{"id":"db8bfbc9-2a08-4c6b-9bfd-50207d1eafd4","arxiv_id":"2507.16610","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"The basis-dependent response of coherence-constrained work extraction is the central claim, but the paper's analytical qutrit formula contradicts its own zero-coherence limit.","lead":"This paper introduces the coherence-constrained maximal work (CCMW), the maximum energy extractable from a quantum battery by unitaries that keep the battery's quantum coherence fixed. It finds that for qubits this work decreases with coherence in the energy eigenbasis but increases when the Hamiltonian has off-diagonal elements, a basis-dependent response that also appears numerically in higher dimensions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's qutrit diagonal-Hamiltonian formula fails the zero-coherence limit: at C=0 it gives 2/√3 instead of the required 2, so the central higher-dimensional closed form is internally inconsistent.","rationale":"The reader's REJECT verdict is supported by the evidence. However, the single most decisive concern is not the pure-state sufficiency assumption identified as the reader's weakest_assumption, but rather an outright mathematical inconsistency in Theorem 3. The paper explicitly asserts that the zero-coherence CCMW for J_z^d equals the difference between the maximum and minimum eigenvalues, which is 2 for Eq. (8) in d=3, while Theorem 3 yields 2/√3 at C=0. This is a concrete failure of the central qutrit closed-form expression, independent of whether pure states suffice to attain the CCMW. The proof's scaled-ellipse equations appear to use the wrong assignment of energy spacings or sign convention, and the tangent-circle argument gives an incorrect extremum. The qubit result and the qualitative idea of basis-dependent response are not refuted by this particular check, so a corrected manuscript might be salvageable, but the current paper does not support its stated higher-dimensional closed-form claims. Therefore I concur with the reader's rejection and recommend no change to the verdict.","tokens_in":22347,"tokens_out":12375,"duration_ms":133825,"concrete_test":"Evaluate Theorem 3 at C=0 and compare to the paper's own zero-coherence baseline: for J_z^3 in Eq. (8), ξ_3(0) must equal 2. Independently solve the d=3 optimization in Eq. (22) with Δϵ_{0,1}=1, Δϵ_{0,2}=2 at C=0, i.e. maximize and minimize X1^2+X2^2 over the ellipse defined by Eq. (16), and check whether the difference is 2. If the formula returns 2/√3, the tangent-circle construction in Eqs. (23)-(24) is using the wrong scaled ellipse and must be re-derived.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing defect is the claimed higher-dimensional closed form, Theorem 3. For the Hamiltonian J_z^3 in Eq. (8), the paper itself states that the CCMW at zero coherence must equal the maximum minus minimum eigenvalue, namely 2. Evaluating Theorem 3 at C=0 gives f3=1 and ξ3(0)=sqrt((1+1+0)/3*(1+1-0))=sqrt(4/3)=2/√3, which does not equal 2. The same contradiction appears in Fig. 1, where the d=3 curve starts at 2. This is not a matter of the unproved pure-state sufficiency assumption; even if Result 1 and ξ^p_3=ξ_3 are granted, the formula is wrong. The origin of the error is in the proof of Theorem 3: Eq. (24) is not the scaled isocoherent ellipse for J_z^3 as defined in Eq. (8). With levels (-1,0,1) and Δϵ=(1,2), the C=0 ellipse contains the point corresponding to |2> at squared distance 2 and the point |0> at the origin, so the correct ξ^p_3(0) is 2. The tangent-circle calculation in Eqs. (23)-(24) instead yields 2/√3. Since Theorem 3 is the main closed-form demonstration in higher dimensions and is used to support the claimed basis-dependent response, this internal inconsistency invalidates the central claim as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines the coherence-constrained maximal work (CCMW) as the maximum energy difference extractable by coherence-preserving unitaries, optimized over all states of fixed l1 coherence in a given dimension. For qubits it derives the closed form ξ2(C) = |h1−h3|√(1−C²) + 2h2C, which decreases with coherence when the Hamiltonian is diagonal in the coherence basis and increases when the Hamiltonian has equal diagonal and nonzero off-diagonal elements. For higher dimensions the paper reports numerical results for d = 3, ..., 6, claims