{"id":"84c8b5d3-f124-4831-b9d1-b15aae9088b1","arxiv_id":"2507.06030","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Phase-sensing precision for a state mixed with a thermal bath is exactly set by its athermality; for light this is the new measure 'latent coherence'.","lead":"This paper derives the exact maximum precision for measuring a phase when a quantum state is combined with a thermal bath under energy-conserving interactions: the precision is fixed by the state's athermality. For light, this yields a new measure of coherence that remains meaningful when the second interferometer input is warm rather than vacuum.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's qubit-bath construction only approximates exponential degeneracy for energy shifts on its lattice; for generic incommensurate system spectra, the proof that Eq. (2) is the true supremum is incomplete.","rationale":"I examined the derivation of Theorem 1 (Eq. 2) and the optical Theorem 3 (Eq. 5). The optical result is exact and its proof is self-contained: the cosine-sine decomposition reduces any passive linear unitary to a single balanced beam splitter, and the QFI evaluation in Appendix E is a direct sum. No concern there. The finite-dimensional theorem is the core. Its proof has two steps: (i) apply Lemma 1 blockwise; (ii) optimize over bath degeneracies via Lemma 2. Step (i) is sound: energy conservation gives a block-diagonal U, and Ref. [40] supplies the blockwise maximum. Step (ii) contains the gap described above: the explicit B' construction only delivers the required degeneracy scaling for lattice-matched energy shifts, while the theorem asserts it for arbitrary HS. The paper's own text acknowledges the need to 'effectively treat the bath as a continuum' but the discrete qubit model does not rigorously implement a continuum limit for exact energy conservation. The step-function integral may well be correct - the append-a-bath argument suggests any competing bath can be beaten by an approximately exponential one - but the proof as written does not settle the incommensurate case. Therefore the correctness of Eq. (2) for generic systems is not fully established. This differs from the reader's stated weakest assumption: the reader assumes the finite-bath approximation works and worries about attainability; I worry that the approximation itself is unproven for generic spectra, which would make Eq. (2) potentially the wrong value of the supremum, not merely an unattained one. The reader's ACCEPT is reasonable for a physics preprint, but the central theorem's proof has a testable gap. I recommend CONDITIONAL: accept subject to either a rigorous continuum-bath proof of Lemma 2 or a numerical demonstration for an incommensurate three-level system. The optical Eq. (5), the qubit special case, and the resource-theoretic properties are unaffected.","tokens_in":27729,"tokens_out":24741,"duration_ms":268305,"concrete_test":"Test the missing limit analytically or numerically. Analytical: replace the discrete qubit bath in Lemma 2 by a bath with a continuous spectrum and a smooth density of states rho(E) proportional to e^{beta E} on a finite interval [0,E_max], re-derive Eq. (B13) as an integral over E, and take E_max to infinity. Check whether the Riemann-sum limit reproduces Eq. (2) with the same chi functions for a system with energies {0,1,sqrt(2)}. Numerical: for that same system, build finite baths from N harmonic oscillators with rationally independent frequencies (e.g., frequencies scaled as 1/N and sqrt(2)/N plus a comb to make the spectrum dense), compute the max QFI by diagonalizing each energy block and using Lemma 1, and study convergence to Eq. (2) as N grows. If the sequence does not approach Eq. (2), the theorem's statement needs qualification.