Counter-examples for Tensorization Property of Strong Data Processing Inequality for Quantum Divergences

Yu Cao

REVIEW 1 major objections 5 minor 16 references

Counter-examples for Tensorization Property of Strong Data Processing Inequality for Quantum Divergences

T0 review · 1 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper constructs explicit finite-dimensional quantum channels where the strong data processing inequality constant of a product channel exceeds both single-channel constants, for a quantum chi-square divergence and for quantum…

desk verdict Chi-square tensorization counterexample is solid; the relative entropy half rests on an unproved strengthening of Hastings' theorem. read the letter →

arxiv 2608.13204 v1 pith:4UDY52K5 submitted 2026-08-13 quant-ph math-phmath.MP

Yu Cao This is my paper · ORCID

classification quant-phmath-phmath.MP MSC 81P4594A17

keywords strongdataprocessinginequalitytensorizationquantumchi-squaredivergencerelativeentropyamplitude-dampingchannelminimumoutputsuperadditivity

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 reports two counterexamples to the tensorization of strong data processing inequality (SDPI) constants for quantum divergences. In the classical setting the SDPI constant of a product channel is always the maximum of the two single-channel constants for every $f$-divergence; the author shows this identity can fail for finite-dimensional quantum channels. The first counterexample uses the quantum chi-square divergence at the endpoint $\kappa_0(x)=\tfrac12(1+x^{-1})$ of the mean-$\alpha$ family, with an amplitude-damping channel on a qubit, and gives an explicit interval of parameters where the product-channel constant strictly exceeds the single-channel one. The second counterexample shows the same failure for quantum relative entropy, using a channel built from a known random-unitary channel with strictly subadditive minimum output entropy and a reference state that approaches a pure state. If correct, these results settle the open question negatively: tensorization is not a general property of quantum SDPI constants.

What carries the argument

The machinery is the ratio form of the SDPI constant. For a divergence with an operator weight, the constant is $\eta_D(\mathcal E,\sigma)=\sup_{H=H^\dagger,\mathrm{tr}H=0}\frac{\mathrm{tr}[\mathcal E(H)\Omega_{\mathcal E(\sigma)}\mathcal E(H)]}{\mathrm{tr}[H\Omega_\sigma H]}$. For the chi-square family $D_\kappa(\rho\|\sigma)=\mathrm{tr}[(\rho-\sigma)\Omega^\kappa_\sigma(\rho-\sigma)]$ with $\Omega^\kappa_\sigma=R_\sigma^{-1}\kappa(L_\sigma R_\sigma^{-1})$, and for $\kappa_0$ this becomes $\mathrm{tr}[\mathcal E(H)^2\mathcal E(\sigma)^{-1}]/\mathrm{tr}[H^2\sigma^{-1}]$, turning the problem into an explicit quadratic-form maximization. The amplitude-damping channel and the bipartite Hermitian test operator $\tilde H=X\otimes X-Y\otimes Y$ supply the inequality. For the relative-entropy part, the key objects are a random-unitary channel with strictly subadditive minimum output entropy, the extended channel $\mathcal N(\hat\rho)=\varrho\pi_N+\mathcal E(\rho)$, the reference state $\hat\sigma_\epsilon$, and a two-case estimate of $\eta_{\mathrm{QRE}}(\mathcal N,\hat\sigma_\epsilon)$ that splits small population $x$ (perturbative quadratic bound) from large $x$ (binary-divergence lower bound) to show $\limsup_{\epsilon\to0}\eta_{\mathrm{QRE}}(\mathcal N,\hat\sigma_\epsilon)\log(1/\epsilon)\le C$.

What would settle it

For the chi-square claim, evaluate the explicit formulas in Section 2.1 for, say, $\gamma=1/2$ and $s=1/2$; if the two-copy lower bound $\frac{(1-\gamma)^2(1+f(s)^2)(1-s^2)^2}{(1+s^2)(1-f(s)^2)^2}$ does not exceed $\frac{(1-s^2)(1-\gamma)}{1-f(s)^2}$, Proposition 2 collapses. For the relative-entropy claim, the deciding check is whether a random-unitary channel with $\mathrm{H}_{\min}(\mathcal E\otimes\bar{\mathcal E})<2\mathrm{H}_{\min}(\mathcal E)$ actually exists; if not, the construction of Section 3 has no foundation.

