Quantum KKL-type Inequalities Revisited

Dejian Zhou; Sijie Luo; Wenlong Lin; Yong Jiao

REVIEW 1 major objections 5 minor 25 references

Quantum KKL-type Inequalities Revisited

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Developing the random restriction method in the quantum setting, this paper proves a quantum Eldan-Gross inequality and related KKL-type inequalities.

desk verdict A solid quantum random-restriction toolbox with two strong new inequalities, but the proof of the headline dimension-free KKL theorem has a real gap at (4.5) that is repairable in principle. read the letter →

arxiv 2411.12399 v1 pith:J55SEAVI submitted 2024-11-19 math.FA

Yong Jiao , Wenlong Lin , Sijie Luo , Dejian Zhou This is my paper · ORCID

classification math.FA MSC 46L5394D1047D07

keywords quantumKKLinequalityEldan-GrossTalagrandisoperimetricrandomrestrictionBooleanfunctionsheatsemigroupFourierspectrumnoncommutativeLpspaces

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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 brings the random restriction technique—a combinatorial method for bounding the influence of Boolean functions on the hypercube—into the quantum setting, where the hypercube is replaced by the algebra $M_{2^n}$ with normalized trace. The authors use it to prove a quantum Eldan-Gross inequality for projections: $\mathrm{var}(T)/\sqrt{\log(1+1/\sum_j \|d_j(T)\|_1^2)} \le K \|(\sum_j |d_j(T)|^2)^{1/2}\|_1$ with a universal constant $K$, together with a dimension-free quantum KKL inequality for $0\le T\le 1$: $\max_j \|d_j(T)\|_p^p \ge \frac14 \exp[ - \frac{K}{2-p} \frac{\sum_j \|d_j(T)\|_p^p}{\mathrm{var}(T)}]$ for $1\le p<2$. These are quantitative influence bounds for quantum Boolean functions, the kind of control needed for the open quantum KKL conjecture. The same machinery yields a quantum Talagrand-type isoperimetric inequality and recovers recent alternative answers to the quantum KKL conjecture based on semigroup and CAR-algebra methods, marking a unification of several quantum influence inequalities under one approach.

What carries the argument

The central object is the noncommutative random restriction operator, defined for a subset $J \subseteq [n]$ by $R^J_j(T) = E_{M_{J^c \cup \{j\}}}(d_j(T))$ if $j \in J$ and $R^J_j(T) = 0$ otherwise, where $E$ is the conditional expectation onto the subalgebra supported on $J^c \cup \{j\}$ and $d_j(T)$ is the $j$-th quantum partial derivative. When $J$ is chosen randomly with selection probability $1/d$, this operator isolates Fourier weight in the band $d \le |\mathrm{supp}(s)| < 2d$, as expressed through the spectral weight $W_{\approx d}(T) = \sum_{d \le |\mathrm{supp}(s)| < 2d} \hat{T}(s)^2$. The proof also uses the difference of two noise operators, $H_d(T) = (1 - 1/(2d))^L(T) - (1 - 1/d)^L(T)$, where $\delta^L(T)$ is the noise operator that shrinks each Fourier coefficient of degree $k$ by $\delta^k$, to extract the same frequency band, and combines the resulting estimates with hypercontractivity and log-Sobolev inequalities for the heat semigroup. This machinery converts the task of bounding influence into a sequence of variance-influence inequalities, yielding Theorems 1.8 and 1.11.

What would settle it

Compute the two sides of equation (4.5) for a concrete Fourier polynomial $T$ and a specific dyadic $d$ (for example, $T = \frac12 + \frac{1}{2\sqrt{n}} \sum_j \sigma_j$) and check whether the relaxed inequality $A W_{\approx d}(H_d) - B \sqrt{\mathrm{Inf}_p(H_d)} \sqrt{W_{\approx d}(H_d)} \ge A W_{\approx d}(T)/16 - (B/4) \sqrt{\mathrm{Inf}_p(H_d)} \sqrt{W_{\approx d}(T)}$ holds. A single violation would invalidate the proof of Theorem 1.8; if no violation is found, the relaxation may be salvageable by a sharper estimate.

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Extended reading notes

Core claim

The paper's central claim is that the random restriction method, previously used for a unified proof of the classical KKL, Talagrand, and Eldan-Gross inequalities, admits a noncommutative version on the quantum hypercube $M_{2^n}$. In that setting, the authors prove the quantum Eldan-Gross inequality (Theorem 1.11): for every projection $T \in M_{2^n}$ there is a universal constant $K$ such that $\mathrm{var}(T) / \sqrt{\log(1 + 1/\sum_j \|d_j(T)\|_1^2)} \le K \|(\sum_j |d_j(T)|^2)^{1/2}\|_1$, and the dimension-free quantum KKL inequality (Theorem 1.8): for $1 \le p < 2$ and $0 \le T \le 1$, $\max_j \|d_j(T)\|_p^p \ge \frac14 \exp[ - \frac{K}{2-p} \frac{\sum_j \|d_j(T)\|_p^p}{\mathrm{var}(T)}]$. These follow from a sequence of Fourier-spectrum estimates for the restriction operator, combined with hypercontractivity and log-Sobolev inequalities for the heat semigroup. If correct, the inequalities give quantitative control of the influence of quantum Boolean functions by their variance and provide an alternative partial answer to the quantum KKL conjecture.