that pure states suffice to attain the CCMW, states a monotonicity theorem for pure states, derives closed-form qutrit expressions (Theorem 3 and Eq. (29)), and characterizes isocoherent passive states. The central higher-dimensional analytical claims are, however, internally inconsistent as written.","tokens_in":22710,"tokens_out":13635,"duration_ms":137163,"significance":"If correct, the qubit result and the basis-dependent response of coherence-constrained work would be a useful addition to quantum battery thermodynamics: the qubit formula is explicit and testable, and the contrast between energy-basis and off-diagonal-basis coherence responses is conceptually interesting. The paper also contains a clear numerical study for higher dimensions and a plausible classification of passive states under state-independent coherence-preserving unitaries. These strengths are outweighed by the fact that the main qutrit closed form is arithmetically inconsistent with the paper's own normalization and figure, and the general monotonicity theorem is not proved by the geometric argument given.","major_comments":[{"comment":"Theorem 3 is internally inconsistent with the normalization introduced after Eq. (8). For d = 3 the Hamiltonian is diag(−1, 0, 1), and the paper states that the CCMW at zero coherence must equal 2. Evaluating the printed expression at C = 0 gives f3 = 1 and ξ3(0) = sqrt((1+1+0)/3)(1+1−0) = 2/√3, not 2. This also contradicts Fig. 1, where the d = 3 curve starts at 2. Moreover, the printed expression does not follow from the proof's own intermediate results: using the proof's x± formulas gives the different expression ξ3(C) = (√(1+C)+√(1−C/3))/√2 · √(1 − C + √(1+C)√(1−C/3)), which at C = 0 equals 2. The theorem as stated must be corrected or removed.","section":"Sec. V.A, Theorem 3"},{"comment":"The proof of Theorem 2 is not valid as written. The argument that the isocoherent ellipse shrinks with increasing coherence does not imply that the difference between its maximum and minimum distances from the origin decreases monotonically: a shrinking ellipse can become more eccentric, and the relevant distances are to an ellipse that is not centered at the origin. The theorem is therefore unproven even for the special Hamiltonian J_z^d, and the numerical evidence in Results 1 and 2 covers only d = 3, 4, 5, 6 for that Hamiltonian, not the general diagonal Hamiltonian ̅H_d considered in the theorem.","section":"Sec. V.A, Theorem 2"},{"comment":"The pure-state sufficiency Result 1 is a numerical observation obtained with the ISRES optimizer, not a proven statement. It is used as a premise for the closed-form Theorem 3 and for Eq. (29), so those formulas are conditional. The paper itself acknowledges in the proof of Theorem 2 that ξ^p_d = ξ_d may fail for d ≥ 6 or for Hamiltonians other than J_z^d; with that caveat, the higher-dimensional closed forms should be presented as numerically supported conjectures rather than established results.","section":"Sec. V.A, Result 1 and its use"},{"comment":"The proof of Theorem 1 asserts without proof that every coherence-preserving unitary on a qubit has the form U_C = σ_x^p exp(iβσ_z) σ_x^q. This parametrization is plausible for the set of unitaries that preserve l1 coherence for all states, but it excludes, for example, arbitrary unitaries acting on states with C = 0, where any unitary preserves zero coherence. Since the theorem's conclusion depends on optimizing over the allowed unitaries, the classification should be proved or its domain of validity stated explicitly.","section":"Sec. IV, proof of Theorem 1"}],"minor_comments":[{"comment":"Result 1 states that pure states suffice for C ∈ [0, (d−1)/2], while the preceding text correctly gives the maximum l1 coherence of a pure d-dimensional state as d−1; the range in Result 1 appears to be a typo.","section":"Sec. V.A, Result 1"},{"comment":"Corollary 1 states that the optimal initial state exists for C ∈ [0, 1/2], but the qubit l1 coherence ranges over [0, 1], and the proof of Theorem 1 uses a ∈ [−√(1−C²), √(1−C²)], so the range should be [0, 1].","section":"Sec. IV, Corollary 1"},{"comment":"The subsection title and text refer to \"incoherent passive\" states, but the intended notion is \"isocoherent passive\" states; the terminology should be made consistent.","section":"Sec. V.A.1"},{"comment":"The sentence \"The detail derivation of this expression is given in Appendix A\" contains a grammatical error and should read \"The detailed derivation...