{"id":"75b19a75-aa8d-4111-af0c-b8db8f0600ea","arxiv_id":"2505.19079","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors derive a quantum Fisher information formula for non-Hermitian systems that adds a term from the changing state norm, and test it on pseudo-Hermitian and PT-symmetric qubits.","lead":"This paper derives formulas for how much information a quantum state carries about an unknown parameter when the system is non-Hermitian, including the effect of the state's changing norm. It applies the formulas to two simple non-Hermitian models and asks whether non-Hermitian effects can be exploited for more precise sensing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (19) does not follow from the paper's own SLD equation: the diagonal matrix element in Eq. (18a) is inconsistent with Eq. (2), so the 16e^(2α)(∂θα)^2 norm term is an artifact.","rationale":"The reader's weakest assumption identifies the unproven identification F=⟨L†L⟩. My stress test finds a sharper, internal problem: the pure-state derivation that produces the headline Eq. (19) is algebraically inconsistent with the very SLD equation that defines L. This is not a matter of differing conventions or missing external comparison; it fails on the paper's own assumptions. All downstream formulas (pure, mixed, generator-based, and the two examples) use Eq. (19)/(20), so the central claim is unsupported. A corrected derivation changes the coefficient of (∂θα)^2 from 16 to 4 (under Hermitian L) or leaves it arbitrary (non-Hermitian L), which would materially change the claimed non-Hermitian enhancement. The paper could be revised by redefining what quantity is being computed and proving a measurement bound, but as written the main result should not be accepted. This moves the reader's CONDITIONAL to REJECT.","tokens_in":14531,"tokens_out":25739,"duration_ms":158185,"concrete_test":"Choose a simple pure state, e.g., |ψ(θ)⟩=(|0⟩+e^(iθ)|1⟩)/√2 and α(θ)=θ. Solve Eq. (2) for L on the support of ρ, first with L=L† and then with a non-Hermitian L that differs in Im⟨ψ|L|ψ⟩, and compute F=⟨L†L⟩ for each. Compare the two values with Eq. (19): the Hermitian solution replaces 16(∂θα)^2 with 4(∂θα)^2, and the non-Hermitian solutions differ among themselves, showing that ⟨L†L⟩ is not determined by Eq. (2). This settles whether Eq. (19) follows from Eq. (15).","verdict_should_be":"REJECT","load_bearing_attack":"Eq. (19) is not a consequence of the defining equation (2). For ρ=e^(2α)|ψ⟩⟨ψ| with ⟨ψ|ψ⟩=1, the diagonal matrix element of Eq. (2) gives Re⟨ψ|L|ψ⟩=2∂θα. Eq. (18a) with k=|ψ⟩ instead requires ⟨ψ|L|ψ⟩=4∂θα; the two are incompatible unless ∂θα=0. Thus Eq. (18a) is only valid for k outside the support of ρ, and the sum leading to Eq. (19) misweights the k=|ψ⟩ term. Moreover, even before this step, Eq. (2) fixes ⟨L†L⟩ only up to the imaginary part of ⟨ψ|L|ψ⟩, so F=⟨L†L⟩ is not unique for non-Hermitian L. Imposing L=L† and solving Eq. (2) gives F=4e^(2α)[⟨∂θψ|∂θψ⟩−|⟨∂θψ|ψ⟩|^2+(∂θα)^2], not Eq. (19). The factor 16 in the norm-velocity term, and every quantitatively different claim in the PT and pseudo-Hermitian examples, is an artifact of this algebraic error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14865,"tokens_out":13603,"duration_ms":101151,"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":[{"comment":"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).","section":"§II, Eqs. (18a)-(19)"},{"comment":"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.","section":"§II, Eq. (15)"},{"comment":"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.","section":"§II, Eqs. (24a)-(27)"},{"comment":"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.","section":"§III, Eq. (41) and Fig. 1"}],"minor_comments":[{"comment":"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.","section":"§II, Eq. (3)"},{"comment":"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.","section":"§IV, Eqs. (51)-(53)"},{"comment":"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.","section":"§III, Eq. (43)"},{"comment":"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.