{"id":"7e29d180-5b78-46ca-bed2-ba91fa614548","arxiv_id":"2505.05058","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"Non-Hermitian sensors cannot outperform Hermitian ones when post-selection success probability is counted, because their effective quantum Fisher information is bounded by the total quantum Fisher information of the Naimark-dilated system.","lead":"This paper shows that non-Hermitian quantum sensors, despite huge apparent sensitivity near exceptional points, have a fundamental limit once the success probability of the post-selection step is included. Using Naimark dilation, the authors prove that the effective quantum Fisher information of such sensors never exceeds the total quantum Fisher information of an extended Hermitian system, and they introduce an efficiency ratio for practical designs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The no-advantage claim rests on an arbitrary Naimark dilation: η(0) is under-specified, and the QFI of the 'Hermitian counterpart' changes with that choice, so Eq. (4) is not a fixed resource bound.","rationale":"The reader's weakest assumption pointed at the single-qubit dilation and the divergence of 1/ν' in the PT-broken phase. I agree, but the sharper problem is that Eq. (4) is not a unique bound: η(0) is a free parameter of the Naimark construction, and changing it changes both F_Q[|Ψ>_SE] and Pd while the proof remains valid. The paper even concedes in the conclusions that the Naimark dilation may not be the minimal extension and that full control of the environment is infeasible. This matters because the headline claim is a universal no-advantage statement, whereas the derivation demonstrates an inequality for one constructed dilation. I found no mathematical error in the central derivation, so the appropriate outcome remains CONDITIONAL rather than ACCEPT or REJECT, with the condition that the authors scope the abstract to 'for the Naimark-dilated counterpart with a specified initial metric' and make the resource accounting precise. The novelty overlap with Ref. [44] flagged by the reader is a separate but reinforcing reason to require revision.","tokens_in":20736,"tokens_out":10214,"duration_ms":104820,"concrete_test":"Fix H_pH in Eq. (5), λ=0.2, θ=π/(4t) with t=2, and construct the dilation for η(0)=k diag(1,λ^{-2}) for k=1,2,10,100 using Eqs. (S21,S18). Compute |Ψ(t)>_SE, Pd, F_nH_Q from Eq. (S28), and F_Q[|Ψ>_SE]; plot F_Q and Pd F_nH_Q/F_Q versus k. If both vary with k, Eq. (4) is dilation-dependent and cannot anchor a universal no-advantage claim without a resource-fixing criterion. Then repeat for the EP Hamiltonian (7) with η(0)=cI for c=10,100,1000 at fixed t, and in the PT-broken case (8) at θ=0.5, r=1, φ=π/4 to quantify how F_Q grows with 1/ν' as t increases; this exposes the hidden resource cost of the 'Hermitian counterpart.'","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central no-advantage statement (abstract; Eq. (4)) is an interpretation of the inequality Pd F_nH_Q[|ψ>_S] ≤ F_Q[|Ψ>_SE], which follows trivially from the post-selection Fisher decomposition (1). The load-bearing question is whether the 'Hermitian counterpart' |Ψ>_SE is a well-defined, fairly resourced object. It is not: the Naimark construction leaves η(0) indeterminate (Eq. (S14); SM Sec. III only requires η(t)-I positive). For the pseudo-Hermitian sensor, every k≥1 with η(0)=k diag(1,λ^{-2}) gives a valid dilated Hamiltonian via Eq. (S21) and a joint state whose ancilla amplitude is sqrt(kλ^{-2}-1); F_Q[|Ψ>_SE], Pd, and hence the tightness of Eq. (4) all depend on k. For the EP sensors the paper itself chooses η(0)=100 (main text) and notes that in the PT-broken phase ν'→0 so the amplification factor 1/ν' diverges (SM Sec. V). Thus Eq. (4) is always true by construction—it is a corollary of Eq. (1)—but it bounds the effective non-Hermitian QFI by the QFI of a counterpart whose resources (ancilla amplitude/coupling) are chosen by the theorist and can be arbitrarily large. Without specifying a minimal physical embedding and counting its implementation cost, the abstract's claim that non-Hermitian sensors 'cannot outperform their Hermitian counterpart' overstates what the proof establishes; it is a bound relative to a particular dilation, not a fundamental resource-inclusive no-go.