\".","section":"Sec. V.B"},{"comment":"The text says the identification ξ^p_d = ξ_d may fail for d ≥ 6, but Result 1 was reported for d = 3, 4, 5, 6; the statement should read d > 6 to be consistent.","section":"Sec. V.A, proof of Theorem 2"}],"recommendation":"reject","confidential_remarks":"The qubit result appears salvageable, but the paper's broader higher-dimensional claims are undermined by the incorrect Theorem 3 and the invalid proof of Theorem 2. I would not encourage acceptance without a substantial rewrite; a revised manuscript that corrects the qutrit formula, restricts or properly proves Theorem 2, and presents the pure-state sufficiency claim as a numerical conjecture could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the one thing you should know: the qubit theorem is fine, and the basis-dependent response is a real observation; but the paper's higher-dimensional closed-form results are not. Theorem 3 as written fails at C=0.\n\nWhat's actually new: CCMW is a sensible definition, the qubit formula closes the loop between coherence and extractable work under coherence-preserving maps, and the contrast between the energy-basis (decreasing) and off-diagonal-basis (increasing) responses is a useful qualitative design principle. The derivation of Theorem 1 is direct and checks out.\n\nWhere it falls apart: Theorem 3's qutrit expression for the diagonal Hamiltonian. At C=0, the CCMW must equal the max-min eigenvalue difference, which is 2 for J_z^3. The formula gives 2/√3. The error sits in the proof: Eq. (24) is not the scaled isocoherent ellipse for J_z^3; the scaling coefficients are swapped, so the tangent-point calculation is done against the wrong curve. Even if you grant the numerically observed pure-state sufficiency, the formula is wrong.\n\nTwo more caveats, both secondary. Result 1 (pure states suffice for d≥3) is only numerical, with no code or search parameters, so it's not reproducible. And the proof of Theorem 2 rests on a handwavy claim that as the ellipse contracts the gap between max and min distance shrinks; plausible, but not established. Also, I'd flag a possible inconsistency between the appendix's off-diagonal qutrit formula, which at C=2 gives 8α/3, and Fig. 3's endpoint, which looks much lower. That deserves a check.\n\nThe upshot: the qualitative message about basis dependence is likely correct, and the qubit result is publishable. But the paper's headline claim of closed-form qutrit relations is unsupported by internal arithmetic. This is not a reject-and-forget situation; the authors should fix the derivations, either by correcting the scaling or by presenting the numerical results as the higher-dimensional evidence.\n\nWho gets value: quantum thermodynamics people, particularly those working on quantum batteries and coherence-constrained operations. They should read it with the Theorem 3 caveat in mind.\n\nI'd send it to referees rather than desk-reject: the error is precise and fixable, and the qubit part deserves an airing.","headline":"The qubit result is real and the basis-dependent response is a useful insight, but the qutrit closed-form fails at C=0, so the higher-dimensional claim as stated does not hold.","tokens_in":23168,"tokens_out":9204,"would_cite":false,"duration_ms":95971,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P40","81P68"],"pacs":[],"model":"deepseek-v4-flash","headline":"For qubit batteries with fixed coherence, maximal extractable work is $\\xi_2(C)=|h_1-h_3|\\sqrt{1-C^2}+2h_2C$, so whether coherence helps or hurts depends on the basis.","keywords":["coherence-constrained maximal work","quantum batteries","ergotropy","quantum coherence","$\\ell^1$-norm of coherence","passive states","work extraction","basis dependence"],"falsifier":"Evaluate the paper's qutrit diagonal-Hamiltonian formula at zero coherence: for the Hamiltonian in Eq. (8), unitarity forces the CCMW to equal 2 (the difference between the maximum and minimum eigenvalues), whereas the formula from Theorem 3 yields $\\sqrt{4/3}$, so this single evaluation would falsify that closed form or the pure-state identification behind it.","tokens_in":22134,"feed_emoji":"🔋","tokens_out":8906,"duration_ms":89810,"temperature":0.7,"pith_summary":"The