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that Eqs. (2) and (5) give the exact optimal QFI under energy-conserving unitaries with a thermal bath. For Eq. (2), 'exact' requires that the supremum over all baths equals the step-function integral. The proof in Appendix B rests on Lemma 2, which asserts that any finite bath can be appended with B' so that the combined degeneracy satisfies D~(E+epsilon) approx D~(E)e^{beta epsilon} for all system energies epsilon, with high probability. The construction in Appendix C takes n qubits with a common gap xi. Its calculation (Eq. C7) assumes epsilon/xi = l is an integer; then D'(E+epsilon)/D'(E) approx e^{beta epsilon}. If a system energy gap epsilon_i is not an integer multiple of xi, then D'(E+epsilon_i)=0 for all lattice energies E of B', so the combined degeneracy D~(E+epsilon_i) also vanishes, while D~(E)e^{beta epsilon_i} > 0. The approximation is then false, not just imprecise. Choosing xi smaller makes the lattice denser but still cannot represent a fixed incommensurate epsilon_i exactly; the block decomposition (B9) then lacks the term |epsilon_i> tensor |E-epsilon_i>_B because E-epsilon_i is not a bath energy, so that channel is simply absent. The text's remark that xi is 'chosen small enough that all gaps are well approximated' does not repair exact energy conservation. Consequently, the proof that the supremum over baths equals Eq. (2) has a missing argument for generic (non-commensurate) system Hamiltonians. This is more than the attainability caveat: if the limit cannot be approached as claimed, Eq. (2) could overestimate the true supremum rather than merely being an unattained bound. It is the single most load-bearing point because it targets the proof of the main theorem itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies phase estimation when a finite-dimensional diagonal probe state is coupled to a thermal bath by an energy-conserving unitary, and defines F_T as the supremum of the quantum Fisher information (QFI) over all baths and all such unitaries. The central result, Eq. (2), expresses F_T as an integral of step functions built from the system energies and beta-ordered Boltzmann-weighted probabilities. For linear optics, Eq. (5) states that the optimal QFI is (hbar omega)^2 (nbar_r + 1) C_r(rho), with C_r(rho) = sum_{n>=1} f(p_n, r p_{n-1}) n, achieved by a single balanced beam splitter with one thermal ancilla mode. The paper also derives a speed-limit analogue Eq. (10), proves monotonicity of F_T under thermal operations, and relates C_r to nonclassicality, to quantum illumination, and to standard optical coherence.","tokens_in":28075,"tokens_out":24820,"duration_ms":283447,"significance":"If the finite-dimensional proof is completed, Eq. (2) is a rare closed-form solution to an optimization over arbitrary thermal baths and all energy-conserving unitaries, and it gives a precise operational meaning to athermality as a resource for phase sensing. The optical result Eq. (5) is self-contained, directly derived, and much stronger: it identifies an explicit optimal protocol (one balanced beam splitter) and a computable, convex, monotone quantity C_r that also serves as a nonclassicality witness and satisfies a clean complementarity relation with quantum illumination (Theorem 5). The paper is carefully structured, with explicit block decompositions, use of the external Lemma 1 from Ref. [40], and several cross-checked temperature limits. These strengths are substantial and make the manuscript potentially valuable for quantum thermodynamics and quantum metrology.","major_comments":[{"comment":"The proof of Lemma 2 only treats bath energy shifts that are integer multiples of the qubit gap xi: Eq. (C7) sets epsilon/xi = l and computes ln[D'(E+epsilon)/D'(E)]. If a system gap epsilon_i is not an integer multiple of xi, then E+epsilon_i is not an energy of B' for any lattice energy E, so D'(E+epsilon_i)=0 and the combined degeneracy ratio is 0 rather than approximately e^{beta epsilon_i}. The sentence in Appendix C that xi is 'chosen small enough that all gaps are well approximated' does not repair this, because [U,H_S+H_B]=0 is an exact constraint: the term |epsilon_i>_S tensor |E-epsilon_i>_B in Eq. (B9) is simply absent when E-epsilon_i is not a bath energy. Consequently the rescaling argument leading from Eq. (B13) to Eq. (B14), and hence Eq. (2) and Eq. (10), is not justified for a generic non-commensurate system spectrum. A complete proof needs either a different bath construction whose spectrum is closed under addition of every system gap (for example, a multi-species qubit bath with independent gaps) with degeneracy slope approaching beta in the relevant energy window, or an explicit limiting argument showing that the supremum equals the step-function integral; alternatively, Theorem 1 