Load-bearing premise

The argument's load-bearing premise is that in equation (4.5) the negative correction term can be taken as small as one quarter of its maximum possible value, even though the proved bounds only constrain the relevant spectral weight within a factor of four.

Editorial extensions

If this is right

For every projection on the quantum hypercube, the quantum Eldan-Gross inequality gives a universal trade-off between variance, the sum of squared $L^1$-influences, and the $L^1$ norm of the gradient vector, extending the classical Eldan-Gross inequality to noncommutative Boolean analysis.
The dimension-free quantum KKL inequality (Theorem 1.8) bounds the maximum $L^p$-influence of any bounded element from below by an exponential decay in the total $L^p$-influence divided by variance; for $p < 2$ this is strong enough to imply the $L^p$-influence KKL bound of Theorem 1.9, $\max_j \|d_j(T)\|_p^p \ge C (2-p) \mathrm{var}(T) \log(n)/n$.
Combined with Fourier-spectrum estimates, the Eldan-Gross inequality yields a quantum Talagrand-type isoperimetric inequality and the balanced-projection bound $\max_j \|d_j(T)\|_1 \ge C \sqrt{\log n}/n$, as well as recovering the CAR-algebra results that were previously obtained by different methods.
The method unifies several recent quantum influence inequalities under one technique, indicating that random restriction is as effective in the quantum hypercube as it is classically.

Reading between the lines

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

If the proof of Theorem 1.8 can be completed (the step at (4.5) currently rests on an unproven relaxation), the theorem would provide a dimension-free quantum KKL inequality that is uniform in $n$; if the step fails, the theorem may still be true but needs a different argument.
The random restriction machinery likely extends to higher-order analogues, such as the quantum Talagrand influence inequality with constant $(2-p)^{-1}$ rather than $(2-p)^{-2}$, as suggested by the independent quantum results cited in the paper.
The Fourier-spectrum estimate of Theorem 5.1 is a noncommutative version of a classical lemma and may be applicable to quantum noise sensitivity and Fourier tail bounds beyond the present inequalities.
A testable extension is to check whether the quantum Eldan-Gross inequality holds for all self-adjoint contractions, not just projections, and whether the universal constant $K$ can be made explicit.

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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 genuinely new content here is the quantum random restriction method and the two inequalities it proves cleanly: the quantum Talagrand-type isoperimetric inequality (Theorem 1.10) and the quantum Eldan-Gross inequality (Theorem 1.11). The proofs of those two are detailed, use standard noncommutative semigroup and Fourier tools, and appear internally consistent. The paper is also honest about overlap: Theorem 1.9 recovers Rouzé--Wirth--Zhang, and the p=1 case of Theorem 1.8 was independently proved by Blecher--Gao--Xu. That is not a defect; the method is the contribution, and the recovered results are useful sanity checks.

The soft spot is real. In the proof of Theorem 1.8, equation (4.5) takes Corollary 3.10 applied to H_d, which contains a negative term proportional to sqrt(W(H_d)), and replaces that negative term by a quarter-size version involving sqrt(W(T)). That replacement requires sqrt(W(H_d)) <= sqrt(W(T))/4, i.e. W(H_d) <= W(T)/16. But Lemma 4.2(iii) only guarantees W(T)/16 <= W(H_d) <= W(T), and the lower endpoint is attainable for d=1 with T=(1+sigma_1)/2, where W(H)=W(T)/4. So the displayed inequality is false as written. The stress-test note is right that this is not fatal: using the valid upper bound W(H_d) <= W(T) on the negative term, together with Inf_p(H_d) <= 2^p Inf_p(T) and the D_>= condition, enlarges the constant rather than breaking the argument. But as written, the proof of Theorem 1.8 is incomplete and must be revised.

The rest of the paper looks sound. The Fourier-spectrum estimates in Section 5, the Buser-type inequality, the local reverse Poincaré inequality, and the derivation of the Eldan--Gross inequality from Theorem 1.10 are all coherent. The citation pattern is appropriate; the self-citation to the CAR algebra paper is for comparison, not assumption. The counterexample in Remark 4.4 correctly shows why p=2 fails even classically.

Who should read this: anyone working on quantum Boolean analysis, noncommutative influences, or functional inequalities on the quantum hypercube. The random restriction method and the Eldan--Gross inequality will be useful even if Theorem 1.8 needs a corrected proof. The paper deserves a serious referee; the referee should focus on (4.5) and ask for a repaired proof or an explicit statement that Theorem 1.8 is conditional pending that fix.

