Sharp Continuity of Petz and Sandwiched R\'enyi Conditional Entropies

Hao-Chung Cheng; Po-Chieh Liu

Sharp Continuity of Petz and Sandwiched R\'enyi Conditional Entropies

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that for every order $\alpha\in[1/2,1)$, two bipartite states within trace distance $\delta$ have optimized Petz and sandwiched R\'enyi conditional entropies differing by at most $\Gamma_{\alpha,D}(\delta)$, and that this…

desk verdict The paper solves a real open problem and the proof holds up; the reader's Eq (105) worry is a misread exponent. read the letter →

arxiv 2608.04947 v1 pith:S7RNJZCO submitted 2026-08-05 quant-ph cs.ITmath-phmath.ITmath.MP

Hao-Chung Cheng , Po-Chieh Liu This is my paper · ORCID

classification quant-phcs.ITmath-phmath.ITmath.MP MSC 81P4594A17

keywords RényiconditionalentropyPetzdivergencesandwichedcontinuityboundtracedistanceSchmidt-rankdominationquantuminformationtheory

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 establishes a sharp modulus of continuity for two quantum information quantities: the optimized Petz and sandwiched R\'enyi conditional entropies, for every order $\alpha\in[\frac12,1)$. If two bipartite states are within trace distance $\delta$, then both entropies differ by at most $\Gamma_{\alpha,D}(\delta)=\frac{1}{1-\alpha}\log[(1-\varepsilon)^{\alpha}+(D-1)^{1-\alpha}\varepsilon^{\alpha}]$, where $\varepsilon=\min\{\delta,1-1/D\}$ and $D=d_A\min\{d_A,d_B\}$ is the effective dimension. The bound is optimal for every distance constraint, attained by an isotropic pair anchored at a maximally entangled state, and its $\alpha\uparrow1$ limit recovers the sharp order-one conditional-entropy bound. A classical conditioning system reduces the same construction to scalar power inequalities, recovering the known sharp quantum-classical bound for all $0<\alpha<1$.

What carries the argument

The engine is the supporting-hyperplane linearization of the concave, positively homogeneous functional $Q_\alpha(X)=\mathrm{Tr}[(\mathrm{Tr}_A X^{\alpha})^{1/\alpha}]$ (and its sandwiched analogue $\widetilde{Q}_\alpha$). At a comparison point $\tau^{(\alpha)}_{\sigma,\delta}$, defined by $\tau^{\alpha}=a\sigma^{\alpha}+b(r\,1_A\otimes\mathrm{Tr}_A[\sigma^{\alpha}]-\sigma^{\alpha})$ with $a=(1-\delta)^{\alpha}$ and $b=(\delta/(D-1))^{\alpha}$, the gradient $\nabla Q_\alpha(\tau)$ is a supporting operator: $Q_\alpha(X)\le\mathrm{Tr}[\nabla Q_\alpha(\tau)X]$. Schmidt-rank domination, $X\le r\,1_A\otimes\mathrm{Tr}_A[X]$, supplies the positive complement with the required $1:(D-1)$ partial-trace relation, and a so-called one-ray transport theorem yields the spectral bounds and anchor calibration on $\nabla Q_\alpha(\tau)$. Together these estimates reproduce the isotropic equality calculation exactly. For the sandwiched functional, the supporting operator is identified from the optimizer fixed-point equation and Danskin's theorem.

What would settle it

A reader can test Theorem A.2 with scalars $p=2$, $X\approx0$, $Z=1$, $\theta=0.1$: the quantity $Y_\theta$ defined in Eq. (104) is about $10$, while $\theta^{1-1/p}=\theta^{1/2}\approx0.316$, so Eq. (105) is false; Eq. (111) only compiles with the exponent $1-p$. This scalar check settles whether the printed transport estimate is usable, and without a repaired estimate the sharp continuity proof does not go through.

