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REVIEW 4 major objections 6 minor 1 cited by

R\'{e}nyi and Shannon mutual information in critical and decohered critical system

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Pith's one-line read The paper introduces a Rényi-2 generalized Shannon mutual information and shows the critical Ising chain keeps its conformal-field-theory central charge under both relaxed measurements and local decoherence.

desk verdict A genuinely new interpolating mutual information, but the central robustness claim rests on four-point fits with no error bars or residuals, so treat the plateau as plausible rather than established. read the letter →

arxiv 2506.18475 v1 pith:J5EGNIO4 submitted 2025-06-23 quant-ph cond-mat.stat-mechcond-mat.str-el

Yoshihito Kuno , Takahiro Orito , Ikuo Ichinose This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-el
keywords Rényi-2generalizedShannonmutualinformationcriticaltransverse-fieldIsingmodelconformalfieldtheoryscalingcentralchargelocaldecoherencemixedstatesdoubledHilbertspacematrixproduct
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 Rényi-2 generalized Shannon mutual information (R2GSMI) that continuously interpolates between the Rényi-2 mutual information ($p_m=0$) and the Rényi-2 Shannon mutual information ($p_m=1/2$) by replacing perfect projective measurements with local decoherence of strength $p_m$. For the critical transverse-field Ising model, it shows numerically that this quantity obeys the conformal-field-theory scaling law $I^{(2)}(A,B,p_m)=\frac{c_2}{4}\ln\!\big(\frac{L}{\pi}\sin\frac{\pi L_A}{L}\big)+b_2$ with $c_2=1$, the Ising value, over a broad region of the parameter plane. Under a global $Y$-decoherence of strength $p_y$ applied to the whole system, the extracted $c_2$ stays close to 1 across a wide $(p_m,p_y)$ region, indicating that the Ising CFT properties are robust against both measurement relaxation and local decoherence. If the claim holds, a noisy or partially decohering measurement protocol can still certify the Ising central charge without requiring perfect projective measurements.

What carries the argument

The central object is the Rényi-2 generalized Shannon mutual information $I^{(2)}(A,B,p_m)=S^{(2)}_{A,M}(p_m)+S^{(2)}_{B,M}(p_m)-S^{(2)}_{A\cup B,M}(p_m)$, built from a subsystem Rényi-2 entropy $S^{(2)}_{A,M}(p_m)=-\log\operatorname{Tr}_A[(\rho^{M,A}_{A,p_m})^2]$ where the pure state is first partially decohered by a local channel of strength $p_m$ in the measurement basis $\hat{M}$. At $p_m=1/2$ this channel acts as a non-selective projective measurement and the quantity reduces to the known Rényi-2 Shannon mutual information; at $p_m=0$ it reduces to the Rényi-2 mutual information. The numerical machinery is the doubled Hilbert space (Choi) representation, in which the density matrix becomes a supervector on a ladder, and a maximal depolarization channel on subsystem $B$ projects out $B$ so that the norm of the supervector gives $\operatorname{Tr}_A[(\rho^M_A)^2]$. The fitted scaling law $I^{(2)}=\frac{c_2}{4}\ln\!\big(\frac{L}{\pi}\sin\frac{\pi L_A}{L}\big)+b_2$ is the diagnostic that converts the numerical data into a central-charge estimate.

What would settle it

Repeat the $(p_m,p_y)$ scan at $L=64$ and $L=128$ with the same fit window: if the extracted $c_2$ drifts with system size or the boundaries of the $c_2\simeq 1$ plateau move, the robustness is a finite-size effect; alternatively, compute the residuals of the Eq. (17) fit as a function of $L_A$—systematic curvature near $p_m\approx 0.1$ would show that the dip in $c_2$ reflects a failure of the scaling ansatz rather than a physical change in the state.