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21048,"tokens_out":20842,"duration_ms":181822,"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":[{"comment":"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.","section":"Pseudo-Hermitian sensor"},{"comment":"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.","section":"Equations (1)-(4)"},{"comment":"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.","section":"EP sensor examples"}],"minor_comments":[{"comment":"In the arXiv title, 'post-select ed' contains an extra space; this should be corrected.","section":"Title"},{"comment":"The sentence 'by considering of the noisy QFI' is ungrammatical and should be rephrased.","section":"After Eq. (4)"},{"comment":"The SM reference is incomplete: 'Supplementary Materials are available on .' should give a URL or DOI.","section":"Reference [89]"},{"comment":"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.","section":"Fig. 3 caption"},{"comment":"The quantities F_d and F_r in Eq. (S1) are used before being explicitly defined; the definitions should be added.","section":"SM Eq. (S1)"}],"recommendation":"major_revision","confidential_remarks":"The paper's core inequality is essentially a restatement of known post-selected metrology bounds (e.g., Zhang, Datta, and Walmsley, PRL 114, 210801 (2015)) applied to Naimark dilation; the novelty lies in the mapping and the worked examples. The pseudo-Hermitian normalization error is a concrete technical issue that must be fixed, and the abstract's overstatement of the no-advantage claim should be carefully qualified. If the authors address these points, the paper could be a useful contribution to the non-Hermitian sensing literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nWhere 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.\n\nThere 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.\n\nI 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.","headline":"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.","tokens_in":21631,"tokens_out":6477,"would_cite":true,"duration_ms":62581,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Non-Hermitian quantum sensors cannot outperform Hermitian ones once post-selection success is counted.","keywords":["non-Hermitian sensing","quantum Fisher information","Naimark dilation","post-selected measurements","exceptional points","weak-value amplification","PT symmetry","quantum metrology"],"falsifier":"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.","tokens_in":20488,"feed_emoji":"⚛️","tokens_out":7767,"duration_ms":72070,"temperature":0.7,"pith_summary":"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.","feed_headline":"Non-Hermitian sensors can't beat Hermitian ones","feed_subtitle":"Count the discarded trials: non-Hermitian sensors stay below their Hermitian counterpart's total QFI.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Naimark dilation theorem that underlies the construction of the Hermitian counterpart.","marker":"[59]"},{"why":"Shows how a non-Hermitian Hamiltonian is simulated by a unitary enlarged-system evolution followed by post-selection.","marker":"[60]"},{"why":"Provides the Naimark dilation of PT-symmetric non-Hermitian systems used in the argument.","marker":"[61]"},{"why":"Gives the Fisher-information decomposition for post-selected measurements, including the inequality that Eq. (4) inherits.","marker":"[66]"},{"why":"Establishes that weak-value-amplification post-selection is suboptimal for estimation, the analogue the paper draws.","marker":"[64]"},{"why":"Introduces the 'when less is more' efficiency perspective that the paper adapts to non-Hermitian sensors.","marker":"[71]"}],"fun_headline_variants":["Post-selection caps non-Hermitian sensor gains","Non-Hermitian sensing bound by post-selection cost","Discarded trials limit non-Hermitian sensor precision","Hermitian limit stands for non-Hermitian sensors","Post-selection explains non-Hermitian sensor limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Post-selection caps non-Hermitian sensor gains","Non-Hermitian sensing bound by post-selection cost","Discarded trials limit non-Hermitian sensor precision","Hermitian limit stands for non-Hermitian sensors","Post-selection explains non-Hermitian sensor limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1303,"prompt_tokens":881,"completion_tokens":422,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":497,"tokens_out":422,"duration_ms":3710,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:15:17.799036+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dressel, S","cited_arxiv_id":null,"evidence_quote":"Supplies the Naimark dilation theorem that underlies the construction of the Hermitian counterpart."},{"cited_title":"Ferrie and J","cited_arxiv_id":null,"evidence_quote":"Gives the Fisher-information decomposition for post-selected measurements, including the inequality that Eq. (4) inherits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the 'when less is more' efficiency perspective that the paper adapts to non-Hermitian sensors."}],"review_version":1}