A note on Bianchi-Don\`a's proof to the variance formula of von Neumann entropy

Lu wei

arxiv: 1906.10303 · v1 · pith:VVN3HYGSnew · submitted 2019-06-25 · 💻 cs.IT · math-ph· math.IT· math.MP

A note on Bianchi-Don\`a's proof to the variance formula of von Neumann entropy

Lu wei This is my paper

Pith reviewed 2026-05-25 16:46 UTC · model grok-4.3

classification 💻 cs.IT math-phmath.ITmath.MP

keywords von Neumann entropyvariance formulaproof comparisonquantum information

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The pith

Bianchi-Donà's proof of the von Neumann entropy variance formula uses the same calculations as the first proof after its initial steps.

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

This note examines a recent proof of the variance formula for von Neumann entropy that was conjectured earlier and first proved in another work. Bianchi and Donà start from a different point but omit the subsequent calculations that reach the result. The note shows those omitted steps are essentially identical to the ones in the original proof. A reader cares because the comparison clarifies that the two routes converge without introducing distinct methods.

Core claim

Despite having a different starting point, the subsequent calculations omitted in Bianchi and Donà's paper are essentially the same as those in the first proof of the variance formula.

What carries the argument

The algebraic steps that convert the initial expressions into the final variance formula for von Neumann entropy.

If this is right

The second proof does not supply an independent derivation path beyond its opening choice.
Verification of the variance formula rests on the same computational sequence in both accounts.
Future work can treat the two presentations as interchangeable for the purpose of extending the formula.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the calculations truly coincide, then any claimed novelty in the second paper is limited to the choice of starting expression.
The note implies that explicit reproduction of the omitted algebra would make the equivalence immediately visible to readers.

Load-bearing premise

The omitted calculations in Bianchi and Donà's paper can be reconstructed from the given outline and shown to match the earlier proof without any additional details supplied.

What would settle it

A full expansion of the omitted steps in Bianchi and Donà's derivation that reveals a material difference from the sequence in the first proof.

read the original abstract

Bianchi and Don\`{a} [1] have recently reported a proof to the variance formula of von Neumann entropy, which was conjectured in [2] and firstly proved in [3]. The purpose of this short note is to show that, despite having a different starting point, the subsequent calculations (omitted in [1]) leading to the result are essentially the same as in [3].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a brief note asserting that two proofs of a known variance formula are essentially the same after their starting points, but without showing the matching steps.

read the letter

The punchline is that this paper adds no new result. It is a short comparative note claiming that the calculations omitted from Bianchi-Donà's proof match those in the earlier proof [3], even though the starting points differ. The author does not reconstruct or display those steps, so the claim stays at the level of an observation rather than a checked identity. That is the one thing a colleague needs to know up front. The note does one small thing well: it flags a possible redundancy between the two proofs for anyone who has already read both papers. If the equivalence holds, readers comparing the works might save a little time. The citations are straightforward and point to the right references—the conjecture, the first proof, and the recent one. The soft spot is exactly where the stress-test note says it is. The central assertion rests on the omitted calculations being “essentially the same,” yet the note provides no term-by-term mapping, no intermediate expressions, and no sketch of how the paths converge. Because the note’s only purpose is to establish that equivalence, the absence of the actual comparison leaves the claim unverified from the text alone. There is no new math or data to check, so the usual questions of derivation or reproducibility do not arise; the issue is simply that the promised demonstration is missing. This note is for the narrow set of readers already tracking proofs of the von Neumann entropy variance formula and curious about their overlap. It does not stand on its own and will not interest a broader audience. I would not bring it to a reading group, would not cite it, and would not send it for peer review. The observation is too slight and too lightly supported to justify referee time.

Referee Report

1 major / 0 minor

Summary. The manuscript is a short note asserting that the omitted calculations in Bianchi and Donà's proof [1] of the variance formula for von Neumann entropy are essentially the same as in the original proof [3], despite different starting points.