must be restricted to commensurate spectra. As written, Eq. (2) is proven for qubits and for spectra commensurate with a single gap, but not for the claimed fully general finite-dimensional case.","section":"Appendix C, Lemma 2; Eq. (C7); Eq. (B14)"},{"comment":"The stated characterization that F_T vanishes precisely when p_i e^{beta epsilon_i} = p_j e^{beta epsilon_j} for every pair, equivalently rho_S = gamma_S, is false when H_S has degenerate energy levels. For example, take d=2 with epsilon_0=epsilon_1=0 and p_0 != p_1; then rho_S != gamma_S but H_S=0, so the phase encoding is trivial and F_T=0, and Eq. (2) indeed gives 0 because chi^down_epsilon is identically zero. The proof in Appendix B, around Eq. (B23), tries to handle epsilon_i=epsilon_j by taking a doubly degenerate bath level, but on the corresponding block H_S is proportional to the identity, so Lemma 1 yields zero QFI. The theorem and the resource-theoretic interpretation should either assume a nondegenerate spectrum or reformulate the zero set of F_T in terms of energy shells (equality of coarse-grained Gibbs occupations).","section":"Main text after Eq. (2); Appendix B, §B5, property (1)"}],"minor_comments":[{"comment":"The equality case of the bound Eq. (8) is said to be characterized by tr(rho a rho a^dagger)=0. As written, this trace vanishes for every number-diagonal state, so it cannot characterize the states with support on only even or odd photon numbers. The intended condition appears to be tr(rho a^dagger rho a)=0, i.e., sum_n p_n p_{n+1}(n+1)=0, which is equivalent to p_n p_{n-1}=0 for all n.","section":"Appendix E and H"},{"comment":"There are several typographical errors that should be corrected: 'intially', 'maxmise', 'annihiliation', 'indispensible', 'enocde', 'miminum', and 'ths work' in the introduction, preliminaries, and conclusion.","section":"Throughout"},{"comment":"The Fisher difference f(a,b) is defined in the main text with f(a,0)=a, but the convention f(0,0)=0 is only stated later in Appendix A; it would help to state this convention at first use so that the step-function integrand is unambiguous.","section":"Introduction, Eq. (2)"},{"comment":"The proof of Theorem 3 implicitly sets hbar omega = 1 in the appendices, while the main-text Eq. (5) restores the factor (hbar omega)^2. A brief note to this effect would prevent confusion when comparing the appendices with the main text.","section":"Appendix E"}],"recommendation":"major_revision","confidential_remarks":"The central finite-dimensional theorem is elegant and likely correct, but the proof as written has a genuine gap for non-commensurate spectra (Lemma 2) and a smaller but definite error in the claimed zero characterization for degenerate Hamiltonians. I do not view these as fatal: a technically more careful bath construction or an explicit commensurability restriction would address the first issue, and a nondegeneracy assumption or shell-wise statement would address the second. The optical section is essentially independent and appears sound, so the paper's most immediately usable results are already in good shape. The self-citations are used appropriately for technical reductions rather than as circular support."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi [Name],\n\nQuick take: this is a genuinely useful paper, not a crank submission. The optical result, Eq. (5), is clean and I'd bet on it. The finite-dimensional formula, Eq. (2), is the headline, and its proof has a gap that needs attention before the claim 'exact for arbitrary finite-dimensional systems' can stand.\n\nWhat's new: a closed-form expression for the optimal QFI for phase sensing with a thermal bath under energy conservation, a resource-theoretic measure of athermality (latent coherence C_r) with a nice nonclassicality witness, and a trade-off with quantum illumination. The appendices are unusually careful: block decomposition, use of Lemma 1 from Fiderer et al., qubit and temperature limits all check out. The optical derivation via one beam splitter is elegant and the Poisson/Fock/thermal examples are useful.