Non-Hermitian sensing from the perspective of post-selected measurements

Franco Nori; Keyu Xia; Neng Zeng; Tao Liu; Yu-Ran Zhang

REVIEW 3 major objections 5 minor 1 cited by

Non-Hermitian sensing from the perspective of post-selected measurements

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

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Pith's one-line read Non-Hermitian quantum sensors cannot outperform Hermitian ones once post-selection success is counted.

desk verdict A correct but mostly corollary bound on non-Hermitian sensing, made useful by an efficiency metric and three worked examples; the abstract overstates the resource-independence of the result. read the letter →

arxiv 2505.05058 v2 pith:O7XBNP2T submitted 2025-05-08 quant-ph

Neng Zeng , Tao Liu , Keyu Xia , Yu-Ran Zhang , Franco Nori This is my paper · ORCID

classification quant-ph

keywords non-HermitiansensingquantumFisherinformationNaimarkdilationpost-selectedmeasurementsexceptionalpointsweak-valueamplificationPTsymmetrymetrology

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

Non-Hermitian sensing, especially near exceptional points, is often advertised as delivering enormous or even divergent quantum Fisher information. This paper establishes that the advertised sensitivity is an artifact of ignoring the measurement record: non-Hermitian evolution is equivalent to post-selecting a Hermitian system–environment evolution, so the quantity that sets the real Cramér–Rao bound is the success-probability-weighted effective quantum Fisher information $P_d F^{nH}_Q$. The main result, Eq. (4), is $P_d F^{nH}_Q[|\psi(t)\rangle_S] \le F_Q[|\Psi(t)\rangle_{SE}]$, where $F_Q$ is the total quantum Fisher information of the Naimark-dilated Hermitian system. The paper checks the bound on a pseudo-Hermitian qubit, two exceptional-point sensors, and a loss-loss sensor, and introduces the ratio $P_d F^{nH}_Q/F_Q$ as an efficiency measure. The upshot is that non-Hermitian sensors are not fundamentally better estimators than Hermitian ones, but they can still be efficient or noise-resilient in specific regimes.

What carries the argument

The load-bearing object is Naimark dilation: a non-Hermitian Hamiltonian $H_S(t)$ on the sensor is lifted to a Hermitian Hamiltonian $H_{SE}(t)$ on a larger system, with the joint state written as $|\Psi(t)\rangle_{SE}\propto |\psi(t)\rangle_S|0\rangle_E+\hat m(t)|\psi(t)\rangle_S|1\rangle_E$, where $\hat m(t)=[\hat\eta(t)-I]^{1/2}$ and $\hat\eta(t)$ is determined by $H_S(t)$ and the initial metric operator. The mechanism then combines this dilation with the post-selection Fisher-information decomposition of Eq. (1), splitting the total information into detected, rejected, and post-selection terms. Since the total Fisher information for any post-selected strategy is bounded by the joint-state QFI, Eq. (2), the same bound transfers to non-Hermitian sensing as Eq. (4). This chain turns a divergent-looking QFI into a finite effective QFI and turns the efficiency question into a ratio $P_d F^{nH}_Q/F_Q$ that can be optimized.

What would settle it

One concrete check is to pick a non-Hermitian Hamiltonian and probe state, construct the Naimark-dilated Hermitian system, and sweep the unknown parameter while comparing $P_d F^{nH}_Q[\psi]$ with $F_Q[\Psi]$. Any parameter value with $P_d F^{nH}_Q > F_Q$ would falsify Eq. (4); in the laboratory, a full two-outcome measurement record whose estimation variance falls below the joint-state Cramér–Rao bound would do the same.

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Extended reading notes

Core claim

The discovery is a resource-accounting bound, not a statement that non-Hermitian sensors are useless. Using the Naimark dilation theorem, the paper embeds any non-Hermitian Hamiltonian evolution in a larger unitary evolution on a system plus one auxiliary qubit; the sensor state is recovered by projecting the environment onto one of two outcomes, and the probability of that outcome is $P_d$. The paper then proves that the effective quantum Fisher information of the sensor state, $P_d F^{nH}_Q$, never exceeds the total quantum Fisher information $F_Q$ of the joint Hermitian state. The same logic explains why raw QFI diverges at exceptional points: the success probability collapses at the same rate, so the product stays finite and bounded. The authors conclude that when the environment is counted as a resource, non-Hermitian sensors are suboptimal estimators, while retaining practical value because extracting all information from the environment is infeasible and post-selection can suppress technical noise.