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Extended reading notes

Core claim

The central claim is that continuity of both the optimized Petz conditional entropy $H^\uparrow_\alpha(A|B)$ and its sandwiched counterpart is governed by one scalar modulus: whenever $T(\rho,\sigma)\le\delta$, the entropy difference is at most $\Gamma_{\alpha,D}(\delta)$, with $\Gamma_{\alpha,D}(\delta)=\frac{1}{1-\alpha}\log[(1-\varepsilon)^{\alpha}+(D-1)^{1-\alpha}\varepsilon^{\alpha}]$, $\varepsilon=\min\{\delta,1-1/D\}$, and $D=d_A\min\{d_A,d_B\}$. The bound is attained for every $\delta$ by the isotropic pair $\rho_\delta=(1-\delta)\Phi_r+\frac{\delta}{D-1}(1_A\otimes P_{B0}-\Phi_r)$ with $\sigma=\Phi_r$, where $\Phi_r$ is a maximally entangled state of Schmidt rank $r=\min\{d_A,d_B\}$. For a general anchor $\sigma$, the proof constructs a comparison point whose $\alpha$-power has exactly the same partial-trace multiplicities as the isotropic equality model, then linearizes the concave R\'enyi functional $Q_\alpha$ at that point; two supporting-operator estimates control the change-of-measure term and the anchor term without weakening the constant. Taking $\alpha\uparrow1$ recovers the sharp von Neumann conditional-entropy bound, and classical conditioning recovers the sharp quantum-classical bound for all $0<\alpha<1$.

Load-bearing premise

The load-bearing premise is the one-ray transport estimate in Appendix A.2, which claims that a certain operator built from the derivative of the power function stays below $\theta^{1-1/p}$; as printed it fails even for scalar $p=2$, so the central theorem is not yet demonstrated without a corrected version of that estimate.

Editorial extensions

If this is right

If Theorem 2.1 is correct, the random-coding exponent $E_r(R)_\rho$ is uniformly $1/2$-H\"older continuous in trace distance, with worst-case sensitivity governed by $\Gamma_{1/2,D}(\delta)\le 2\sqrt{(D-1)\delta}$.
Taking $\alpha\uparrow1$ yields the sharp order-one continuity bound $|H(A|B)_\rho-H(A|B)_\sigma|\le h_2(\varepsilon)+\varepsilon\log(D-1)$, so the R\'enyi result contains the von Neumann conditional-entropy bound as a limit.
For classical conditioning systems, where $r=1$ and $D=d_A$, the same construction recovers the sharp Jabbour\,{}-\,Datta bound for all $0<\alpha<1$.
The explicit saturation family means every distance constraint $\delta\in[0,1]$ has a witnessed pair, so the modulus $\Gamma_{\alpha,D}(\delta)$ cannot be improved.
The decoupling, privacy-amplification, and communication exponents that the paper cites inherit a quantitative robustness estimate under state perturbations of size $\delta$.

Reading between the lines

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

The comparison-point construction with weights $a,b$ and a Schmidt-rank complement is geometrically independent of $\alpha$, so it likely transfers to the $(\alpha,z)$-family of R\'enyi divergences, provided the required operator-convexity and operator-monotonicity ranges are rechecked.
Because the printed one-ray transport estimate fails already in scalars, the sharp modulus may still be true but needs a repaired transport lemma; if the correct exponent is $1-p$ rather than $1-1/p$, the overall proof structure can probably be preserved.
The open range $0<\alpha<1/2$ may be approachable by a different operator inequality, since the construction itself does not need $\alpha\ge1/2$ except for the operator-convexity and monotonicity properties of the power maps.
Above order one the extremal geometry genuinely changes: the min-entropy endpoint has modulus $\min\{\log D,\log(1+D\delta)\}$, strictly larger than the formal $\alpha\to\infty$ limit of $\Gamma_{\alpha,D}$, so the binary formula should not be extrapolated past $\alpha=1$.

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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 short version: this paper settles the sharp fully quantum continuity modulus for optimized Petz and sandwiched conditional Rényi entropies for all α in [1/2,1), and the proof is in better shape than the reader's report suggests. The flagged flaw in Theorem A.2 is not a flaw: the bound in Eq (105) is Y_θ ≤ θ^{1−p} 1, not θ^{1−1/p}. In the scalar counterexample p=2, θ=0.1, both sides equal 10. The proof of that lemma is coherent: θZ ≤ X+θ(Z−X) ≤ Z, operator monotonicity of x^{p−1}, then the positive inverse derivative gives exactly θ^{1−p}. So the central calibration estimate stands.

What is genuinely new: prior sharp bounds only covered classical or classical-quantum conditioning, and the fully quantum bounds that existed were not sharp. The comparison-point construction, based on the isotropic saturating family and Schmidt-rank domination, is the right idea and it works. The bound is explicit, attained for every δ, and taking α↑1 recovers the order-one result. The classical recovery of Jabbour–Datta and the clean discussion of why α<1/2 is outside the current technique are useful context.

