Smooth Perturbations to R\'enyi Entropy

Andrew Buchanan

REVIEW 2 major objections 5 minor 20 references

Smooth Perturbations to R\'enyi Entropy

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For a Gaussian state obtained by a smooth perturbation of the massless vacuum on a ball, every coefficient of the Rényi entropy difference can be computed exactly, for all Rényi parameters in odd dimensions and for integer parameters in…

desk verdict A genuinely useful extension of the perturbative Rényi entropy method, but the stated analyticity assumption is not enough to justify the finite-rank machinery and the theorem needs repair. read the letter →

arxiv 2411.19312 v1 pith:INQOU33B submitted 2024-11-28 hep-th hep-lat

Andrew Buchanan This is my paper · ORCID

classification hep-thhep-lat

keywords RényientropyentanglementGaussianstatesmasslessscalarfieldmutualinformationthermalperturbationserieslatticetheory

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 presents a method for computing the Rényi entropy difference between the massless scalar vacuum on a ball and a smoothly perturbed Gaussian state, order by order in a perturbation parameter that has units of energy. It claims that, whenever the difference of the two-point correlation functions is analytic in position and in the perturbation parameter, every coefficient of the Taylor expansion of the Rényi entropy difference is finite and exactly computable for all Rényi parameters in odd spatial dimensions and for integer Rényi parameters in even spatial dimensions. The field-theoretic trace is reduced to finite-dimensional linear algebra together with a small family of closed-form integrals, so arbitrarily many orders become accessible for a fixed dimension. The procedure is applied to two physical settings: the large-distance expansion of the Rényi mutual information of two distant balls, and the low-temperature expansion of the entropy of a thermal massless scalar field on a ball.

What carries the argument

The engine of the calculation is a basis for polynomial two-vectors adapted to the ball. The paper defines vectors built from spherical harmonics extended to harmonic polynomials, together with the operator $H=(1/\pi)\arccot(J\sigma_0)$, which is also the differential operator $\frac{1}{2}(\nabla\cdot(p\nabla\varphi_2)-(d-1)\varphi_2,\;p\varphi_1)$ with $p=1-|x|^2$; this $H$ is the modular Hamiltonian of the vacuum on the ball. The set formed by applying powers of $H$ and the swap operator $J$ to these vectors is a basis, and each perturbative order decomposes into finitely many coefficients in this basis. Rotational equivariance, via Schur's lemma, reduces all resolvent matrix elements to functions that depend only on one angular-momentum label and on the total number of $H$ factors; these are computed as single integrals involving a product of $\Gamma$ functions and a hyperbolic cosecant, evaluated in closed form by the standard $\beta$ integral. The remaining contour integral of the derivative of the Rényi-entropy function times these rational functions of $\arctan(z)/\pi$ is then evaluated using Bernoulli polynomials in odd dimensions and residue sums or zeta-value formulas in even dimensions.

What would settle it

One concrete check: independently evaluate the coefficient of $(rT)^7$ in the $d=4$, $\alpha=2$ thermal Rényi entropy difference by direct numerical contour integration of Eq. (18); the paper's listed value is $-64\zeta(7)/(35\pi)$. A mismatch at this order would show that the finite-rank trace reduction or the contour-integral evaluation is in error.

Watch

Extended reading notes

Core claim

The central claim is that the expansion of the Rényi entropy difference of a smoothly perturbed Gaussian state on the ball is governed by a finite-rank trace in the continuum. Starting from the lattice identity for Rényi entropy as a trace of a matrix function of the skewed correlation matrix, the paper passes to the continuum by replacing the finite matrix with a $2\times2$ kernel acting on polynomial two-vectors. Unit analysis fixes the degree of each order: if the perturbation parameter has energy units in $d$ spatial dimensions, the entries of the $k$-th kernel are homogeneous polynomials of specific degrees, so each order of the perturbation has finite rank. The paper claims that, under the analyticity assumption on the correlation functions, this reduction makes every coefficient of the entropy-difference series finite and computable for all orders. The remaining contour integrals are evaluated in closed form for all Rényi parameters in odd dimensions and for integer Rényi parameters and the min-entropy limit in even dimensions, with the max-entropy limit divergent. The applications yield three nonzero terms for distant-ball Rényi mutual information in dimensions two through six and five nonzero terms for thermal entropy differences in dimensions three through six.

Load-bearing premise

The load-bearing premise, stated in Section 2 as Eq. (11), is that the differences of the two-point correlation functions are analytic in $x$, $y$, and $t$ at $t=0$; if the kernel is only smooth, the finite-rank reduction and degree counting that make every trace well defined can fail.