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Extended reading notes

Core claim

The central claim is that the CFT scaling of the R2GSMI and its Rényi-2 central charge $c_2$ are robust properties of the critical TFIM ground state, not artifacts of a particular measurement limit. For the $\hat{M}=X$ conformal basis, $c_2=1$ for every $p_m$ from 0 to 1/2, meaning that relaxing the projective measurement all the way to the Shannon limit leaves the extracted central charge unchanged. For the $\hat{M}=Z$ basis, $c_2=1$ both at $p_m=0$ (the Rényi-2 mutual information, where the exact Ising result is known) and for $p_m\ge 0.2$, with a continuous dip near $p_m\approx 0.1$; the paper takes this as a partial but not complete breakdown of robustness in that basis. When the whole system is subjected to local $Y$-decoherence, the fitted $c_2$ remains near 1 over a broad region of the $(p_m,p_y)$ plane, deviating only for strong decoherence $p_y\gtrsim 0.3$ and especially at $p_y=1/2$. The paper concludes that the Ising CFT properties observed through both the conventional Rényi-2 Shannon mutual information and Rényi-2 mutual information survive in this generalized mutual information under local decoherence.

Load-bearing premise

The load-bearing premise is that the sine-form CFT scaling law with constant $c_2$ and $b_2$ holds at the numerically accessible size $L=32$ for every $(p_m,p_y)$ considered, so that fitting only four subsystem sizes ($L_A=8,12,14,16$) in the decohered case yields the true $c_2$; if the ansatz or the fit window fails at those sizes, the claimed robustness is not established.

Editorial extensions

If this is right

  • Because the R2GSMI connects the Rényi-2 mutual information and the Rényi-2 Shannon mutual information as limits, any experimental setup that realizes partial decoherence of strength $p_m$ can interpolate between the two quantities and should still detect $c_2=1$ for the Ising critical point.
  • In the $\hat{M}=X$ basis the extracted central charge is $c_2=1$ for all $p_m$, so the CFT fingerprint is immune to relaxing the measurement in that basis.
  • In the $\hat{M}=Z$ basis the robustness has a window: $c_2\simeq 1$ for $p_m\ge 0.2$ and at $p_m=0$, with a dip near $p_m\approx 0.1$, so the plateau region defines where noisy Z-basis measurements remain reliable.
  • For a globally $Y$-decohered critical state, $c_2\simeq 1$ persists up to $p_y\sim 0.3$ in the $(p_m,p_y)$ plane, indicating that local environmental decoherence does not immediately destroy the Ising CFT signature.
  • The doubled-space depolarization scheme provides a practical route to compute such generalized mutual informations for mixed states, not only for the pure ground state treated here.

Reading between the lines

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

  • Editorial inference: the dip in $c_2$ near $p_m\approx 0.1$ for the $Z$ basis may reflect a crossover between measurement-dominated and entanglement-dominated information, but a finite-size artifact is equally plausible; a system-size scan at $L=64,128$ would distinguish the two.
  • Editorial inference: if the same interpolation is applied to a bosonic CFT such as the XXZ chain (a direction the paper flags as future work), a similar robustness plateau would suggest that weak local decoherence generically preserves CFT central charges, not just the Ising one.
  • Editorial inference: the $p_m$-tuning offers an experimental handle: a measurement apparatus with controlled noise could still certify the central charge as long as its effective $p_m$ stays inside the robust plateau, turning measurement noise from a nuisance into a probe.
  • Editorial inference: replacing $\hat{M}=Z$ or $X$ by other local operators in the R2GSE definition may produce a phase diagram of robustness that tracks whether the chosen basis is a conformal boundary condition; this is testable with the same numerical scheme.
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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 / 6 minor

Summary. The paper introduces a Rényi-2 generalized Shannon mutual information (R2GSMI) that interpolates between the Rényi-2 mutual information (pm = 0) and the Rényi-2 Shannon mutual information (pm = 1/2). The authors propose a doubled-Hilbert-space matrix product state scheme for computing the R2GSMI, apply it to the L = 32 critical transverse-field Ising chain, and fit the CFT scaling ansatz of Eq. (17) to extract the Rényi-2 central charge c2. They report that c2 stays close to 1 for both Z-basis and X-basis measurements as pm varies, and that a broad interior region of the (pm,py) plane remains near c2 = 1 under Y-decoherence, from which they conclude that Ising CFT properties are robust against local decoherence.

