Distribution Relationship of Quantum Battery Capacity

Tinggui Zhang; Xiaofen Huang; Yiding Wang

REVIEW 4 major objections 5 minor 63 references

Distribution Relationship of Quantum Battery Capacity

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Quantum battery capacity obeys a monogamy bound for X-states: the whole system always stores at least as much as the sum of its parts.

desk verdict A plausible two-qubit capacity monogamy result with a real proof gap in the n-qubit extension; refereeable, but the n-qubit theorem is not established as written. read the letter →

arxiv 2502.02860 v1 pith:2WD7QPDL submitted 2025-02-05 quant-ph

Yiding Wang , Xiaofen Huang , Tinggui Zhang This is my paper · ORCID

classification quant-ph PACS 04.70.Dy03.65.Ud04.62.+v

keywords quantumbatterycapacitymonogamyofX-statesresidualmajorizationunitaryevolutionthree-qubitX-stateGHZstate

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

This paper tries to establish a monogamy-type inequality for quantum battery capacity: for an n-qubit X-state, the battery capacities of the individual qubits add up to no more than the capacity of the whole system. The authors prove the two-qubit case in full, define the leftover amount as residual battery capacity (RBC), split it into coherent and incoherent parts, and show that a global unitary can move part of it into the subsystems. They assert the n-qubit generalization as a theorem and give stronger three-qubit inequalities, together with counterexamples to some intuitively plausible three-qubit relations. The interest is practical: if true, it bounds how much storable energy can be localized in parts of a quantum battery and suggests unitary redistribution protocols.

What carries the argument

The workhorse is the battery capacity functional $C(\rho;H)=\sum_{i=0}^{d-1}\varepsilon_i(\lambda_i-\lambda_{d-1-i})$ for eigenvalues sorted decreasingly, which is Schur-convex in the state's spectrum. Lemma 1 (diagonal majorization by eigenvalues) turns any X-state inequality into an incoherent-state inequality; Lemma 2 performs the explicit two-qubit incoherent calculation using the four Hamiltonian eigenvalues $\pm\sqrt{(\varepsilon_A+\varepsilon_B)^2+\gamma^2}$ and $\pm\sqrt{(\varepsilon_A-\varepsilon_B)^2+\gamma^2}$. Lemma 3, the unproved $n$-qubit extension of Lemma 2, is the step that Theorem 3 leans on.

What would settle it

Take a specific three-qubit incoherent X-state, such as diagonal entries $(8,7,4,3,2,1,6,5)/36$, and compute $\sum_i C(\tau_{A_i};H_{A_i})$ versus $C(\tau;H)$ for small $\gamma>0$. If any such state violates Eq. (14), the claimed $n$-qubit monogamy relation is false; the equality-ordering conditions can also be checked by brute force for $n=3$.

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Extended reading notes

Core claim

The central claim is that battery capacity distributes like an entanglement-type monogamy: $C(\rho_{A_1};H_{A_1})+\cdots+C(\rho_{A_n};H_{A_n}) \le C(\rho;H)$ for every $n$-qubit X-state $\rho$ under the Hamiltonian $H = \sum_i \varepsilon_{A_i}\sigma_3^{(i)} + \gamma\sigma_1^{\otimes n}$. For two qubits the proof runs through majorization: the diagonal of $\rho$ is majorized by its spectrum, so the decohered state $\tau$ has no larger capacity, and an explicit incoherent-state inequality for $\tau$ gives the bound. Equality occurs for incoherent states with certain diagonal orderings when $\gamma=0$. The difference $\Delta C$ is decomposed as $\mathrm{RBC}_{\mathrm{ic}} + \mathrm{RBC}_{\mathrm{c}}$, and there is always a unitary that raises the subsystem capacity sum, at an efficiency that decreases as the interaction parameter $\gamma$ grows.