paper introduces a quantity called the coherence-constrained maximal work (CCMW): the largest energy a quantum battery can release through unitaries that preserve a fixed amount of quantum coherence, with the initial state also constrained to have that coherence. For a qubit, it proves a closed-form law, $\\xi_2(C)=|h_1-h_3|\\sqrt{1-C^2}+2h_2C$, expressed in the basis where the coherence is fixed. The law has a basis-dependent message: when coherence is measured in the Hamiltonian's energy eigenbasis, the CCMW decreases as coherence grows, whereas when the Hamiltonian has equal or zero diagonal entries and a nonzero off-diagonal entry in that basis, the CCMW increases linearly with coherence. The same qualitative response is reported numerically for dimensions 3 through 6, and closed-form qutrit expressions are derived under the numerically observed assumption that pure states already attain the CCMW. If correct, the result turns coherence into a protocol-relative resource for quantum batteries rather than an unconditional advantage.","feed_headline":"More coherence can raise or lower extractable battery work","feed_subtitle":"For qubits, energy-basis coherence cuts maximal work while off-diagonal coherence boosts it linearly.","key_machinery":"The machinery is the pair of quantities $\\xi_d(C)$ and $\\xi^p_d(C)$: the CCMW over all states, and its restriction to pure states, with coherence measured by the $\\ell^1$-norm $C(\\rho)=\\sum_{i\\ne j}|\\rho_{ij}|$ in a fixed basis. For qubits, every fixed-coherence state is written with a population imbalance $a$ and a phase $\\theta$, and every coherence-preserving unitary has the form $\\sigma_x^p e^{i\\beta\\sigma_z}\\sigma_x^q$, so the optimization separates into an independent population term, $|h_1-h_3|\\sqrt{1-C^2}$, and a phase term, $2h_2C$. In higher dimensions, pure fixed-coherence states are parameterised by points on isocoherent ellipses, the intersection of the unit sphere with the plane $\\sum_i x_i=\\sqrt{1+C}$; as coherence grows the ellipse shrinks, making the difference between farthest and nearest points, and hence the extractable work, monotonically smaller. The closed-form qutrit results are obtained by computing those extremal distances on the scaled ellipse.","core_discovery":"The central discovery is that, under coherence-preserving work extraction, the resource character of coherence is not fixed: it is decided by the relation between the coherence basis and the Hamiltonian. Theorem 1 states that for a qubit the CCMW is $\\xi_2(C)=|h_1-h_3|\\sqrt{1-C^2}+2h_2C$, where $h_1,h_3$ are the diagonal entries and $h_2$ the absolute off-diagonal entry of the Hamiltonian in the coherence basis. This yields two contrasting regimes: an energy-eigenbasis coherence decreases the maximal extractable work, while a basis in which the Hamiltonian is purely off-diagonal (equal or zero diagonal entries) gives $\\xi_2(C)=2h_2C$, a linear increase with coherence. For higher dimensions the paper reports the same basis-dependent response, proved for pure states in the diagonal case via the shrinking of isocoherent ellipses and completed by numerical optimization for $d=3,4,5,6$; closed-form expressions are given for qutrit batteries in both diagonal and off-diagonal Hamiltonian settings. The paper also identifies isocoherent passive states, from which no work can be drawn while coherence is preserved.","pith_inferences":["Worth testing beyond this paper: the basis-dependent response may be an artefact of the $\\ell^1$-norm; replacing it with another coherence quantifier could weaken or reverse the monotonicities, since the closed forms rely on the $\\ell^1$ geometry.","A protocol designer could deliberately choose the coherence basis so that the Hamiltonian has off-diagonal structure, turning coherence into a work-enhancing resource; the paper does not itself propose such a protocol.","The pure-state sufficiency found numerically for $d=3,4,5,6$ suggests mixed-state preparation is unnecessary for maximal extraction in those cases; if a counterexample is found for $d\\ge 6$ or non-equispaced Hamiltonians, the paper's closed-form qutrit formulas would become lower bounds rather than exact CCMW values."],"forward_implications":["For qubits with coherence fixed in the energy eigenbasis, the CCMW equals $|h_1-h_3|\\sqrt{1-C^2}$ and therefore