\n\nThe soft spot is Lemma 2. The construction uses n qubits with gap xi, and the proof of exponential degeneracy assumes epsilon/xi = l is an integer. For a system with an incommensurate energy gap, D'(E+epsilon)=0 for every lattice energy E of the appended bath, so the combined degeneracy D~(E+epsilon) also vanishes. Making xi small doesn't repair this: energy conservation is exact, and the missing channels simply aren't there. So the argument that finite baths can approach the supremum in Eq. (2) arbitrarily closely is incomplete for generic (non-commensurate) system Hamiltonians. This is not just a note about attainability; the supremum could in principle be lower than Eq. (2). I don't have a counterexample, and the formula may well be correct, but the proof as written doesn't establish it. The optical result is unaffected because there the bath modes have the same frequency, so commensurability is built in.\n\nMinor issues: the proof that FT=0 iff rho=gammaS is sketchy in the degenerate-energy case, and there's no numerical cross-check. Both are minor.\n\nBottom line: worth a serious referee. The referee should be asked specifically to either fix Lemma 2 or restrict the theorem's statement to commensurate spectra / continuous baths. I'd accept for review on the strength of the optical result and the resource-theoretic framework even if the finite-dimensional theorem needs qualification.\n\nBest,\n[Name]","headline":"Strong paper with a real proof gap: the optical result is solid, but the finite-dimensional formula's proof needs a fix for incommensurate spectra.","tokens_in":28656,"tokens_out":7938,"would_cite":true,"duration_ms":101527,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the optimal phase-sensing precision of a quantum probe combined with a thermal bath is governed exactly by the probe's athermality, and gives closed-form bounds in finite dimensions and in linear optics.","keywords":["quantum Fisher information","phase sensing","athermality","thermal operations","interferometry","latent coherence","nonclassicality witness","quantum speed limit"],"falsifier":"Take the qubit case with a finite ladder bath of $K$ levels: the derivation in Appendix B shows the QFI saturates $f(p_0e^{-\\beta\\epsilon},p_1)\\epsilon^2$ only as $K\\to\\infty$ through the probability factor $P_B(E<E_{\\rm max}-\\epsilon)$. If a finite or physically constrained bath were demonstrated to exceed this bound, or if the best achievable QFI in an engineered finite bath fell systematically below it for finite excitation probabilities, the claim that Eq. (3) is the optimal precision would be falsified.","tokens_in":27521,"feed_emoji":"🌟️","tokens_out":10306,"duration_ms":119851,"temperature":0.7,"pith_summary":"Thermal backgrounds are normally treated as a nuisance in interferometry: the second input port is assumed to be vacuum. This paper shows that when the background is genuinely thermal, the athermality of the probe state is the resource that sets the phase-sensing precision. For any finite-dimensional system, the optimal quantum Fisher information under energy-conserving interactions with a thermal bath is an exact step-function integral of a \"Fisher difference\" over Boltzmann-weighted populations; in linear optics it reduces to a latent coherence $C_r(\\rho)=\\sum_{n\\ge 1} f(p_n, r p_{n-1})\\,n$ achieved by one balanced beam splitter. The same technique bounds the speed of system–bath evolution. A sympathetic reader should take away that brightness and mean photon number are replaced, at nonzero temperature, by a rigorously defined notion of athermality, one that can even witness optical nonclassicality.","feed_headline":"Athermal probes set the phase-precision limit in thermal baths","feed_subtitle":"Exact Fisher-information bounds tie phase sensitivity to a probe's departure from thermal equilibrium.","key_machinery":"The carrying object is the Fisher difference $f(a,b)=(a-b)^2/(a+b)$, used in a blockwise optimization over conserved total energy. Because an energy-conserving unitary is block-diagonal in total energy, the maximal-QFI lemma from Ref. [40] applies independently to each block; optimizing the bath then leads to idealized baths whose degeneracies scale exponentially, $D(E+\\epsilon)=D(E)e^{\\beta\\epsilon}$, which turns exact sums into the step-function integral of Eq. (2). In the optical setting the same machinery reduces to the latent coherence $C_r$, a convex, thermal-operation-monotone functional of the photon-number distribution that pairs