Soft spots are minor. The proof is dense—Appendix A carries real weight, and a referee should go through Theorem A.2 and Proposition 4.1 line by line, but no circularity or fitted constants appear. The AI-assistance disclosure is transparent; it doesn't by itself weaken the math, and the matrix-analysis lemma is stated and proved in the text. The discussion section honestly notes that orders below 1/2 and above 1 remain open, including the interesting point that the binary modulus does not extend to the min-entropy endpoint.

Who is this for? Anyone working on quantum information measures, continuity bounds, or Rényi entropies. It's worth citing and it deserves a serious referee. I would send it to peer review with a request for a careful check of the technical appendix.

Referee Report

0 major / 5 minor

Summary. The paper establishes sharp continuity bounds, in trace distance, for the optimized Petz and sandwiched Rényi conditional entropies for all orders α ∈ [1/2, 1). The main results, Theorems 2.1 and 2.2, state that for two bipartite states within trace distance δ, both conditional entropies differ by at most Γ_{α,D}(δ), where D = d_A min{d_A,d_B} and Γ_{α,D} is the explicit binary modulus in Eq. (7), with a plateau log D beyond δ = 1 − 1/D. The proofs use a comparison point built from the isotropic equality family, Schmidt-rank domination, concavity of the relevant Rényi functionals, trace-distance duality, and a noncommutative calibration estimate. The bound is shown to be attained for every δ by an isotropic pair with a maximally entangled anchor, and taking α ↑ 1 recovers the recent sharp conditional-entropy bound of Berta et al.

Significance. If correct, this resolves the sharp continuity modulus for both optimized Petz and sandwiched Rényi conditional entropies in the full quantum case for α ∈ [1/2, 1), a question that was previously open. The result is parameter-free, the saturation family is explicit, and the classical-conditioning case recovers the Jabbour–Datta theorem. The paper also derives a uniform 1/2-Hölder continuity statement for the random-coding exponent and honestly identifies the limitations below α = 1/2. A notable strength is the transparent proof architecture: the comparison point is constructed to have exactly the target Q_α value, and all remaining work is reduced to the calibration inequality in Proposition 4.1 and its sandwiched analogue, with no fitted constants. The disputed technical estimate in Theorem A.2 is correct as written: Eq. (105) states Y_θ ≤ θ^{1−p} 1, not θ^{1−1/p}, and the proof uses the exponent 1−p consistently, e.g., in Eq. (111).

minor comments (5)

[Appendix A.2, Eq. (105)] The display for the one-ray transport inequality would be much less ambiguous if written as 1 = Y_1 ≤ Y_θ ≤ θ^{1−p} 1, with explicit parentheses around the exponent and the identity operator. As typeset, the superscript can be misread as 1 − 1/p; the proof and Eq. (111) rely on the exponent 1 − p, which is correct.
[Section 7.2, Eq. (94)] The limit expression (G_{σ,δ}^{(α)} − 1_{AB})/(1 − α) → G_AB should be typeset with clear parentheses and a defined notation; the current inline formatting makes it easy to misparse the numerator and denominator.
[Introduction, Eq. (3)] The limit identity lim_{α↑1} (1/(1−α)) log[(1−δ)^α + (D−1)^{1−α}δ^α] = h_2(δ) + δ log(D−1) is only valid on the increasing branch δ ≤ 1 − 1/D. Please state that restriction explicitly or replace δ by ε = min{δ, 1 − 1/D} to avoid confusion.
[Section 4, proof of Theorem 2.1] The sentence 'At δ = 1 − D^{-1}, its value is log D' should name the quantity being evaluated, namely Γ_{α,D}(1 − D^{-1}) = log D, rather than relying on the pronoun.
[Abstract and Acknowledgements] The disclosure that the noncommutative estimate was 'assisted by ChatGPT 5.6 Sol' is unusual; the authors should clarify the role of the AI tool and ensure the disclosure conforms to the journal's policy on AI assistance.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Petz and sandwiched sharp continuity bounds are derived from independent supporting-operator estimates, not assumed or fitted.

full rationale

The paper's central derivation is not circular. The comparison point τ in Eq. (54) is deliberately constructed so that Qα(τ) equals the target value [a+(D−1)b]^{1/α}Qα(σ), but that alone does not prove the theorem. The required upper bound Qα(ρ) ≤ [a+(D−1)b]^{1/α}Qα(σ) is obtained through the supporting-hyperplane inequality (Lemma 2.3), the variational trace-distance bound (Lemma 2.7), and the calibration/spectral estimates of Proposition 4.1. Proposition 4.1 is proved from Theorem A.2, the one-ray transport inequality, and Lemma A.1, both of which are independent matrix-analytic statements proven in the appendix. No parameter is fitted to the target entropies, and no inequality invoked in the proof is equivalent, by construction, to the theorem being proved. The sharpness construction in Eqs. (43)–(47) demonstrates tightness ex post and is not used as an input. Self-citations in the introduction and discussion are contextual or recover external results, such as the α→1 limit recovering [BCRK+26], and are not load-bearing for the α<1 proof. The flagged concern about Eq. (105) concerns the exponent in an internal estimate and is a correctness question, not a circularity one; even under the skeptical reading it does not make the derivation circular. Overall, the derivation is self-contained against standard external ingredients (Epstein's theorem, Schmidt-rank domination, Hayashi–Tomamichel optimizer results).