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Extended reading notes

Core claim

The paper's central claim is that the identity $\eta_D(\mathcal E_1\otimes\mathcal E_2,\sigma_1\otimes\sigma_2)=\max\{\eta_D(\mathcal E_1,\sigma_1),\eta_D(\mathcal E_2,\sigma_2)\}$ fails for general quantum divergences. For the quantum chi-square divergence with $\kappa_0(x)=\tfrac12(1+x^{-1})$, the author takes $\mathcal E_\gamma$ to be the amplitude-damping channel and $\sigma_s=\tfrac12(I+sZ)$ with $s$ in an explicit interval; a direct computation gives $\eta_{\kappa_0}(\mathcal E_\gamma,\sigma_s)=\frac{1-s^2}{1-f(s)^2}(1-\gamma)$ with $f(s)=\gamma+s-\gamma s$, while the two-copy constant is bounded below by $\frac{(1-\gamma)^2(1+f(s)^2)(1-s^2)^2}{(1+s^2)(1-f(s)^2)^2}$, which is larger in that interval. Since $\kappa_\alpha$ varies continuously, the failure persists for all $\alpha$ close enough to $0$. For quantum relative entropy, the author constructs channels $\mathcal N,\bar{\mathcal N}$ on $\mathcal H_{N+1}\to\mathcal H_N$ from a random-unitary channel satisfying $\mathrm{H}_{\min}(\mathcal E\otimes\bar{\mathcal E})<2\mathrm{H}_{\min}(\mathcal E)$, and a full-rank reference state $\hat\sigma_\epsilon=(1-\epsilon)|0\rangle\langle0|+\epsilon\pi_N$. As $\epsilon\to0$, the single-channel constant stays at most about $C/\log(1/\epsilon)$ times the reference entropy scale, while the product channel's ratio is at least $\beta C/(\log N+\log(1/\epsilon))$ with $\beta>1$, forcing $\eta_{\mathrm{QRE}}(\mathcal N\otimes\bar{\mathcal N},\hat\sigma_\epsilon\otimes\hat\sigma_\epsilon)>\eta_{\mathrm{QRE}}(\mathcal N,\hat\sigma_\epsilon)$ for small enough $\epsilon$.

Load-bearing premise

The relative-entropy counterexample depends on the existence of a random-unitary channel whose minimum output entropy is strictly subadditive, while the chi-square counterexample is self-contained and relies only on its explicit calculation.

Editorial extensions

If this is right

For generic quantum chi-square divergences there is no tensorization identity: product channels can have a strictly larger SDPI constant than either factor, so the contraction ratio of a tensor product is not determined by the local constants.
The positive tensorization result at $\kappa_{1/2}$ is isolated; the failure at $\kappa_0$ extends by continuity to a whole neighborhood $[0,\alpha_*)$, so only special quantum chi-square divergences tensorize.
For quantum relative entropy, the SDPI constant fails to tensorize for general channels, answering the open question left by earlier work on contraction coefficients.
The failures occur with full-rank reference states, so they are not artifacts of singular or boundary states.
Tensorization may still hold for restricted channel classes such as quantum-classical channels and for special divergences, but it cannot be expected for general quantum channels.

Reading between the lines

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

Because the chi-square violation is given by explicit rational functions, it can serve as a ready numerical benchmark: for any damping strength and any $s$ in the stated interval, a direct evaluation of the two constants should reproduce the inequality.
The relative-entropy counterexample is existential rather than constructive; combining the underlying existence theorem with quantitative dimension estimates could turn it into a concrete small-dimensional pair of channels.
The pattern, with contraction coefficients already known to fail tensorization and now SDPI constants failing as well, suggests that tensorization is a commutative, classical phenomenon; one may expect failures for most noncommuting divergences, with the $\kappa_{1/2}$ case and quantum-classical channels as the exceptions.
A natural testable extension is to replace relative entropy by Rényi divergences in the same construction; if the $\log(1/\epsilon)$ scaling is the only driver, analogous violations should appear for a range of Rényi orders.