adjacent Fock coefficients through $f(p_n, r p_{n-1})$; a single balanced beam splitter suffices to attain it, and the associated optimal measurement is a reweighted quadrature observable that reduces to homodyne detection at zero temperature.","core_discovery":"The central claim is that the optimal phase sensitivity obtainable by mixing a phase-invariant probe state with a thermal bath, under only global unitarity and energy conservation, is exactly a function of the probe's athermality. For a finite-dimensional system with Hamiltonian $H_S=\\sum_i \\epsilon_i |\\epsilon_i\\rangle\\langle\\epsilon_i|$ and state $\\rho_S=\\sum_i p_i |\\epsilon_i\\rangle\\langle\\epsilon_i|$, the maximal QFI is $F_T(\\rho_S,H_S)=\\frac{1}{2}\\int_0^{Z_S} dx\\, f(\\chi^\\downarrow_q(x),\\chi^\\uparrow_q(x))(\\chi^\\downarrow_\\epsilon(x)-\\chi^\\uparrow_\\epsilon(x))^2$, where $f(a,b)=(a-b)^2/(a+b)$ and the step functions $\\chi^\\downarrow,\\chi^\\uparrow$ encode Boltzmann-weighted probabilities and energies ordered against the partition function $Z_S$. This vanishes if and only if $\\rho_S$ is thermal, and is a monotone under thermal operations. For a single optical mode at frequency $\\omega$ mixed with thermal modes through linear optics, the result is $(\\hbar\\omega)^2(\\bar n_r+1)C_r(\\rho_S)$ with $C_r(\\rho_S)=\\sum_{n\\ge 1} f(p_n, r p_{n-1})\\,n$, $r=e^{-\\beta\\hbar\\omega}$, attained by one balanced beam splitter; at high temperature $C_1(\\rho)=\\tfrac{1}{2}F(\\rho,x)$, making the same quantity a witness of $P$-function nonclassicality via $C_r(\\rho_{\\rm cl})\\le (1-r)\\langle N\\rangle + r$. The same derivation gives the maximal interaction speed, $F^{\\rm int}_T(\\rho_S,H_S)=\\frac{1}{2}\\int_0^{Z_S} dx\\, f(\\chi^\\downarrow_q(x),\\chi^\\uparrow_q(x))$, so that athermality alone bounds how fast a system and bath can evolve together.","pith_inferences":["Because the optimum requires a bath with exponentially growing degeneracies, realistic finite reservoirs will leave a gap; computing finite-bath corrections (e.g., truncating the energy ladder) could turn Eqs. (2), (5), and (10) into practically useful upper bounds and test how close engineered baths can come.","The latent coherence $C_r$ may be directly measurable in microwave or circuit-QED interferometers, where $\\bar n_r$ is sizeable, using the reweighted quadrature strategy; this would extend interferometric phase sensing to regimes where vacuum-input assumptions fail.","The same formalism could be re-run for multiple copies or multi-mode resource states, where mode correlations (like those in filtered laser light) may modify the trade-off between phase sensing and illumination precision.","The paper leaves open whether the athermality speed limit is tight for practical open-system examples; studying specific master equations could settle that, and would connect the result to concrete clock and thermalization processes."],"forward_implications":["If $F_T$ is the true optimum, then no energy-conserving unitary with any thermal bath at temperature $T$ can yield phase precision beyond Eq. (2); any reported sensitivity above it would indicate a missing resource such as coherence or an external phase reference.","The optical bound is tight for a single thermal mode and a balanced beam splitter, so a standard Mach–Zehnder with a thermal second port and a reweighted homodyne measurement is already optimal among all linear-optics networks.","The monotonicity of $F_T$ under thermal operations means partial thermalization can only degrade sensing precision, so probe preparation and the sensing interaction should be kept separate from any thermalizing contact.","At any nonzero background temperature, the latent coherence gives a nonclassicality witness: a measured $C_r$ above $(1-r)\\langle N\\rangle + r$ certifies $P$-function nonclassicality without requiring an external phase reference.","The athermality speed limit bounds system–bath evolution speed purely from the initial state's temperature mismatch, giving a thermodynamic speed limit for open-system dynamics."],"supporting_citations":[{"why":"Supplies Lemma 1, the maximal quantum Fisher information for a fixed spectrum under any unitary, applied blockwise to each total-energy subspace.","marker":"[40]"},{"why":"Provides the thermal-operations