Editorial extensions

If this is right

For any fixed spatial dimension the series can be pushed to arbitrarily high order: the only inputs are finite matrix traces and a short list of closed-form integrals, so no new physics input is needed beyond the correlation kernels.
The analyticity assumption implies the Rényi entropy difference is infinitely differentiable at the origin, and the paper notes it is plausibly analytic, although convergence of the series is not proven.
The distant-ball mutual information series is now known through three nonzero terms in dimensions two to six; in even dimensions, inverse powers of pi begin to appear at the 4(d-1)-th power and beyond.
The low-temperature thermal entropy difference is known through five nonzero terms in dimensions three to six, with leading terms matching earlier results and, for von Neumann entropy, only powers that are whole-number combinations of d-1 and d+1 appearing.

Reading between the lines

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

The same finite-rank reduction should extend to multiple expansion parameters with energy units, such as temperature, mass, and inverse separation; the paper lists this as a natural extension but does not carry it out.
In even dimensions, the restriction to integer Rényi parameters and the min-entropy limit is an artifact of the contour-integral evaluation rather than of the trace reduction, so a closed-form formula for rational Rényi parameters would complete the even-dimensional case.
The practical bottleneck for mutual information is spherical versus cylindrical symmetry; a cylindrical version of the decomposition should push the computation past six dimensions, where the paper reports computation time becoming intractable.
Because the whole series is built from vacuum two-point functions, an independent lattice computation at fixed lattice spacing could be extrapolated to test the claim that the continuum trace equals the finite-rank polynomial trace, giving a numerical check of the analyticity assumption.

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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

Short version: the paper has real content—new coefficients for Rényi mutual information and thermal entropy, plus a general framework for computing them—but the central theorem as stated overreaches. The analyticity assumption in Eq. (11) does not imply the polynomial, finite-rank structure that the method requires.

The genuinely new pieces are the reduction of the trace computation to finite-dimensional linear algebra and closed-form contour integrals, and the explicit formulas for Rényi entropy in odd dimensions for all α and even dimensions for integer α and α=∞. The appendices are detailed and self-contained, and the checks against [3–9] at leading and some subleading order are appropriate. The paper is also honest in its footnotes, e.g., the caveat that the series may not be analytic in t.

The soft spot is load-bearing. The argument in Section 2.2 jumps from analyticity of δX, δP, δV_off in (x,y,t) to “unit analysis” forcing each K^(k) to be a homogeneous polynomial. Analyticity alone gives you Taylor coefficients that are analytic functions; it does not give you polynomials. Unit analysis fixes scaling degrees only when there is no other dimensionful scale, and even then a homogeneous function need not be a polynomial. The counterexample δX = t/(1-|x|^2) on the unit ball satisfies the stated analyticity but gives an infinite-rank σ^(1). So the theorem as written is overbroad.

The two applications are likely fine: in each case the perturbing kernel is known explicitly and its Taylor coefficients are indeed polynomials. But the paper does not state that stronger condition as part of the method, and it provides no independent numerics or code to verify the new higher-order coefficients. The lower-order agreements are reassuring, not conclusive.

For a reader who works on free-field Rényi entropies, the coefficient tables are useful and worth checking. The paper deserves serious refereeing, but the referee should ask for a repaired statement of the theorem—say, add polynomiality/finite-rank as an explicit assumption—and ideally an independent check of at least one new coefficient.

Referee Report

2 major / 5 minor

Summary. The paper presents a method for computing Taylor coefficients of the Rényi entropy difference S_α(ρ(t))−S_α(ρ(0)) for Gaussian perturbations of the massless scalar vacuum on a ball. The strategy, extending the authors' earlier work [9], uses a lattice-field-theory representation of the entropy as a contour trace and then takes a continuum limit in which the correlation kernels are represented as operators on polynomial two-vector fields. Under an analyticity assumption on the correlation functions (Eq. 11), the paper claims that each order of the expansion can be reduced to finite-dimensional linear algebra and closed-form integrals, yielding computable coefficients for all orders in odd dimensions and for integer α and α=∞ in even dimensions. The method is applied to two physical settings: Rényi mutual information of two distant balls in d=2,...,6 and low-temperature thermal entropy on a ball in d=3,...,6. Appendices provide detailed derivations of matrix elements, contour integrals, and explicit expansions, with several checks against prior literature [3–9].