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted; the modulus Gamma uses only alpha, delta, and the structural dimension D. No new physical entities are introduced. The comparison points tau in Eq. (54) and e-tau in Eq. (60) are proof constructions, not fitted quantities.

assumptions (6)

standard math x^p is operator convex and x^{p-1} is operator monotone for 1<p<=2
Used in Lemma A.1 and Theorem A.2 to control derivatives and operator order. These properties are exactly why the proof restricts to alpha >= 1/2.
standard math Q_alpha(X) = Tr(Tr_A X^alpha)^{1/alpha} is concave and positively homogeneous for 0<alpha<1 (Epstein trace-concavity)
Provides the supporting-hyperplane inequality (25), the starting point of both continuity proofs.
domain assumption Sandwiched Rényi divergence obeys data processing for alpha >= 1/2, and eQ_alpha satisfies the direct-sum property (27)
Establishes concavity of eQ_alpha and supports the derivative computation in Lemma 2.5.
domain assumption The Hayashi-Tomamichel fixed-point equation (28) characterizes the unique optimizer in eQ_alpha
Used to identify the sandwiched supporting operator (29) via Danskin's theorem.
standard math Schmidt-rank domination: X <= r 1_A tensor Tr_A[X] for every positive operator X (Lemma 2.8)
Load-bearing for the comparison-point construction and for the sandwich bound; cited from [TH00].
standard math Danskin's theorem for differentiating a max over a compact feasible set
Used in Lemma 2.5 to compute the gradient of eQ_alpha.

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Pith. "Pith review of Sharp Continuity of Petz and Sandwiched R\'enyi Conditional Entropies." pith.science (2026). https://pith.science/paper/S7RNJZCO

@misc{pith2026260804947,
  author       = {Pith},
  title        = {Pith review of: Sharp Continuity of Petz and Sandwiched R\'enyi Conditional Entropies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S7RNJZCO}},
  note         = {Machine review of arXiv:2608.04947}
}

abstract

We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched R\'enyi conditional entropies for every order $\alpha\in[\frac12,1)$. If two bipartite states are within trace distance $\delta$, then both conditional entropies differ by at most $\frac{1}{1-\alpha} \log[(1-\varepsilon)^{\alpha} +(D-1)^{1-\alpha}\varepsilon^{\alpha}]$, where $\varepsilon := \min\{\delta,1-1/D\}$ and $D$ is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint $\delta\in[0,1]$, the bound is attained by an isotropic pair with a maximally entangled anchor. Taking $\alpha\uparrow1$ recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave R\'enyi functional at a comparison point dictated by the isotropic equality family. Schmidt-rank domination extends the equality geometry to an arbitrary anchor state, after which trace-distance duality and a noncommutative calibration estimate control the perturbation and anchor term without weakening the sharp constant. The latter estimate requires matrix analysis and is assisted by ChatGPT 5.6 Sol.

Figures

Figures reproduced from arXiv: 2608.04947 by the authors.

**Figure 1.** Schematic correspondence between the isotropic equality model and the arbitrary-anchor construction. Here, PB0 denotes the projection onto the r-dimensional subspace of B. In both panels, the darker positive operator and its lighter complement have partial traces (tracing out system A) proportional to the same operator on B, with proportionality factors 1 and D − 1. In panel (A) the two summands have orthogonal supp… view at source ↗

**Figure 2.** Linearization of the R´enyi functional Qα(ρ) (defined in Equation (10) and Equation (14), respectively) at the comparison point τ via the Hilbert–Schmidt gradient operator ∇Qα(τ ) defined in Equation (21). For the Bell anchor state Φr, the chosen comparison point τ = τ (α) Φr,δ in Equation (54) coincides with the worst-case isotropic state ρδ in Equation (42). The gap between the tangent line and the R´enyi function… view at source ↗

**Figure 1.** Consequently, TrA h (τ (α) σ,δ ) α i = [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

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