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper addresses a relevant problem and contains interesting applications, but the central algebraic derivation of Eq. (19) is internally inconsistent, and the same defect propagates to the mixed-state formulas and to the applications. The claimed non-Hermitian enhancement is not a robust prediction of the model as written. I recommend rejection, though a substantially revised version with a correct SLD treatment and a properly normalized QFI could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I checked the derivation of Eq. (19) carefully, and the stress-test note is right. The step in Eq. (18a) is only valid for k outside the support of the density matrix. For k = |ψ>, the term ⟨k|ψ⟩⟨ψ|L†|ψ⟩ that Eq. (18a) drops is nonzero, so using it in the completeness sum misweights the k = |ψ> component. That is precisely where the factor 16 comes from. If you instead require L = L†, the standard SLD condition, the pure-state QFI is 4e^{2α}[⟨∂θψ|∂θψ⟩ − |⟨∂θψ|ψ⟩|² + (∂θα)²], not Eq. (19). In addition, F = ⟨L†L⟩ is not well-defined for non-Hermitian L: Eq. (2) fixes the real part of ⟨ψ|L|ψ⟩ but leaves the imaginary part free, so the norm-velocity term and every quantitative result that depends on it are artifacts of an implicit choice. This is not a minor technicality; it kills the paper's main claim that non-Hermiticity enhances QFI through the norm factor.\n\nThat said, the paper is trying to solve a real problem. The projected-Hilbert decomposition and the idea that the time-dependent norm contains parameter information are sensible. The Hermitian limit α = 0 does reduce correctly, and the Naimark dilation comparison is a legitimate way to benchmark non-Hermitian versus Hermitian sensing. The PT-symmetric examples would be useful if the formulas were right, because comparing unbroken and broken phases is a natural question.\n\nThe soft spots go beyond the algebraic error. The n-dependence of the QFI in Eq. (41) is a red flag: the QFI for a ray should be independent of the arbitrary normalization n, and the fact that it depends on n shows the formalism is gauge-sensitive. The PT section also has typos (sinα vs sinx) and the claim about the exceptional point is confused. The paper promises a comparison with Ref. [53] but never delivers one.\n\nThis is a load-bearing flaw, not a patchable detail. As written, the paper should not be published. But the topic is timely, the framework is salvageable, and the error is instructive. I would send it to a referee who can pinpoint the SLD definition problem, but the authors need to redo the derivation and the examples before this is acceptable.","headline":"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.","tokens_in":15370,"tokens_out":19402,"would_cite":false,"duration_ms":140553,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P50","81Q12"],"pacs":["03.65.Ta","03.65.-w"],"model":"deepseek-v4-flash","headline":"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.","keywords":["quantum Fisher information","non-Hermitian systems","pseudo-Hermitian systems","PT symmetry","parameter estimation","Naimark dilation","projected Hilbert space","quantum metrology"],"falsifier":"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.","tokens_in":14348,"feed_emoji":"🎯","tokens_out":5390,"duration_ms":40842,"temperature":0.7,"pith_summary":"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.","feed_headline":"Non-Hermitian states gain a norm-driven term in estimation precision","feed_subtitle":"New QFI formula adds 16e^{2α}(∂θα)², reduces to the standard bound when the norm is fixed, and is tested on PT-symmetric qubits.","key_machinery":"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.","core_discovery":"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 $m=\\pm 1$, $\\phi=\\pi$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the definition of quantum detection and estimation theory and the symmetric logarithmic derivative framework that the paper extends to non-Hermitian systems.","marker":"[19]"},{"why":"Establishes QFI as the maximum Fisher information over all measurements, the baseline definition used in Eq. (1).","marker":"[20]"},{"why":"Provides the QFI formula for density matrices with arbitrary ranks, used for the pure- and mixed-state reductions.","marker":"[21]"},{"why":"Prior result on