decreases monotonically with coherence, vanishing at maximal coherence.","For qubits with a Hamiltonian whose diagonal entries in the coherence basis are equal or zero, the CCMW equals $2h_2C$ and grows linearly with coherence.","In higher dimensions ($d=3,4,5,6$) the same basis-dependent response is observed numerically: energy-basis coherence lowers the CCMW, while off-diagonal-basis coherence raises it.","When pure states suffice, the qutrit diagonal-Hamiltonian CCMW has the closed form of Theorem 3, and the off-diagonal Hamiltonian case has the piecewise linear form of Eq. (29).","Isocoherent passive states exist for diagonal Hamiltonians: in the energy basis their populations are ordered oppositely to the energy levels, so no coherence-preserving unitary can extract work from them."],"supporting_citations":[{"why":"Defines ergotropy and the unitary work-extraction framework the CCMW is built on.","marker":"[42]"},{"why":"Defines the $\\ell^1$-norm of coherence used throughout as the fixed resource.","marker":"[57]"},{"why":"Supplies the standard characterisation of passive states used for the isocoherent passive-state analysis.","marker":"[43]"},{"why":"Gives the general passive-state conditions for finite quantum systems from which the isocoherent form is adapted.","marker":"[44]"},{"why":"Supplies the rearrangement inequality used to conclude that passive populations are ordered oppositely to the Hamiltonian.","marker":"[67]"},{"why":"Provides the prior link between quantum coherence and ergotropy that this paper extends to the coherence-constrained setting.","marker":"[60]"}],"fun_headline_variants":["Basis decides if coherence helps or hurts battery work","Energy-basis coherence cuts battery work; off-diagonal boosts","Quantum battery: coherence's impact flips based on basis","Coherence can slash or boost battery work depending on basis"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In higher dimensions, the derivation of the closed-form CCMW formulas assumes that the best pure state always gives the same maximal work as the best mixed state with the same coherence, a fact observed numerically for $d=3,4,5,6$ with one family of Hamiltonians but not proven in general.","fun_headline_variants_meta":{"raw":{"variants":["Basis decides if coherence helps or hurts battery work","Energy-basis coherence cuts battery work; off-diagonal boosts","Quantum battery: coherence's impact flips based on basis","Coherence can slash or boost battery work depending on basis"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000294,"raw_usage":{"total_tokens":1745,"prompt_tokens":1011,"completion_tokens":734,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":667}},"tokens_in":627,"tokens_out":734,"duration_ms":8235,"temperature":1.0,"reasoning_tokens":667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:10:33.561160+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the paper's qutrit diagonal-Hamiltonian formula at zero coherence: for the Hamiltonian in Eq. (8), unitarity forces the CCMW to equal 2 (the difference between the maximum and minimum eigenvalues), whereas the formula from Theorem 3 yields $\\sqrt{4/3}$, so this single evaluation would falsify that closed form or the pure-state identification behind it.","supporting_citations":[{"cited_title":"Fault-tolerant quantum simulation of materi- als using bloch orbitals,","cited_arxiv_id":null,"evidence_quote":"Defines ergotropy and the unitary work-extraction framework the CCMW is built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the $\\ell^1$-norm of coherence used throughout as the fixed resource."},{"cited_title":"Versatile millikelvin hybrid cooling platform for superconductivity research,","cited_arxiv_id":null,"evidence_quote":"Supplies the standard characterisation of passive states used for the isocoherent passive-state analysis."},{"cited_title":"Experimental verification of quantum battery capacity with an optical platform,","cited_arxiv_id":null,"evidence_quote":"Gives the general passive-state conditions for finite quantum systems from which the isocoherent form is adapted."},{"cited_title":"Colloquium: Quan- tum coherence as a resource,","cited_arxiv_id":null,"evidence_quote":"Supplies the rearrangement inequality used to conclude that passive populations are ordered oppositely to the Hamiltonian."}],"review_version":1}