Load-bearing premise

The proof assumes the environment can be represented by a single auxiliary qubit whose initial state can be adjusted so that the Naimark dilation is Hermitian, and that counting that qubit's information captures the true resource cost of a real multi-mode environment.

Editorial extensions

If this is right

Exceptional-point sensors do not offer a fundamental precision advantage over Hermitian sensors once the success probability is included; their raw QFI divergence is cancelled by a vanishing success probability.
The pseudo-Hermitian sensor can be efficient, with $P_d F^{pH}_Q$ approaching $F_Q$ near a specific parameter value, meaning most of the information is carried by few successful runs.
For the two exceptional-point sensors and the loss-loss sensor analyzed here, most information resides in rejected outcomes and the post-selection record itself, so their effective sensitivity is far below the joint-state QFI.
Optimizing non-Hermitian sensors means optimizing the ratio $P_d F^{nH}_Q/F_Q$ and treating the environment as a resource, not minimizing the raw QFI of the reduced state.

Reading between the lines

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

The paper does not say this, but the same accounting applies to any metrological scheme with a small success probability, so the framework could benchmark critical quantum sensors and quantum-jump-based protocols.
A testable prediction following from the paper's setup is that in a real multi-mode environment the effective sensitivity should be even lower than the bound, because additional environment modes carry information that is not post-selected; checking this would require a full spectral decomposition of the environment.
One further design consequence is that the efficiency ratio can serve as a practical selection criterion: for a given technical-noise level, choose the post-selection window that maximizes $P_d F^{nH}_Q/F_Q$, treating the ratio as an information yield rather than a precision limit.

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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

Read this if you work on non-Hermitian sensing. The paper proves that the effective QFI of a non-Hermitian sensor, weighted by post-selection success, is bounded above by the QFI of a Naimark-dilated Hermitian system. That inequality is true, but it is essentially a corollary of the known post-selection bound from Zhang, Datta, and Walmsley plus the Naimark construction. The genuinely new piece is the efficiency ratio Pd F_nH / F_tot and the comparison of three concrete sensors, which gives a practical way to discuss how much of the total information is actually used.

The math is clean. The derivation of Eq. (4) from the post-selection Fisher decomposition is straightforward and correct. The pseudo-Hermitian example is the most instructive: the divergent QFI is tamed by the success probability, and the efficiency ratio is high, echoing the weak-value 'when less is more' story. The EP examples are worked consistently, and the supplementary material is honest about the need for large η(0) in those cases.

Where the paper is soft is in the interpretation, not the proof. The abstract says non-Hermitian sensors 'cannot outperform their Hermitian counterpart when all information is harnessed.' That phrasing is too broad. The right-hand side of Eq. (4) is the QFI of a specific dilated system, and that QFI depends on the choice of η(0). For the EP sensors the authors pick η(0)=100 without a principled reason; for the PT-broken phase the amplification factor diverges. So the bound is always true by construction, but it can be made arbitrarily loose by choosing a dilation with a large ancilla amplitude. The statement that a non-Hermitian sensor cannot beat 'its' Hermitian counterpart is only meaningful once you specify which counterpart, and the paper does not always do that carefully. The pseudo-Hermitian case with η(0)=ζ/ν_ζ is a defensible minimal choice, but that should be said explicitly and the abstract should be aligned with that scope.

There is also a novelty-citation issue the authors should address head-on. Ref. [44] already gave fundamental sensitivity limits for non-Hermitian sensors. The paper cites it but does not clearly separate what is new beyond that work. The efficiency-ratio analysis is likely enough to justify a paper, but the authors should make the distinction explicit.

I would send this to peer review. It is a sound, carefully argued paper with correct examples, and the efficiency metric may be genuinely useful for experimental design. A good referee should ask for an abstract rewrite and a discussion of how the bound depends on the dilation choice, but the core result does not need to be redone.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a framework for understanding the sensitivity of non-Hermitian quantum sensors by mapping them, via Naimark dilation, to post-selected measurements on a larger Hermitian system. The authors derive the inequality P_d F_nH_Q ≤ F_Q, where P_d is the post-selection success probability, F_nH_Q is the QFI of the non-Hermitian sensor state, and F_Q is the QFI of the dilated Hermitian state. They interpret this as showing that non-Hermitian sensors cannot outperform their Hermitian counterparts when all resources are accounted for, and they illustrate the framework with three examples: a pseudo-Hermitian qubit, an EP-based Brillouin-ring sensor, and a PT-symmetric two-level sensor, with a loss-loss model in the SM. The paper also introduces an efficiency ratio analogous to weak-value amplification.