Load-bearing premise

The n-qubit incoherent inequality (Lemma 3) is asserted without proof; Theorem 3 collapses if for some $n\ge 3$ an incoherent X-state has subsystem capacity sum larger than total capacity.

Editorial extensions

If this is right

Within X-states, the battery capacity that can be stored in individual qubits is limited by the total capacity, so no local charging protocol can make subsystem capacities exceed the whole.
The equality conditions for $\gamma=0$ identify incoherent states whose entire battery capacity is stored locally, with no residual capacity.
The decomposition of residual battery capacity into coherent and incoherent parts gives a quantitative measure of how much storage is due to coherence versus incoherent population differences.
A global unitary can transfer part of the residual capacity into subsystems without changing the total capacity, providing an explicit redistribution protocol.
For three-qubit X-states, stronger distributive relations hold: the total capacity dominates each bipartite-plus-single capacity plus the coherent residual.
Some intuitively plausible three-qubit relations, such as $C(\rho_{ABC}) \ge C(\rho_{AB})+C(\rho_{AC})-C(\rho_A)$, fail, and explicit counterexamples are given.

Reading between the lines

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

If Lemma 3 fails for some $n>2$, the claimed $n$-qubit monogamy relation would collapse; a direct numerical scan of random incoherent three-qubit X-states could test Eq. (14) independently of the two-qubit proof.
The capacity-redistribution unitary in Theorem 4 resembles a state-correlation relocation operation; it is plausible that similar gains exist for non-X states, though the present proof does not cover them.
The equality-order conditions for $n>2$ might characterize states with vanishing coherent residual capacity, which would give a clean operational meaning to zero coherent RBC beyond three qubits.
The capacity-transfer efficiency formula in Example 3 suggests an experimentally testable trade-off: increasing the coupling $\gamma$ reduces how much residual capacity can be extracted into subsystems, which could be measured in a small multi-qubit battery.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline is that the paper has a genuine, modest result: for two-qubit X-states, battery capacity is monogamous under the Hamiltonian in Eq. (2), and the proof via majorization (Lemma 1) plus Schur-convexity (Prop. 1) is essentially one line and correct. The RBC split into coherent/incoherent parts is a useful way to account for where the capacity difference lives, and the unitary-optimization examples are concrete. For that part, the paper earns a serious look.

The soft spots are in the generalizations. Lemma 2's proof contains a false bound: the inequality 2(α1+α3−α2−α4)εB ≥ 2|ρ11+ρ33−ρ22−ρ44|εB does not hold for all labelings; the reader's counterexample with diagonal (10,8,9,1)/28 works. The lemma itself still looks true with a different ordering argument, so this is a fixable proof bug, not a false theorem. More important, Lemma 3 (the n-qubit incoherent-state inequality) is asserted with no proof: the paper says "the result in Lemma 2 still holds" and moves on. Theorem 3 rests directly on it, so the n-qubit monogamy claim is not established as written. The stress-test note gives an independent convexity argument that the lemma is true, which supports the conclusion but does not appear in the paper. In a revision, that proof has to be supplied. Theorem 5's proof also maximizes C(τAB)+C(τC) at a particular grouping without justifying that maximum; the claim may be true but the step is not rigorous as written.

The citation pattern is fine; [23] is the legitimate source of the capacity definition and Schur-convexity, and the self-citations are contextual.

Bottom line: this is for the quantum-battery community and deserves a serious referee. The two-qubit result is correct and the n-qubit extension is probably correct but currently proofless at a load-bearing point. I would send it to peer review with major-revision expectations: fix Lemma 2's proof, prove Lemma 3, and tighten Theorem 5.