Significance. If the numerical evidence were conclusive, the paper would offer a useful new observable that connects two established information measures and provides a practical way to test central charge robustness under decoherence. The doubled-Hilbert-space method is clearly described, and the endpoint benchmarks pm = 0 and pm = 1/2 agree with earlier work. However, the central claim rests on two-parameter fits with only four subsystem sizes at a single system size, with no reported residuals, error bars, or goodness-of-fit statistics; the anomalous dip near pm ≈ 0.1 for the Z basis is also left unexplained. The result is plausible but currently underdetermined by the presented evidence.

major comments (4)
  1. [Section V.B, Eq. (17), Fig. 4] The entire (pm, py) phase map, including the claimed broad c2 ≈ 1 plateau, is generated from two-parameter fits of Eq. (17) using only LA = 8, 12, 14, 16 at fixed L = 32. This leaves only two degrees of freedom per fit, and the paper provides no residuals, chi-squared values, bootstrap uncertainties, or stability checks. Since the interior of the parameter plane is the new content beyond the previously studied pm = 0 and pm = 1/2 lines, the authors should provide per-point goodness-of-fit diagnostics and demonstrate that the extracted c2 is stable under small changes in the fit range, such as omitting one of the four subsystem sizes.
  2. [Section V.A, Fig. 2(b)] The nonmonotonic dip in c2 near pm ≈ 0.1 for the Z basis is reported without explanation. If the fit ansatz of Eq. (17) is poor precisely in this region, then the claim that 'all cases are well-fitted' is not supported, and the robustness statement is weakened. The authors should quantify the fit quality for the points in the dip, for example by showing residuals or a chi-squared value, and should discuss whether the dip is a physical effect of the interpolation between R2MI and R2SMI or an artifact of the fitting procedure.
  3. [Section V, numerical setup] All central charge extractions are performed at a single system size, L = 32, with no finite-size scaling analysis. A universal central charge estimate normally requires either a check that results are stable with L or a controlled large-L extrapolation. Without such a check, the reported c2 ≈ 1 plateaus could reflect a finite-size crossover rather than the Ising CFT value, especially in the decohered regions where the deviation grows at larger py.
  4. [Section V.B] The paper states that the four subsystem sizes LA = 8, 12, 14, 16 give 'sufficient accurate results,' but no justification for this specific choice is given, and it is not consistent with the earlier statement that data for 8 ≤ LA ≤ 24 were used. The authors should either use the full available range or explain why the restricted range is appropriate for Eq. (17), and they should report the resulting fit uncertainties.
minor comments (6)
  1. [Throughout] There are several typographical and terminology inconsistencies, including 'R´enyi' for 'Rényi', 'R2RSMI' used interchangeably with 'R2GSMI', and 'docohred' instead of 'decohered'. The authors should carefully proofread the manuscript.
  2. [Section V.A] The sentence 'In 0.2≤ pm≤ 0, 5' contains a clear typo; this should be corrected to a proper interval such as '0.2 ≤ pm ≤ 0.5'.
  3. [Figure 4] The color map is generated by cubic interpolation from discrete data points, which can create visually large plateaus and suppress the actual sampling density. Showing the data points explicitly or using a scatter plot would make the evidence for the plateau more transparent.
  4. [Section III.C] The phrase 'Stein-spring representation' should be 'Stinespring representation'.
  5. [Abstract and Introduction] The claim that the R2GSMI 'can offer more experimentally accessible alternative' to entanglement entropy would benefit from a concrete experimental protocol or reference; as written, the accessibility claim is not quantified.
  6. [References] Reference [18] reports c2 = 1.02 for the XY model, which is mentioned only in the text; the numerical comparison to that value is useful and could be highlighted as an internal benchmark.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: c2 is fit as a free parameter against external benchmarks; the R2GSMI interpolation is numerically computed, not defined into existence.

full rationale

The central claim is that the coefficient c2 extracted by fitting Eq. (17) remains near 1 across a broad (pm, py) region. This is not circular: the fit treats c2 as a free parameter and never injects the benchmark value c2 = 1 into the data. The benchmark is external and explicitly labeled a conjecture: "By the conjecture [16–19], the central charge c is related to the Rényi-2 central charge as c2 = 2c," with Refs. [16–19] being independent prior works. The new quantity R2GSMI is defined in Eq. (12) and reduces at pm = 0 and pm = 1/2 to R2MI and R2SMI, but the interpolating values are computed numerically from the channel in Eqs. (4)–(7); nothing forces the fitted c2 to remain constant between those limits. The same-author citations ([25], [34], [38–40]) support numerical methodology and earlier related studies, not the load-bearing CFT scaling law or the external c2 = 1 benchmark. The paper's own admitted limitation—"this analytical derivation is difficult"—concerns the absence of an analytic proof, which is a numerical/correctness caveat rather than circular reasoning. Concerns about fitting only four subsystem sizes without error bars affect the reliability of the extracted c2, but they do not mean the conclusion was assumed by construction. No equation reduces to another equation by definition, and no fitted parameter is renamed as a prediction. The derivation chain is therefore self-contained, with circularity score 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on fitting c_2 to the CFT ansatz, which assumes the scaling form; this is a modeling assumption rather than an invented physical entity. No new particles, forces, or dimensions are introduced.