framework and β-ordering used to construct the optimal exponentially-degenerate bath and to prove F_T is a thermal-operation monotone.","marker":"[2]"},{"why":"Shows any passive linear-optical unitary on signal plus thermal modes reduces to a single beam splitter with one ancilla, making Eq. (5) an exact maximum.","marker":"[41, 42]"},{"why":"Gives the upper bound on QFI for classical phase-invariant states used to prove the latent-coherence nonclassicality witness.","marker":"[43]"},{"why":"Provides the quantum-illumination metrology QFI whose complementarity with latent coherence yields the trade-off relation.","marker":"[49]"},{"why":"Establishes the quantum Cramér–Rao bound that makes QFI the operational measure of phase precision.","marker":"[31]"},{"why":"Supplies the QFI expression and monotonicity properties used in the block-decomposition derivations.","marker":"[61]"},{"why":"Gives the state-space metric interpretation of square-root QFI, which justifies treating the interaction speed as a QFI quantity.","marker":"[24]"}],"fun_headline_variants":["Athermality alone sets quantum phase-sensing bound","Out-of-equilibrium probe fixes phase sensitivity limit","Phase precision bound from probe's athermality","Athermal resource dictates quantum phase limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bounds are suprema over idealized thermal baths whose degeneracies grow exponentially with energy, $D(E+\\epsilon)=D(E)e^{\\beta\\epsilon}$, a property no finite physical bath can exactly satisfy, so for real finite baths the formulas are approached only in the limit of large, finely tuned reservoirs.","fun_headline_variants_meta":{"raw":{"variants":["Athermality alone sets quantum phase-sensing bound","Out-of-equilibrium probe fixes phase sensitivity limit","Phase precision bound from probe's athermality","Athermal resource dictates quantum phase limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000562,"raw_usage":{"total_tokens":2720,"prompt_tokens":1051,"completion_tokens":1669,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":667,"completion_tokens_details":{"reasoning_tokens":1609}},"tokens_in":667,"tokens_out":1669,"duration_ms":15366,"temperature":1.0,"reasoning_tokens":1609,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:18:26.146593+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the qubit case with a finite ladder bath of $K$ levels: the derivation in Appendix B shows the QFI saturates $f(p_0e^{-\\beta\\epsilon},p_1)\\epsilon^2$ only as $K\\to\\infty$ through the probability factor $P_B(E<E_{\\rm max}-\\epsilon)$. If a finite or physically constrained bath were demonstrated to exceed this bound, or if the best achievable QFI in an engineered finite bath fell systematically below it for finite excitation probabilities, the claim that Eq. (3) is the optimal precision would be falsified.","supporting_citations":[{"cited_title":"Thermodynamically consistent collisional master equation in a low-density gas with internal structure","cited_arxiv_id":"2506.21394","evidence_quote":"Supplies Lemma 1, the maximal quantum Fisher information for a fixed spectrum under any unitary, applied blockwise to each total-energy subspace."},{"cited_title":"Narasimhachar, S","cited_arxiv_id":null,"evidence_quote":"Gives the upper bound on QFI for classical phase-invariant states used to prove the latent-coherence nonclassicality witness."},{"cited_title":"Lloyd, Enhanced Sensitivity of Photodetection via Quantum Illumination, Science 321, 1463 (2008)","cited_arxiv_id":null,"evidence_quote":"Provides the quantum-illumination metrology QFI whose complementarity with latent coherence yields the trade-off relation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the quantum Cramér–Rao bound that makes QFI the operational measure of phase precision."},{"cited_title":"Quantum thermodynamics with coherence: Covariant Gibbs-preserving operation is characterized by the free energy","cited_arxiv_id":"2406.06234","evidence_quote":"Supplies the QFI expression and monotonicity properties used in the block-decomposition derivations."},{"cited_title":"Morris, B","cited_arxiv_id":null,"evidence_quote":"Gives the state-space metric interpretation of square-root QFI, which justifies treating the interaction speed as a QFI quantity."}],"review_version":1}