Referee Report

4 major / 5 minor

Summary. The manuscript studies the distribution of quantum battery capacity, defined in Eq. (1) from ref. [23], for X-states and the Hamiltonian (2)/(13). For two-qubit X-states it claims C(ρ_A;H_A)+C(ρ_B;H_B) ≤ C(ρ_AB;H_AB) (Theorem 1), with equality for incoherent states whose diagonal elements satisfy one of the orderings in Eq. (10). The difference is introduced as residual battery capacity (RBC) and split into incoherent and coherent parts. The paper then claims an n-qubit generalization (Theorem 3, Eq. (16)), stronger three-qubit inequalities (Theorem 5), and unitary protocols that transfer residual capacity to subsystems (Theorems 2 and 4). The proofs rely on Schur-convexity (Prop. 1), the majorization of diagonal elements by eigenvalues (Lemma 1), and two lemmas on the incoherent-state capacity inequality (Lemmas 2 and 3). Examples with Bell-diagonal and white-noise GHZ states illustrate the concepts.

Significance. The paper introduces a useful conceptual tool, the residual battery capacity and its coherent/incoherent decomposition, and proposes a monogamy-type inequality for battery capacity, a quantity of growing interest in quantum thermodynamics. If rigorously established, the inequalities would be a new distribution relation for a thermodynamic resource and would complement existing monogamy results for entanglement and other correlations. The paper is largely self-contained, uses standard majorization techniques, and provides concrete worked examples. The main weakness is rigor: the proof of the key two-qubit lemma contains an invalid inequality, and the n-qubit generalization is asserted without proof. The framework is promising, but the central theorems are not established as written.

major comments (4)

[Section II, Lemma 2 (proof)] The proof of Lemma 2 contains a false coefficient-wise inequality. After sorting the diagonal elements as α1 ≥ α2 ≥ α3 ≥ α4, the paper claims 2(α1+α3−α2−α4)ε_B ≥ 2|ρ11+ρ33−ρ22−ρ44|ε_B. This is not true in general. For example, take (ρ11,ρ22,ρ33,ρ44) = (10,8,9,1)/28, so that α1=10/28, α2=9/28, α3=8/28, α4=1/28. Then α1+α3−α2−α4 = 8/28, while |ρ11+ρ33−ρ22−ρ44| = 10/28. Thus the chain of inequalities in Lemma 2 is invalid, and because Theorem 1 invokes Lemma 2, the proof of Theorem 1 is incomplete as written. The statement of Lemma 2 may be salvageable, but a correct proof is required; the subsequent claim that Eq. (9) is equivalent to the four orderings in (10) also depends on this invalid step and should be re-derived.
[Section III, Lemma 3 and Theorem 3] Lemma 3, the n-qubit generalization of the incoherent-state inequality, is stated without proof: the text says 'the result in Lemma 2 still holds in the n-qubit system' and then asserts Eq. (14). This lemma is load-bearing: Theorem 3's proof uses Lemma 3 as the second inequality to obtain Eq. (16). The spectrum of the n-qubit Hamiltonian (13) does not reduce to the simple two-pair structure used in the two-qubit proof, so the generalization is not automatic. A complete proof of Lemma 3 must be supplied, or Theorem 3 should be reformulated as a conjecture supported by evidence.
[Section III, Theorem 4] Theorem 4 is stated without a proof; the sentence 'Similar to the proof of Theorem 2' is not a derivation. The theorem claims that for any n-qubit X-state there exists a unitary evolution that yields a subsystem capacity gain. The two-qubit proof relies on an explicit unitary, but the n-qubit case requires a general construction or existence argument, including the claim that a product of unitaries can always transform the diagonal elements into one of the 2^n orderings that saturate Eq. (14). This should be proved explicitly.
[Section III, Theorem 5 (proof)] In the proof of Theorem 5, the maximization step is asserted without proof: 'the case where C(τ_AB;H_AB)+C(τ_C;H_C) takes the maximum value, which corresponds to µ1+µ2 = ω1+ω2+ω3+ω4, ...' and the subsequent expression for C(τ_C;H_C). Since the theorem's three inequalities (22) depend directly on this characterization of the maximum, a rigorous argument (or a citation) is needed to show that the proposed pairing of the diagonal elements indeed maximizes the sum over all permutations.