free parameters (2)
  • c_2 (Rényi-2 central charge) = ~1 in the robust regime
    Fitted coefficient of the log-term in the CFT scaling ansatz Eq. (17) for each (p_m, p_y) point.
  • b_2 (non-universal offset) = varies with p_m, p_y
    Second fitted parameter in the scaling ansatz Eq. (17).
assumptions (5)
  • domain assumption The critical 1D TFIM ground state is described by the Ising CFT with central charge c = 1/2.
    Used throughout to compare extracted c_2 to the exact Ising value; standard result, e.g., Refs. [6,7].
  • domain assumption The R2GSMI obeys the single-log CFT scaling ansatz of Eq. (17).
    Assumed for all p_m and p_y; no derivation is given, and goodness-of-fit is not quantified.
  • domain assumption The Rényi-n central charge conjecture c_n = c n/(n-1) (n>1), hence c_2 = 2c = 1, holds for the Ising CFT.
    Taken from Refs. [16,18] and used as the benchmark value.
  • standard math The doubled Hilbert space and depolarization identities in Sec. IV correctly compute the purities.
    Uses standard Choi isomorphism and maximal depolarization channel results.
  • ad hoc to paper The MPS numerical parameters (D = 200-240, truncation 1e-7, energy convergence 1e-6) give converged results at L = 32.
    Stated in Sec. V without convergence tests with respect to bond dimension or system size.

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Pith. "Pith review of R\'{e}nyi and Shannon mutual information in critical and decohered critical system." pith.science (2026). https://pith.science/paper/J5EGNIO4

@misc{pith2026250618475,
  author       = {Pith},
  title        = {Pith review of: R\'enyi and Shannon mutual information in critical and decohered critical system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5EGNIO4}},
  note         = {Machine review of arXiv:2506.18475}
}
abstract

We investigate a critical many-body system by introducing a R\'{e}nyi generalized mutual information, connecting between R\'{e}nyi mutual information and R\'{e}nyi Shannon mutual information. This R\'{e}nyi generalized mutual information can offer more experimentally accessible alternative than the conventional entanglement entropy. As a critical many-body state, we focus on the critical transverse-field Ising model (TFIM) described by the Ising conformal field theory (CFT). We show that even if we modify the non-selective projective measurement assumed in R\'{e}nyi Shannon mutual information by replacing the measurement into decoherence by environment, the R\'{e}nyi generalized Shannon mutual information maintains the CFT properties such as subsystem CFT scaling law and its central charge observed through both the conventional R\'{e}nyi Shannon mutual information and R\'{e}nyi mutual information. Furthermore, we apply a local decoherence to the critical ground state of the TFIM and numerically observe the R\'{e}nyi generalized mutual information by changing the parameter controlling environment effect (corresponding to the strength of measurement) in the R\'{e}nyi generalized mutual information and the strength of the decoherence to which the entire system subjects. We find that R\'{e}nyi-$2$ type central charge connected to the central charge is fairly robust, indicating the strong robustness of the Ising CFT properties against local decoherence by environment.

Figures

Figures reproduced from arXiv: 2506.18475 by the authors.

Figure 1
Figure 1. FIG. 1. Relation between the mutual information considered in this [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) shows the scaling data for typical pm’s. We find that all cases are well-fitted with the ansatz of Eq. (17). That is, the CFT scaling form is robust against the relaxation by pm. However, we find that the estimated Renyi- ´ 2 central charge c2 exhibits an interesting pm-dependence as shown in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical extracted R [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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  1. Succession of Ising criticality and its threshold in critical quantum Ising model subject to symmetric decoherence

    quant-ph 2025-08 conditional novelty 6.0 of 10

    A decohered critical Ising state under X+ZZ noise retains Ising CFT exponents (c=1/2, eta=0.25, nu=1) until a threshold where strong-to-weak spontaneous symmetry breaking appears.

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