Significance. If the central claim were established in the strong form stated, the paper would provide a unifying resource-theoretic perspective on non-Hermitian sensing, connecting it to the well-studied post-selection metrology literature. The main inequality is a straightforward corollary of the post-selection Fisher-information decomposition and is mathematically correct for the particular dilation constructed; the examples are worked in considerable detail in the SM. The connection to weak-value amplification is conceptually appealing and could be useful for designing noise-resilient protocols. However, the strength of the no-advantage claim is not supported by the proof, because the bound depends on an arbitrary choice of the initial metric operator η(0) and hence on a non-unique Hermitian counterpart. One of the three main examples also contains a normalization inconsistency that affects its quantitative conclusions. The framework is a useful contribution if these issues are addressed, but as written the paper overstates its main theorem.

major comments (3)

[Pseudo-Hermitian sensor] The dilated state |Ψ(t)> written as |ψ(t)>|0> + i(1-λ^2)^{1/2} sin(θt)|1>|1> with the normalized sensor state |ψ(t)> is not the solution of the Schrödinger equation for the dilated Hamiltonian HSE in Eq. (6). Solving i∂t|Ψ>=HSE|Ψ> for HSE = θλ σ_x⊗I - θ√(1-λ^2) σ_y⊗σ_y with initial state |0>|0> gives |Ψ(t)> = cos(θt)|0>|0> - iλ sin(θt)|1>|0> - i√(1-λ^2) sin(θt)|1>|1>, which corresponds to using the unnormalized sensor state in the dilation formula. The stated success probability P_d = [1+(1-λ^2) sin^2(θt)]^{-1} is therefore not the post-selection probability of the constructed dilated state; the correct value is P_d = cos^2(θt)+λ^2 sin^2(θt). This changes the effective QFI and invalidates the quantitative claim that P_d F_pH ≈ F_Q at θ≈0.785 (with the correct P_d the ratio is λ^2/[cos^2+λ^2 sin^2], which equals 1 only at θt=π/2). The example and Fig. 2 need to be recomputed with consistent normalization.
[Equations (1)-(4)] The inequality P_d F_nH_Q ≤ F_Q is a direct consequence of the post-selection decomposition (1) and the bound (2); it is always true for the particular Naimark dilation chosen. However, F_Q depends on the arbitrary initial metric operator η(0), which the paper itself states is indeterminate (SM Sec. II) and which is fixed ad hoc in the examples (η(0)=100 for the EP sensors). Therefore the abstract's claim that non-Hermitian sensors 'cannot outperform their Hermitian counterpart when all information is harnessed' is not established as a fundamental, resource-inclusive no-go theorem; it is a bound relative to a chosen dilation. A different valid dilation changes F_Q and hence the tightness of the bound, and no minimization over dilations is provided. The final paragraph's caveat that the Naimark dilation may not be minimal partially acknowledges this, but the abstract and introduction still state the stronger claim. The paper should either qualify the no-advantage statement throughout, or prove a dilation-independent resource bound.
[EP sensor examples] The choice η(0)=100 in the EP sections is arbitrary, and the SM itself notes that in the PT-broken phase the required amplification factor 1/ν' diverges, so the 'Hermitian counterpart' can carry unbounded resource cost as t→∞. This reinforces the concern in the previous comment: the no-advantage conclusion is not robust under changes of the dilation. The paper should discuss the behavior of the bound under minimization over η(0) or justify a canonical choice (e.g., the minimal dilation) before drawing general conclusions about the impossibility of non-Hermitian advantage.

minor comments (5)

[Title] In the arXiv title, 'post-select ed' contains an extra space; this should be corrected.
[After Eq. (4)] The sentence 'by considering of the noisy QFI' is ungrammatical and should be rephrased.
[Reference [89]] The SM reference is incomplete: 'Supplementary Materials are available on .' should give a URL or DOI.
[Fig. 3 caption] In Fig. 3(b1,b2) and (c1,c2), the notation for the rejected-state effective QFI is inconsistent (P′_r Q′_r appears twice), and the relation between F_post and F′_post in the caption is not explained.
[SM Eq. (S1)] The quantities F_d and F_r in Eq. (S1) are used before being explicitly defined; the definitions should be added.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central inequality is a deductive corollary of the post-selection Fisher-information decomposition applied to a Naimark dilation, not an input assumed as a conclusion.