Significance. Demonstrating the equivalence of the two proofs would be useful for understanding the development of the result. However, the note does not provide the explicit reconstruction or comparison needed to substantiate this claim, so its significance is limited.

major comments (1)

The paper's purpose is to show the equivalence of the omitted calculations, but no such calculations or mapping are presented in the manuscript. The claim remains an assertion rather than a demonstrated result, which is the central issue for a note whose sole purpose is this comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the major comment below.

read point-by-point responses

Referee: The paper's purpose is to show the equivalence of the omitted calculations, but no such calculations or mapping are presented in the manuscript. The claim remains an assertion rather than a demonstrated result, which is the central issue for a note whose sole purpose is this comparison.

Authors: We agree that the manuscript presents the equivalence as an assertion without including the explicit calculations or mapping between the two proofs. To substantiate the claim and fulfill the note's purpose, a concise outline of the shared subsequent steps is needed. We will revise the manuscript to add this comparison. revision: yes

Circularity Check

0 steps flagged

No derivation chain present; comparative note only

full rationale

The paper is a short note whose sole purpose is to assert that omitted calculations in Bianchi-Donà [1] match those in the prior proof [3]. No equations, derivations, or first-principles steps are introduced or walked through in the manuscript. The claim of equivalence is an assertion about external papers rather than a reduction of any result to its own inputs by construction. This matches the default expectation of no significant circularity for papers without a derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper introduces no new parameters, axioms, or entities as it is a meta-note on proof similarity rather than a mathematical derivation.

pith-pipeline@v0.9.0 · 5586 in / 935 out tokens · 27908 ms · 2026-05-25T16:46:41.932765+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Typical entropy of a subsystem: P age curve and its variance,

E. Bianchi and P. Donà, “Typical entropy of a subsystem: P age curve and its variance,” arXiv:1904.08370v2, 2019

work page arXiv 1904
[2]

Random pure states: Q uantifying bipartite entanglement beyond the linear statistics,

P. Vivo, M. P. Pato, and G. Oshanin, “Random pure states: Q uantifying bipartite entanglement beyond the linear statistics,” Physical Review E, 93, 052106, 2016

work page 2016
[3]

Proof of Vivo-Pato-Oshanin’s conjecture on the ﬂuctuation of von Neumann entropy,

L. Wei, “Proof of Vivo-Pato-Oshanin’s conjecture on the ﬂuctuation of von Neumann entropy,” Physical Review E, 96, 022106, 2017

work page 2017
[4]

A verage entropy of a subsystem,

D. N. Page, “A verage entropy of a subsystem,” Physical Review Letters, vol. 71, no. 9, Aug. 1993

work page 1993
[5]

Kapadia and R

D. Kapadia and R. Germundsson, private communication, W olfram Research, June 3, 2019. 7

work page 2019

[1] [1]

Typical entropy of a subsystem: P age curve and its variance,

E. Bianchi and P. Donà, “Typical entropy of a subsystem: P age curve and its variance,” arXiv:1904.08370v2, 2019

work page arXiv 1904

[2] [2]

Random pure states: Q uantifying bipartite entanglement beyond the linear statistics,

P. Vivo, M. P. Pato, and G. Oshanin, “Random pure states: Q uantifying bipartite entanglement beyond the linear statistics,” Physical Review E, 93, 052106, 2016

work page 2016

[3] [3]

Proof of Vivo-Pato-Oshanin’s conjecture on the ﬂuctuation of von Neumann entropy,

L. Wei, “Proof of Vivo-Pato-Oshanin’s conjecture on the ﬂuctuation of von Neumann entropy,” Physical Review E, 96, 022106, 2017

work page 2017

[4] [4]

A verage entropy of a subsystem,

D. N. Page, “A verage entropy of a subsystem,” Physical Review Letters, vol. 71, no. 9, Aug. 1993

work page 1993

[5] [5]

Kapadia and R

D. Kapadia and R. Germundsson, private communication, W olfram Research, June 3, 2019. 7

work page 2019