fundamental sensitivity limits for non-Hermitian quantum sensors, used as the comparison point for the Naimark-dilated Hermitian system.","marker":"[43]"},{"why":"Schwarz inequality is the starting point of the derivation leading to the non-Hermitian QFI identification $F=\\langle L^{\\dagger}L\\rangle$.","marker":"[50]"},{"why":"Supplies the projected Hilbert space decomposition $|\\Psi\\rangle=e^{\\alpha+i\\beta}|\\psi\\rangle$ used to separate norm and phase dynamics.","marker":"[51]"},{"why":"Supports the treatment of non-Hermitian QFI through optimal measurements and normalized projective states, a comparison point for the paper's definition.","marker":"[53]"},{"why":"Provides the Naimark-dilation construction that embeds the pseudo-Hermitian qubit into a Hermitian two-qubit system.","marker":"[56]"},{"why":"Supplies the PT-symmetric evolution operator and eigenstates used in the Sec. IV calculations.","marker":"[59]"}],"fun_headline_variants":["Quantum precision gets a norm boost in non-Hermitian systems","Non-Hermitian QFI gains norm-velocity term from state evolution","New term in quantum Fisher info from norm speed","Non-Hermitian metrology: norm speed sets new precision bound","Quantum estimation non-Hermitian: norm derivative appears in QFI"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quantum precision gets a norm boost in non-Hermitian systems","Non-Hermitian QFI gains norm-velocity term from state evolution","New term in quantum Fisher info from norm speed","Non-Hermitian metrology: norm speed sets new precision bound","Quantum estimation non-Hermitian: norm derivative appears in QFI"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1301,"prompt_tokens":986,"completion_tokens":315,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":228}},"tokens_in":602,"tokens_out":315,"duration_ms":3222,"temperature":1.0,"reasoning_tokens":228,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:21:04.502953+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Quantum detection and estimation theory,","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of quantum detection and estimation theory and the symmetric logarithmic derivative framework that the paper extends to non-Hermitian systems."},{"cited_title":"Statistical dis- tance and the geometry of quantum states,","cited_arxiv_id":null,"evidence_quote":"Establishes QFI as the maximum Fisher information over all measurements, the baseline definition used in Eq. (1)."},{"cited_title":"Quantum fisher information for density matrices with arbitrary ranks,","cited_arxiv_id":null,"evidence_quote":"Provides the QFI formula for density matrices with arbitrary ranks, used for the pure- and mixed-state reductions."},{"cited_title":"Fundamen- tal sensitivity limits for non-hermitian quantum sensors,","cited_arxiv_id":null,"evidence_quote":"Prior result on fundamental sensitivity limits for non-Hermitian quantum sensors, used as the comparison point for the Naimark-dilated Hermitian system."},{"cited_title":"On the inequality of sums of squares,","cited_arxiv_id":null,"evidence_quote":"Schwarz inequality is the starting point of the derivation leading to the non-Hermitian QFI identification $F=\\langle L^{\\dagger}L\\rangle$."},{"cited_title":"Dynamics of 2×2 matrix non-hermitian quan- tum systems on bloch sphere,","cited_arxiv_id":null,"evidence_quote":"Supplies the projected Hilbert space decomposition $|\\Psi\\rangle=e^{\\alpha+i\\beta}|\\psi\\rangle$ used to separate norm and phase dynamics."},{"cited_title":"Quantum parameter estima- tion of non-hermitian systems with optimal measurements,","cited_arxiv_id":null,"evidence_quote":"Supports the treatment of non-Hermitian QFI through optimal measurements and normalized projective states, a comparison point for the paper's definition."},{"cited_title":"Naimark-dilatedPT- symmetric brachistochrone,","cited_arxiv_id":null,"evidence_quote":"Provides the Naimark-dilation construction that embeds the pseudo-Hermitian qubit into a Hermitian two-qubit system."},{"cited_title":"Complex extension of quantum mechanics,","cited_arxiv_id":null,"evidence_quote":"Supplies the PT-symmetric evolution operator and eigenstates used in the Sec. IV calculations."}],"review_version":1}