minor comments (5)

[Section II, Eq. (7)] The symbol LD is used for the lower bound without being defined; please introduce it explicitly.
[Section II, after Eq. (9)] The phrase 'equivalent to' for the equality conditions should specify the parameter regime (e.g., generic ε_A, ε_B with ε_A>ε_B, or with an explicit note about degenerate cases such as ε_A=ε_B) because the equality condition may fail to be necessary in degenerate cases.
[Section III, Observation 1] The 'genuine battery capacity' C_{A_i} is never defined. The bounds in Eq. (20) are not meaningful unless C_{A_i} is precisely specified (e.g., as a maximization over unitaries on the full system) or the observation is removed.
[Throughout] There are numerous grammatical and typographical issues, including 'satisﬁed its diagonal elements' in Theorem 3, 'the equal sign' instead of 'the equality', missing accents in references, and inconsistent notation such as ρ_i^A versus ρ_{A_i}; a careful proofreading pass is needed.
[Section III, Example 3] The formula for ∑_i C(ρ̃_{A_i};H_{A_i}) = (1.2+0.2n)β is correct for the stated choices of ε_Ai, but the derivation is terse; a short explanation would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the n-qubit monogamy claim rests on an unproved generalization (Lemma 3) and Lemma 2's proof contains an invalid inequality, but no step reduces to its own input or to a fitted/self-cited premise.

full rationale

The central derivation is not circular. The capacity functional and its Schur-convexity are imported from the external Ref. [23] (Proposition 1), and Lemma 1 is a standard majorization fact proved in the text. Theorem 1 follows from Lemma 1 plus Proposition 1 plus Lemma 2; the RBC decomposition after Eq. (12) is a definitional accounting identity (Eqs. (18)-(19)), not a fitted prediction. The n-qubit Theorem 3 and three-qubit Theorem 5 use the same external input, and the self-citations to Refs. [43] and [45] appear only as contextual discussion and in the illustrative Bell-diagonal example, not as load-bearing premises. The real weaknesses are completeness/correctness gaps: Lemma 3 is asserted for general n without proof ('the result in Lemma 2 still holds in the n-qubit system'), and Lemma 2's proof contains an inequality that is not valid for every diagonal ordering (e.g., diagonal entries (10,8,9,1)/28 give α1+α3−α2−α4 = 8/28 while |ρ11+ρ33−ρ22−ρ44| = 10/28). These gaps should be fixed, but they are not circularity: no parameter is fitted to force Eq. (16), no uniqueness theorem from the authors' prior work is invoked, and the target inequality is not assumed as an input. Score 1 rather than 0 only to acknowledge the minor non-load-bearing self-citations and the definitional RBC split; the derivation itself is independent of its conclusions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper's central claims rest on standard majorization theory and Schur-convexity, plus two unproved or under-justified assumptions: Lemma 3 for n-qubit incoherent states and the maximization step in Theorem 5. No free parameters are fitted to data; the Hamiltonian parameters are physical inputs. The RBC and genuine capacity are new definitions rather than independently evidenced entities.

assumptions (4)

standard math Capacity C(ρ;H) is Schur-convex in ρ (Proposition 1 from ref. [23]).
Used in Theorem 1 and 3 to go from majorization of diagonals by eigenvalues to capacity inequality.
standard math Diagonal elements of a positive semidefinite matrix are majorized by its eigenvalues (Lemma 1).
This is the Hardy-Littlewood-Pólya theorem; used throughout the proofs.
ad hoc to paper For n-qubit incoherent states, the sum of subsystem capacities is bounded by the total capacity (Lemma 3).
Stated without proof in Section III; the two-qubit proof does not generalize trivially because the spectrum of γσ1^{⊗n} is more complex.
ad hoc to paper The maximization of C(τ_AB;H_AB)+C(τ_C;H_C) over permutations is achieved by the pairing (ω1,ω2),(ω3,ω4),(ω5,ω6),(ω7,ω8) (Theorem 5 proof).
Assumed without justification in the proof of Eq. (24); this underlies the three-qubit stronger inequalities.