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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

Quick take on arXiv:2608.13204. There are two results here, and they are not in the same shape.

The chi-square part is the real news. For the endpoint κ0 of the mean-alpha chi-square family, the author takes the amplitude-damping channel on a qubit with reference σ_s and exhibits an explicit bipartite trial state X⊗X − Y⊗Y showing η_{κ0}(E⊗E, σ_s⊗σ_s) > η_{κ0}(E, σ_s) for a range of s. The algebra is fully explicit and I checked it; the conclusion holds. This answers the open question from [2] and shows the positive tensorization result for κ_{1/2} is a special case. The continuity corollary giving a neighborhood [0, α*) is sound. This half alone is a solid, citeworthy negative result.

The relative entropy part is more fragile. The construction follows Hastings: pick a random-unitary channel E with H_min(E⊗Ē) < 2H_min(E), extend it to a channel N on one extra dimension, and connect the SDPI constant to a ratio of relative entropies against (1−ε)|0><0| + ε π_N. The architecture of the proof is plausible, and the bounds in Lemma 6 look right. But the paper needs the strict inequality in Eq. (7): H(E⊗Ē(|Ψ_N><Ψ_N|)) < 2H_min(E). That does not follow from the quoted Hastings theorem, which is a statement about the minimum over all inputs. The minimum could be attained on a non-maximally-entangled input. The paper says 'in particular' and moves on; no proof is given. This is load-bearing: Lemma 5 and Proposition 4 both need β > 1, which needs (7). If (7) is not a direct consequence of known results, the relative entropy counterexample collapses. I don't see it in [6] or in the equivalence paper [13] as an immediate corollary; the author should either prove it or cite a source that states it. The acknowledgment that AI models generated the proofs makes hand-checking this exact step more important, though that alone doesn't invalidate anything.

My take: the chi-square counterexample stands and deserves a serious referee. The relative entropy counterexample is conditional on closing the (7) gap. I would send the paper to peer review with a clear request to fix or justify that step, rather than desk reject. For a reading group, it's worth a slot to discuss the gap and the algebraic construction.

Referee Report

1 major / 5 minor

Summary. The manuscript studies the tensorization property of strong data processing inequality (SDPI) constants for quantum divergences and reports two negative results. Section 2 considers the one-parameter family of quantum chi-square divergences introduced by Temme et al. For the endpoint kappa0(x)=(1+x^{-1})/2, it computes exactly the single-channel SDPI constant for the amplitude-damping channel E_gamma with reference state sigma_s=(I+sZ)/2, and lower-bounds the bipartite constant eta_{kappa0}(E_gamma⊗E_gamma, sigma_s⊗sigma_s) using the trial Hermitian operator H~=X⊗X−Y⊗Y. The resulting strict inequality holds for an explicit interval of s, giving a concrete counterexample to tensorization; Corollary 3 extends the failure to a neighborhood of alpha=0 in the mean-alpha family by continuity. Section 3 constructs channels N and Nbar from H_{N+1} to H_N out of a Hastings random-unitary channel E and its complex conjugate, with reference state sigma_epsilon=(1−epsilon)|0><0|+epsilon pi_N. Lemma 5 converts the asserted strict inequality for the maximally entangled input into beta>1, and Lemma 6 bounds the single-channel SDPI constant by C/log(1/epsilon) up to lower-order terms. Combining these bounds yields eta_{QRE}(N⊗Nbar, sigma_epsilon⊗sigma_epsilon)>eta_{QRE}(N,sigma_epsilon) for sufficiently small epsilon.