Significance. If the central claim holds, this is a useful technical advance: it converts a field-theoretic trace computation into a finite-dimensional matrix problem plus a few families of analytically known integrals, and it produces new coefficients for Rényi mutual information and thermal entropy that agree with known results where available. The paper is unusually transparent about its own limitations (footnotes 4 and 5; the discussion of possible non-analyticity) and ships a large amount of explicit algebra in the appendices, which is a genuine strength. However, the generality of the claimed theorem is not yet established, because the key step that turns analyticity into polynomial finite-rank kernels is unjustified. The two applications are likely to satisfy a stronger condition, but that condition is neither stated nor proven as part of the general method.

major comments (2)

[Section 2.2, Eq. (11)] The inference that analyticity of the correlation functions plus 'unit analysis' implies that each K^(k)(x,y) is a homogeneous polynomial of the stated degrees is not valid. Analyticity of K(x,y;t) at t=0 only implies that each Taylor coefficient K^(k)(x,y) is analytic in x,y; such functions are not generally finite sums of monomials. For example, on the unit ball the kernel δX(x,y;t)=t/(1-|x|^2) is analytic in x,y,t at t=0, but K^(1)(x,y)=1/(1-|x|^2) is not a polynomial, so σ^(1) is not finite rank and the matrices A_k in Eq. (34) are undefined. Thus the claim that the coefficients are finite and computable for all orders under Eq. (11) alone is not established. The paper needs to either (i) add an explicit scale-covariance/no-hidden-length-scale condition that rules out such examples and prove that this condition implies the stated polynomial degrees, or (ii) state the theorem only for perturbations whose kernels are polynomial in x,y of those degrees, and verify that condition for the two applications.
[Section 2.2 and Section 2.4] The finite-rank trace interpretation is the foundation of the method: Eq. (21) decomposes each σ^(k) as a finite sum over the basis B, and Eq. (34) constructs finite matrices A_k. This decomposition presupposes that each σ^(k) is a finite-rank polynomial kernel. Since the polynomiality claim is not proven under Eq. (11) (see previous comment), the trace in Eq. (18) and the reduction to Eq. (37) are not justified for the general theorem as stated. The two physical examples may satisfy the needed condition, but the paper does not demonstrate it; it merely asserts analyticity. This is a load-bearing gap, not a cosmetic issue.

minor comments (5)

[Section 4, Eq. (71)] The expansion variable is written as (r/R)^N in ΔS_{d,α}(r,T), but the expansion is in rT as used in the surrounding text and results. This typo should be corrected to (rT)^N.
[Abstract and Section 2.6] The text says the result holds 'for integer α' in even dimensions, but Appendix D.3.5 shows that the α=0 (max-entropy) limit is divergent in even dimensions. The claim should be qualified to positive integers α (or α≥1) in the abstract and conclusion.
[Section 2.5, Eq. (40)] The summation bound is written as j+j'=k-d-2ℓ+1, but the paper itself notes after Eq. (21) that |ℓ| must be used in two dimensions. The correct bound is j+j'=k-d-2|ℓ|+1; otherwise the block decomposition in Eq. (44) is incorrect for negative ℓ.
[Eq. (31) and Appendix D.1, Eq. (105)] The factor 'γ_m' in Eq. (31) appears as 'γ^m' in Eq. (105); the notation should be made consistent and the power clarified.
[Footnote 5] The statement that S_α(ρ(t))−S_α(ρ(0)) is 'almost certainly analytic' is not a logical consequence of computing its Taylor coefficients to all orders. The paper should either state explicitly that the series may be asymptotic or prove convergence; as written, the remark could mislead readers about the strength of the result.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the computation chain is self-contained given the prior continuum-limit framework; [9] self-citations are not fitted inputs, and the new coefficients are checked against independent literature.

full rationale

The paper's central computation is not equivalent to its inputs. Equation (11) specifies analytic perturbations of the correlation functions; Equations (18)-(20) express the entropy coefficients as contour integrals of traces of resolvent products; Equations (34)-(37) reduce those traces to finite-dimensional matrix algebra with matrix elements computed independently in Appendix D. No entropy coefficient is used to define the correlation kernels, the modular Hamiltonian identification, or the H-eigendecomposition imported from [9]. The self-citations to [9] (continuum trace formula, H = (1/pi) arccot(J sigma_0), eigendecomposition, thermal kernels, identical-ball mutual-information formula) are prior derivations rather than fits renamed as predictions, and the leading and new terms are validated against independent results [3-9] where such results exist. The main caveat is Section 2.2's inference from analyticity plus unit analysis to homogeneous polynomial kernels and finite rank; a counterexample such as delta X = t/(1-|x|^2) is analytic but not polynomial, so the theorem as stated appears overbroad. That is a correctness or justification gap, not circularity, because the output coefficients are not fed back into the input kernels. Footnote 5 similarly concedes an unsolved convergence question. These concerns lower confidence in the universality claim but do not make the derivation circular.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new particles, forces, or dimensions. Its free parameters are the physical Rényi parameter alpha and the spatial dimension d. The main burden is the analyticity assumption and the use of prior results from [9].