invented entities (2)

Residual battery capacity (RBC) and its coherent/incoherent split
purpose: Quantify the gap in the monogamy inequality and its origins
New definitions in Section II; they reorganize existing quantities rather than predict new phenomena.
Genuine battery capacity C_{A_i}
purpose: Bound the extractable capacity of a subsystem in a multipartite setting
Introduced in Observation 1 without an operational definition independent of C(ρ_{A_i};H_{A_i}).

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Pith. "Pith review of Distribution Relationship of Quantum Battery Capacity." pith.science (2026). https://pith.science/paper/2WD7QPDL

@misc{pith2026250202860,
  author       = {Pith},
  title        = {Pith review of: Distribution Relationship of Quantum Battery Capacity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WD7QPDL}},
  note         = {Machine review of arXiv:2502.02860}
}

read the original abstract

The distribution relationship of quantum battery capacity is investigated. First, it is proved that for two-qubit X-states, the sum of the subsystem battery capacities does not exceed the total system's battery capacity, and the conditions are provided under which they are equal. Then define the difference between the total system's and subsystems'battery capacities as the residual battery capacity (RBC) and show that this can be divided into coherent and incoherent components. Furthermore, it is observed that this capacity monogamy relation for quantum batteries extends to general n-qubit X states and any n-qubit X state's battery capacity distribution can be optimized to achieve capacity gain through an appropriate global unitary evolution. Specifically, for general three-qubit X states, stronger distributive relations are derived for battery capacity. Quantum batteries are believed to hold significant potential for outperforming classical counterparts in the future. These findings contribute to the development and enhancement of quantum battery theory.

Figures

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**Figure 3.** FIG. 3: The relationship between [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: When [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: The proportion of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

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So in this case, the residual battery capacity △C(̺) = C(̺; HAB), the RBC of incoherent part RBC ic = |a3| √ (ǫA + ǫB)2 + γ2 + |a3| √ (ǫA − ǫB)2 + γ2, and the RBC of coherent part RBC c = (|a1| + |a2| − |a3|) √ (ǫA + ǫB)2 + γ2 + (|a1| − |a2| − |a3|) √ (ǫA − ǫB)2 + γ2. From the expression of RBCc, we can see that the RBC of coherent part is directly propor...

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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From the expression of RBCc, we can see that the RBC of coherent part is directly proportional to |a1| and |a2|

So in this case, the residual battery capacity △C(̺) = C(̺; HAB), the RBC of incoherent part RBC ic = |a3| √ (ǫA + ǫB)2 + γ2 + |a3| √ (ǫA − ǫB)2 + γ2, and the RBC of coherent part RBC c = (|a1| + |a2| − |a3|) √ (ǫA + ǫB)2 + γ2 + (|a1| − |a2| − |a3|) √ (ǫA − ǫB)2 + γ2. From the expression of RBCc, we can see that the RBC of coherent part is directly propor...

work page

[3] [2]

Therefore, we have △C(ρβ ) = C(ρβ ; H) − n∑ i=1 C(ρAi β ; HAi ) = 2β √ (0.6 + 0.1n)2 + γ2, RBC ic = C(τβ ; H) − n∑ i=1 C(ρAi β ; HAi ) = ( √ (0.6 + 0.1n)2 + γ2 + √ (0.4 + 0.1n)2 + γ2)β, RBC c = C(ρβ ; H) − C(τβ ; H) = ( √ (0.6 + 0.1n)2 + γ2 − √ (0.4 + 0.1n)2 + γ2)β. From the expressions for RBCic and RBCc, it can be seen that as β approaches 1, both RBCic...

work page

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