Significance. If the proofs are correct, the paper resolves an open question in quantum information theory: unlike the classical f-divergence case, SDPI constants for quantum divergences do not tensorize in general. The chi-square counterexample is especially valuable because it is fully explicit: all matrices, the trial operator, and the inequalities are elementary and checkable, and the single-channel constant is computed in closed form. The relative-entropy construction is also conceptually interesting, as it connects Hastings' superadditivity of minimum output entropy to SDPI constants through an epsilon-perturbed reference state, and the paper honestly acknowledges that this part is nonconstructive and depends on an external existence theorem. The paper also gives proper credit to the earlier positive result [2] and clearly states the scope of the negative results.

major comments (1)

[Section 3.1, Eq. (7)] The proof of Lemma 5 and hence Proposition 4 hinges on the strict inequality H(E⊗Ebar(|Psi_N><Psi_N|)) < 2 H_min(E), displayed as Eq. (7). The preceding line states Hastings' theorem only in the minimum-output-entropy form H_min(E⊗Ebar) < 2 H_min(E). That statement concerns a minimum over all input states and does not by itself imply the inequality for the specific maximally entangled input; the minimizer could in principle be a different state. The paper says 'in particular' but gives no derivation. Please quote the precise statement of [6, Theorem 1] if it indeed proves the stronger inequality at the maximally entangled state, or supply a short argument that the Hastings construction yields Eq. (7). Since beta>1 in Lemma 5 and the entire relative-entropy counterexample depend on Eq. (7), this is load-bearing.

minor comments (5)

[Proposition 2 and Abstract] The phrase 'for a generic kappa in K' overstates what is proved: the construction is for kappa0, and Corollary 3 extends it to a neighborhood of alpha=0. Rephrase as 'there exists kappa in K' or 'for kappa0 and nearby mean-alpha members' to match the actual argument.
[Section 3.3, invariance proof] The proof of eta_{QRE}(N,sigma_epsilon)=eta_{QRE}(Nbar,sigma_epsilon) appears to have lost overbars: the printed statements 'E(rho)=E(rho)' and 'N(rho)=N(rho)' would assert equality of a channel with its complex conjugate, which is not true in general. State the argument via the bijection rho -> rho^T and the invariance of D_QRE(·||pi_N) under transposition.
[Corollary 3] The continuity assertion for eta_kappa in alpha is plausible but not proved. Since the counterexample at alpha=0 uses a fixed traceless H and a strict inequality, a one-line stability argument would make the corollary self-contained.
[Lemma 6, Case 1] The quantity D_2 is defined by log tr(sigma^{-1/2} rho sigma^{-1/2} rho), which coincides with the sandwiched Renyi 2-divergence only for scalar sigma. The proof uses sigma=pi_N, which is scalar, but this should be stated explicitly to avoid confusion with the standard Renyi definition used in the cited monotonicity theorem.
[Section 3.1, Shor equivalence] The sentence invoking the equivalence theorem [13] is not used in the proof of Lemma 5 or Proposition 4. It could be removed or replaced with a precise statement of which equivalence is meant.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the counterexamples are direct calculations or rest on external theorems, not on the paper's own claims.

full rationale

The paper derives two negative results. The quantum chi-square counterexample (Proposition 2) is an explicit calculation: the single-channel SDPI constant for the amplitude-damping channel is computed in closed form, and the bipartite constant is lower-bounded by an explicit two-qubit Hermitian witness. The inequality comparing the two expressions is algebraic and self-contained; the only citation in this section, [2], is used as context for the previously known positive result and is not an input to the counterexample. The relative entropy counterexample (Proposition 4) is built on Hastings' external existence theorem [6] that some random-unitary channel E satisfies H_min(E tensor Ebar) < 2 H_min(E), together with Shor's equivalence theorem [13]. These are external, machine-checkable or published results independent of the present paper, and the paper's construction N, Nbar and reference state sigma_epsilon is an explicit lift that reduces the SDPI ratio computation to the entropy quantities from [6]. The auxiliary Lemma 6 is proved in the text by a direct data-processing bound and a Pinsker-type estimate, not by assuming the conclusion. The only potentially questionable step is Eq. (7), where the paper asserts 'in particular' that H(E tensor Ebar(|Psi_N><Psi_N|)) < 2 H_min(E) from the quoted minimum-output-entropy violation; that is a possible gap in justifying the strengthening from a minimum over inputs to the maximally entangled witness, but it is a correctness or hypothesis-verification concern, not a circularity: the claim is not being assumed through self-citation or definitional equivalence. Because the load-bearing ingredients are external benchmarks and the chi-square example is a direct calculation, the appropriate circularity score is 0.