free parameters (2)

alpha (Renyi parameter) = continuous; integer or infinity in even dimensions
Alpha is a free parameter of the entropy measure, not fitted. It is part of the definition of the quantity being computed.
d (spatial dimension) = integer 2..6 for presented results
Dimension is an input parameter of the theory, not fitted.

assumptions (3)

domain assumption Analyticity of correlation function differences delta X, delta P, delta V_off in x, y and t at t=0
This is the key assumption introduced in Section 2 (Eq. 11). It is needed to make the continuum-limit trace finite-rank and is asserted to hold for the examples.
domain assumption The continuum limit of the lattice entropy formula is well defined and equals the field-theoretic entropy.
The lattice contour formula is taken to the continuum via the finite-rank argument in Section 2.2. This is a physical limit and not mathematically proven in the paper.
domain assumption The modular Hamiltonian of the vacuum on the ball is (1/pi) arccot(J sigma_0), as given by Casini-Huerta.
This identification, used in Section 2.2, is imported from [9, 10].

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Cite this review

Pith. "Pith review of Smooth Perturbations to R\'enyi Entropy." pith.science (2026). https://pith.science/paper/INQOU33B

@misc{pith2026241119312,
  author       = {Pith},
  title        = {Pith review of: Smooth Perturbations to R\'enyi Entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INQOU33B}},
  note         = {Machine review of arXiv:2411.19312}
}

abstract

A method is presented for computing the R\'enyi entropy of a perturbed massless vacuum on the ball via a comparison with lattice field theory. If the perturbed state is Gaussian with smoothly varying correlation functions and the perturbation parameter has units of energy, I show the coefficients for R\'enyi entropy are analytically computable for all R\'enyi parameter $\alpha$ in odd dimensions and for integer $\alpha$ in even dimensions. I apply this procedure to compute coefficients for the large distant expansion for the R\'enyi mutual information of distant balls and the low temperature expansion for the entropy of a thermal field.

Figures

Figures reproduced from arXiv: 2411.19312 by the authors.

**Figure 2.** A disconnected contour close to the portion of the i [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** A double keyhole contour which avoids the branch cu [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: A double keyhole contour enclosing the portion of t [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [9]

Bramante and A

J. Bramante and A. Buchanan, Perturbative method for mutual information and thermal entropy of scalar quantum ﬁelds , 2023

work page 2023

[2] [1]

Srednicki, Entropy and area , Physical Review Letters 71 (Aug, 1993) 666–669

M. Srednicki, Entropy and area , Physical Review Letters 71 (Aug, 1993) 666–669

work page 1993

[3] [2]

Entanglement Entropy of Two Spheres

N. Shiba, Entanglement Entropy of Two Spheres , JHEP 07 (2012) 100, [ 1201.4865]

work page Pith review arXiv 2012

[4] [3]

Cardy, Some results on the mutual information of disjoint regions i n higher dimensions , J

J. Cardy, Some results on the mutual information of disjoint regions i n higher dimensions , J. Phys. A 46 (2013) 285402, [ 1304.7985]

arXiv 2013

[5] [4]

Agón and T

C. Agón and T. Faulkner, Quantum Corrections to Holographic Mutual Information , JHEP 08 (2016) 118, [ 1511.07462]

arXiv 2016

[6] [5]

Holographic Mutual Information of Two Disjoint Spheres

B. Chen, Z.-Y. Fan, W.-M. Li and C.-Y. Zhang, Holographic Mutual Information of Two Disjoint Spheres , JHEP 04 (2018) 113, [ 1712.05131]

work page Pith review arXiv 2018

[7] [6]

B. Chen, L. Chen, P.-x. Hao and J. Long, On the Mutual Information in Conformal Field Theory, JHEP 06 (2017) 096, [ 1704.03692]

arXiv 2017

[8] [7]

Chen and J

B. Chen and J. Long, Rényi mutual information for a free scalar ﬁeld in even dimen sions, Phys. Rev. D 96 (Aug, 2017) 045006

work page 2017