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REVIEW 4 major objections 5 minor 69 references

Isocoherent Work Extraction from Quantum Batteries: Basis-Dependent Response

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.16610 v1 pith:6SAYDOX2 submitted 2025-07-22 quant-ph

Shuva Mondal , Debarupa Saha , Ujjwal Sen This is my paper · ORCID
classification quant-ph MSC 81P4081P68
keywords coherence-constrainedmaximalworkquantumbatteriesergotropycoherence$\ell^1$-normofpassivestatesextractionbasisdependence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Sec. V.A, Theorem 3] 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.
  2. [Sec. V.A, Theorem 2] 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.
  3. [Sec. V.A, Result 1 and its use] 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.
  4. [Sec. IV, proof of Theorem 1] 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.
minor comments (5)
  1. [Sec. V.A, Result 1] 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.
  2. [Sec. IV, Corollary 1] 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].
  3. [Sec. V.A.1] The subsection title and text refer to "incoherent passive" states, but the intended notion is "isocoherent passive" states; the terminology should be made consistent.
  4. [Sec. V.B] The sentence "The detail derivation of this expression is given in Appendix A" contains a grammatical error and should read "The detailed derivation...".
  5. [Sec. V.A, proof of Theorem 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the CCMW is defined and computed from first principles, and self-citations are not load-bearing; the main caveat is a correctness issue in the qutrit closed form, not circularity.

full rationale

The paper defines the coherence-constrained maximal work directly as an optimization over states of fixed l1 coherence and coherence-preserving unitaries, and the qubit formula in Theorem 1 is obtained by explicit parameterization and term-by-term maximization rather than by fitting or by defining the answer into the input. The higher-dimensional analysis rests on the numerically observed pure-state sufficiency premise, which the paper itself flags as an assumption and as potentially failing for d>=6 or non-J_z Hamiltonians, so the qutrit closed forms are conditional but not circular. The self-citations in the references are background citations and do not supply any load-bearing premise, uniqueness theorem, or ansatz. The notable concern is not circularity: Theorem 3's expression evaluated at C=0 gives 2/sqrt(3), whereas the paper's own normalization statement requires the CCMW at zero coherence to equal the maximum-minus-minimum eigenvalue difference, namely 2; this indicates an algebraic error in the derivation around Eq. (24), but an internal inconsistency is distinct from an output being equivalent to its inputs by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on the chosen coherence measure, an unproved characterization of coherence-preserving unitaries for qubits, the numerically observed pure-state sufficiency, and an unproved geometric monotonicity property. No novel physical entities are introduced. The normalization 2/(d-1) is a hand-chosen scaling factor that sets the zero-coherence baseline to 2.

free parameters (1)
  • Hamiltonian normalization 2/(d-1) = 2/(d-1)
    Chosen by hand in Eq. (8) so that the CCMW at zero coherence equals 2 for every dimension, providing a common baseline. This scaling affects the numerical value of the CCMW but not the qualitative basis-dependent behavior.
assumptions (4)
  • domain assumption The l1-norm of coherence is the operative measure of quantum coherence throughout.
    The paper explicitly chooses the l1-norm in Sec. II A and builds all results on it. Other coherence measures could lead to different conclusions.
  • ad hoc to paper For a qubit, every coherence-preserving unitary has the form U_C = σ_x^p exp(iβσ_z) σ_x^q.
    This set is stated without proof in Sec. IV and underlies Theorem 1. While the set is likely complete for the optimization performed, the paper does not justify the characterization for a fixed initial state.
  • ad hoc to paper Pure states suffice to attain the CCMW in higher dimensions (Result 1 of Sec. V A).
    This is numerically observed for d=3,4,5,6 with the equispaced Hamiltonian J_z, and is used to derive Theorem 3 and the qutrit off-diagonal expression. The paper itself flags that this may fail for d≥6 or other Hamiltonians.
  • ad hoc to paper The shrinking isocoherent ellipse implies a monotonic decrease of the CCMW.
    In the proof of Theorem 2, the paper claims that as the ellipse contracts with increasing coherence, the difference between the maximum and minimum squared distances from the origin decreases. Area shrinkage alone does not guarantee this, and no rigorous argument is provided.

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Pith. "Pith review of Isocoherent Work Extraction from Quantum Batteries: Basis-Dependent Response." pith.science (2026). https://pith.science/paper/6SAYDOX2

@misc{pith2026250716610,
  author       = {Pith},
  title        = {Pith review of: Isocoherent Work Extraction from Quantum Batteries: Basis-Dependent Response},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SAYDOX2}},
  note         = {Machine review of arXiv:2507.16610}
}
read the original abstract

We identify a connection between quantum coherence and the maximum extractable work from a quantum battery, and to this end, we define the coherence-constrained maximal work (CCMW) as the highest amount of work extractable via coherence-preserving unitaries, optimized over all quantum states with fixed coherence in a given dimension. For qubit systems, we derive an analytical relation between the CCMW and the input coherence, defined with respect to an arbitrary fixed basis. Strikingly, we find that for fixed quantum coherence in the energy eigenbasis, the maximal extractable work decreases with increase of coherence. In contrast, when quantum coherence is with respect to a basis for which the Hamiltonian possesses off-diagonal elements, and has equal diagonal elements, the CCMW increases with the level of quantum coherence. We numerically observe that the basis-dependent response of the CCMW also persists in higher-dimensional quantum systems. Moreover, we show that even in higher dimensions one can derive closed-form relations between the CCMW and the input quantum coherence within certain numerically-assessed conclusions. We also comment on the structure of passive states in an isocoherent scenario, that is, states from which no energy can be extracted under coherence-preserving unitaries.

Figures

Figures reproduced from arXiv: 2507.16610 by the authors.

Figure 1
Figure 1. Unitarily-extracted CCMW from qudit battery with co￾herence, given in energy eigenbasis. The plot depicts the variation of the CCMW, ξd(C˜), with increasing scaled coherence C˜ for battery dimensions d = 2, 3, 4, 5, 6, shown in different colors. It can be ob￾served that, for all the considered dimensions, the CCMW decreases monotonically with increasing coherence. Thus, this plot illustrates that whenever coherence … view at source ↗
Figure 2
Figure 2. Isocoherence ellipses for a qutrit system. The figure shows isocoherent ellipses corresponding to several fixed coherence values C for a qutrit system. Each ellipse is plotted in a different color according to C and is defined by the state parameters x1, x2, and the fixed coherence C. As C increases, the area of each ellipse decreases and eventually vanishes at maximum coherence C = 2. Both axes in the plot are dime… view at source ↗
Figure 3
Figure 3. Variation of ξ p d (C˜) with fixed value of scaled coher￾ence C˜, for qudit systems. The figure depicts the variation of ξ p d (C˜) with the scaled coherence C ∈ ˜ [0, 1], for battery dimension d = 2, 3, 4, 5, 6, plotted in different colors. As it can be seen in the plot that for d = 2, ξ p d (C˜) increases with increase in C ∈˜ [0, 1]. For d ≥ 3, the plots depicts a generic feature exhibiting that ξ p d (C˜) mostly… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Coherence dependence of ξ p d (C˜) for various α¯ values. The figures show how ξ p d (C˜) varies with the scaled coherence C ∈˜ [0, 1] for dimensions d = 2, 3, 4, 5, 6, with coherence fixed in the basis where the Hamiltonian is given by Eq. (30). The four panels corres…

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