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

No numerical fitting is used; the chi-square counterexample uses explicit channels and open parameter ranges, and the relative entropy counterexample inherits the external Hastings channel with an epsilon-to-zero limit. The listed axioms are standard quantum information inequalities plus the external Hastings theorem.

assumptions (4)

domain assumption Hastings' existence theorem: there is a random-unitary channel E with H_min(E tensor Ebar) < 2 H_min(E), yielding strict inequality (7) for the maximally entangled state.
External published theorem [6, Theorem 1]; load-bearing for the relative entropy counterexample in Section 3.
domain assumption Data processing inequality for quantum relative entropy and for the chi-square divergences.
Used throughout; D(E rho || E sigma) <= D(rho || sigma), including for the dephasing channel P in the proof of Lemma 6.
domain assumption Quantum Pinsker inequality: D(rho || sigma) >= (1/2) || rho - sigma ||_1^2.
Used in Case 1 of Lemma 6 to convert trace-norm bounds into relative entropy bounds.
domain assumption Monotonicity of quantum Renyi divergences in alpha, specifically D_1 <= D_2.
Used to upper bound quantum relative entropy by D_2 in Case 1 of Lemma 6.

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Cite this review

Pith. "Pith review of Counter-examples for Tensorization Property of Strong Data Processing Inequality for Quantum Divergences." pith.science (2026). https://pith.science/paper/4UDY52K5

@misc{pith2026260813204,
  author       = {Pith},
  title        = {Pith review of: Counter-examples for Tensorization Property of Strong Data Processing Inequality for Quantum Divergences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4UDY52K5}},
  note         = {Machine review of arXiv:2608.13204}
}

abstract

The data processing inequality is a fundamental property that describes the loss of information through noisy channels. A more refined description is characterized by the strong data processing inequality (SDPI). In classical information theory, the tensorization of strong data processing inequality holds for a whole family of $f$-divergences. However, its quantum counterpart is less known. The tensorization of SDPI was shown only for some special cases previously, and the general understanding about the tensorization property of SDPI for quantum divergences remains open. In this work, we report two negative results: the tensorization property fails for certain quantum chi-square divergences, and it also does not hold for the quantum relative entropy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [2]

Cao and J

Y. Cao and J. Lu,Tensorization of the strong data processing inequality for quantum chi-square divergences, Quantum3(2019), 199

work page 2019

[2] [6]

M. B. Hastings,Superadditivity of communication capacity using entangled inputs, Nature Physics5(2009), no. 4, 255–257

work page 2009

[3] [13]

3, 453–472

P.W.Shor,EquivalenceofAdditivityQuestionsinQuantumInformationTheory,CommunicationsinMathemat- ical Physics246(2004), no. 3, 453–472

work page 2004

[4] [1]

S. T. Belinschi, B. Collins, and I. Nechita,Almost One Bit Violation for the Additivity of the Minimum Output Entropy, Communications in Mathematical Physics341(2016), no. 3, 885–909

work page 2016

[5] [3]

Caputo, Z

P. Caputo, Z. Chen, Y. Gu, and Y. Polyanskiy,Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time, 2024. arXiv:2409.07689

arXiv 2024

[6] [4]

1, 111–143

M.Fukuda,C.King,andD.K.Moser,CommentsonHastings’AdditivityCounterexamples,Communicationsin Mathematical Physics296(2010), no. 1, 111–143

work page 2010

[7] [5]

George and M

I. George and M. Tomamichel,A Unified Approach to Quantum Contraction and Correlation Coefficients. arXiv:2505.15281

arXiv

[8] [7]

Hiai and M

F. Hiai and M. B. Ruskai,Contraction coefficients for noisy quantum channels, Journal of Mathematical Physics 57(2016), no. 1, 015211

work page 2016