer rather than a self-citation or ansatz-smuggling step. The self-citations in the reference list (e.g., Refs. [13], [22], [26]-[28]) are background references to the authors' other work and do not carry the thermometry argument. The exponential fit D(t)=exp(-t/a) in the Fig. 6 caption is a descriptive fit to already-computed decoherence curves, not a fitted parameter renamed as a prediction, and it does not feed back into the QFI calculation. The paper does have serious reproducibility and internal-consistency problems: the explicit APT decoherence factor Gamma_APT(T,t) is never stated despite being the quantity that determines the APT QFI curves, and Sec. V.B reports APT as having the lowest QSNR while Sec. IV.B reports APT as having the highest QFI even though Q_T = T^2 H(T) by Eq. (10). These are correctness or derivation-gap concerns, not circular reductions: no equation in the paper defines the sought result in terms of itself, and no fitted parameter is presented as an independent prediction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard QFI theory, a standard dephasing decoherence function, and an unshown Dyson map for APT. No new entities are introduced. The fit of the exponential decay time constant is the only fitted quantity.

free parameters (2)

Decay time constant a_sym = a=0.685 (Hermitian), 1.688 (PT), 3.141 (APT) for s=1
Exponential fit D(t)=exp(-t/a) to computed decoherence curves, reported in Fig. 6 caption; used to support the claim that APT decoheres slowest.
Model parameters a,b,c,d = a=1, b=c=d=0.6
Hand-picked values for all figures; the central qualitative claim is said to hold across regimes but is not shown analytically for other values.

assumptions (4)

standard math Quantum Fisher information formula Eq. (8) for density matrices
Invoked in Section II as the standard expression for QFI from eigenvalues and eigenvectors.
domain assumption Pure dephasing decoherence factor Gamma(T,t) for a qubit in a thermal bosonic bath, Eq. (38)
Adopted from the open quantum systems literature (Breuer-Petruccione, Razavian et al.) without derivation; underlies Eq. (37).
domain assumption Time-dependent Dyson map maps anti-PT-symmetric qubit to an equivalent Hermitian system
Introduced in Section III via reference [38]; the exact mapping and resulting Gamma_APT are not shown.
domain assumption Normalized non-Hermitian evolution rho(t)=U rho U^dagger / Tr[U rho U^dagger] with U non-unitary
Assumed in Eq. (29) as the dynamical law for the open non-Hermitian system.

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Cite this review

Pith. "Pith review of Hermitian vs non-Hermitian quantum thermometry." pith.science (2026). https://pith.science/paper/RSX7OBY4

@misc{pith2026250910840,
  author       = {Pith},
  title        = {Pith review of: Hermitian vs non-Hermitian quantum thermometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSX7OBY4}},
  note         = {Machine review of arXiv:2509.10840}
}

read the original abstract

We investigate the dephasing dynamics of a qubit as an effective mechanism for estimating the temperature of its surrounding environment for different symmetrizes. 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 Cramer-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. Notably, optimal estimation does not occur when the qubit reaches a stationary state nor under conditions of complete dephasing.

Figures

Figures reproduced from arXiv: 2509.10840 by the authors.

**Figure 1.** FIG. 1: Bloch sphere trajectories of Hermitian (top left), PT-symmetric (top right), and APT-symmetric (bottom) qubits, showing [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** FIG. 2: A quantum thermometric protocol is proposed, uti [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** for details). Inserting Γ(t, ωc) as given in Eq. (6) into the preceding equation yields an analytical formula for the decoherence coefficient for a specified value of s. To optimize the estimation, we search for the interaction time t that maximizes the QFI versus temperature T (cf [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4: We show the QFI [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5: We report the optimal interaction times [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: Fig.7.(a-c). Initially, [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 6.** Figure 6: FIG. 6: The decoherence [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7: The von Neumann entropy [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8: Plot of the QSNR evaluated at the optimal interaction time for different symmetry [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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