full rationale

The paper's central result, Eq. (4) (Pd F_nH_Q[|ψ(t)>_S] ≤ F_Q[|Ψ(t)>_SE]), is obtained by applying the standard post-selection Fisher-information decomposition of Eq. (1) and the inequality F_tot ≤ F_Q of Eq. (2) to the Naimark-dilated joint state. This is an explicit derivation from known quantum-estimation identities, not an assumption equivalent to the desired conclusion. No parameter is fitted to data and then renamed as a prediction: the Naimark construction defines H_SE and |Ψ(t)>_SE directly from H_S, and the inequality is then evaluated, not tuned. The choice η(0)=100 for the EP examples is a modeling choice that affects the tightness of the bound and the physical fairness of the 'Hermitian counterpart' comparison, but it does not make the derivation circular: Eq. (4) remains a logical consequence of Eqs. (1) and (2) for any valid dilation. Concerns about whether the dilated ancilla properly counts the resources of a real environment are correctness or interpretation issues, not circularity. The self-citations present in the paper (e.g., experimental models of EP sensors) are background examples and are not load-bearing for the mathematical chain leading to Eq. (4). The central claim is therefore self-contained against external benchmarks and no circular step can be exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central inequality relies on standard theorems from quantum estimation and dilation theory; no experimental data or fitted constants are used. The example-dependent parameters (λ, η(0), etc.) do not enter the proof of Eq. (4).

free parameters (3)

η(0) (initial metric operator scale) = 100
Chosen in the main text before Fig. 3 to ensure [η(t)-I] positive for t≤15; affects the numerical QFI and success probability in the EP examples but not the inequality.
λ (pseudo-Hermitian coupling ratio) = 0.5
Used in Fig. 2 to illustrate the pH sensor; the efficiency behavior depends on λ, but the bound holds for all λ∈(0,1].
Ω_EP, ω_ccw, r, φ = Ω_EP = ω_ccw; φ = π/4
Model parameters of the EP sensor examples; set for numerical plots, not fitted to data.

assumptions (5)

standard math Naimark dilation theorem: any non-unitary evolution can be embedded as a post-selected branch of a unitary evolution on a larger Hilbert space
Used in the main text to construct H_SE(t) in Eq. (3); standard result, see Refs. [59-61].
standard math Total FI decomposition and bound F_tot ≤ F_Q for post-selected measurements (Eqs. (1,2))
From Ref. [66]; used to derive Eq. (4).
domain assumption The sensor state after post-selection is the renormalized branch |ψ_d> ∝ E⟨0|Ψ⟩, and QFI is computed for this pure state
Standard post-selection ansatz; treats non-Hermitian evolution as conditional evolution.
domain assumption Existence of a Hermitian η(0) making η(t)-I positive for all relevant times, so m(t) is Hermitian
SM Sec. III provides the construction but requires choosing η'(0) and a large amplification factor; for EP sensors this is only achieved with η(0)=100 for finite t.
domain assumption The environment is a two-level ancilla; post-selection is projection onto |0>_E
The Naimark dilation uses a qubit ancilla; the physical environment may have larger dimension, but the bound is derived for this representation.

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Cite this review

Pith. "Pith review of Non-Hermitian sensing from the perspective of post-selected measurements." pith.science (2026). https://pith.science/paper/O7XBNP2T

@misc{pith2026250505058,
  author       = {Pith},
  title        = {Pith review of: Non-Hermitian sensing from the perspective of post-selected measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O7XBNP2T}},
  note         = {Machine review of arXiv:2505.05058}
}

read the original abstract

By employing the Naimark dilation, we establish a fundamental connection between non-Hermitian quantum sensing and post-selected measurements. The sensitivity of non-Hermitian quantum sensors is determined by the effective quantum Fisher information (QFI), which incorporates the success probability of post-selection. We demonstrate that non-Hermitian sensors cannot outperform their Hermitian counterpart when all information is harnessed, since the total QFI for the extended system constrains the effective QFI of the non-Hermitian subsystem. Moreover, we quantify the efficiency of non-Hermitian sensors with the ratio of the effective QFI to the total QFI, which can be optimized within the framework of post-selected measurements with minimal experimental trials. Our work provides a distinctive theoretical framework for investigating non-Hermitian quantum sensing and designing noise-resilient quantum metrological protocols.

Figures

Figures reproduced from arXiv: 2505.05058 by the authors.

**Figure 2.** FIG. 2. Pseudo-Hermitian (pH) sensor with the Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Two EP-based sensors with Hamiltonians in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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