Referee Report

1 major / 5 minor

Summary. The paper develops a noncommutative random restriction method on the matrix algebra M_{2^n} and applies it to prove several quantum analogues of Boolean analysis inequalities. The main results are a dimension-free quantum KKL inequality for L^p-influences with 1 ≤ p < 2 (Theorem 1.8), a related quantum KKL-type inequality (Theorem 1.9), a quantum Talagrand isoperimetric inequality (Theorem 1.10), and the quantum Eldan-Gross inequality (Theorem 1.11). The paper also derives applications to CAR-algebra KKL-type results and a stability result for the L^1-influence KKL theorem. The proofs of Theorems 1.10 and 1.11 are based on the quantum Ornstein-Uhlenbeck semigroup, a Buser-type inequality, a local reverse Poincaré inequality, and Fourier-spectrum estimates, while the proof of Theorem 1.8 uses a dyadic decomposition of the Fourier spectrum combined with Lemmas 4.2 and 4.3.

Significance. If the main results are correct, the paper provides a unified random-restriction framework for quantum Boolean analysis, recovers recent results of Rouzé-Wirth-Zhang and Jiao-Luo-Zhou, and gives a new proof of the quantum Eldan-Gross inequality. The paper is generally careful with constants and includes a useful counterexample (Remark 4.4) showing that the p=2 analogue of Theorem 1.8 fails even in the commutative hypercube. However, the proof of Theorem 1.8 contains an invalid inequality at (4.5), and Theorem 1.9 is derived from Theorem 1.8; therefore the central KKL-type claims need repair before the results can be accepted.

major comments (1)

[Section 4, Eq. (4.5)] The step in the proof of Theorem 1.8 that replaces sqrt(W_{≈d}(H_d(T))) by sqrt(W_{≈d}(T)) in the negative term is unjustified. Corollary 3.10 applied to H_d(T) gives a negative contribution proportional to -Kp/16 sqrt(Inf_p(H_d(T))) sqrt(W_{≈d}(H_d(T))); the paper instead writes -Kp/64 sqrt(Inf_p(H_d(T))) sqrt(W_{≈d}(T)). This replacement is valid only if sqrt(W_{≈d}(H_d(T))) ≤ sqrt(W_{≈d}(T))/4, i.e. W_{≈d}(H_d(T)) ≤ W_{≈d}(T)/16. Lemma 4.2(iii) gives W_{≈d}(H_d(T)) ≥ W_{≈d}(T)/16, and Lemma 4.2(iv) gives W_{≈d}(H_d(T)) ≤ W_{≈d}(T); both inequalities are compatible with values strictly larger than W_{≈d}(T)/16. For example, for d=1 and T=(1+σ_1)/2, one has W_{≈1}(H_1(T)) = W_{≈1}(T)/4, so the required endpoint behavior fails. Since the negative term is being relaxed in the wrong direction, the subsequent lower bound '≥ 4Kp Inf_p(T)/var(T) W_{≈d}(T)' does not follow. Because Theorem 1.9 is proved by invoking Theorem 1.8, the proof of Theorem 1.8 must be revised; the gap appears repairable by keeping W_{≈d}(H_d(T)) in the negative term and using the valid upper bound W_{≈d}(H_d(T)) ≤ W_{≈d}(T), at the cost of adjusting the constants.

minor comments (5)

[Abstract] The name 'Motanaro' should be 'Montanaro'.
[Section 2, Proposition 2.5] Items (i) and (iii) of Proposition 2.5 are stated without proof, and the proof of (ii) relies on (i). Since these curvature identities are used in the proof of Theorem 5.5, it would be helpful to include the short Fourier-expansion verification.
[Section 4, Theorem 4.5] Theorem 4.5 is stated as a theorem but its proof is omitted and deferred to reference [2]. The paper also notes that (4.6) follows from Theorem 1.5, so the result should be labeled as a consequence or remark rather than a new theorem.
[Section 5.3, proof of Theorem 1.11] In the display after (5.12), the denominator appears as 'sqrt(log(1 + 1/log(M(T))))'; this should be 'sqrt(log(1 + 1/M(T)))' to match the Eldan-Gross inequality and the subsequent use of the bound 1 + log(1/M(T)) ≤ 2 log(1/M(T)).
[Section 5.4, Proposition 5.2] The proof of Proposition 5.2 is lengthy but well organized; however, a short sentence in the proof after (5.34) says 't0 ≥ (4e)^{d/2}' while Lemma 5.23 only requires t0 > (2e)^{d/2}; the stronger bound follows from (5.34) and M_J(T) ≤ M(T) ≤ e^{-2d}, but this implication could be made explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main inequalities are proved from semigroup, hypercontractivity, and random-restriction estimates; the self-citations are motivational and non-load-bearing.

full rationale

After walking the derivation chain, I find no circular step. The main inequalities (Theorems 1.8, 1.9, 1.10, 1.11) are proved from the quantum heat semigroup, hypercontractivity (2.5), the log-Sobolev and modified log-Sobolev inequalities (Lemmas 2.3 and 2.4), the curvature estimates (Proposition 2.5), and the newly introduced noncommutative random restriction estimates (Section 3), rather than from the target inequalities. The self-citation to the authors' CAR-algebra paper [11] appears only as motivation and as a result to be recovered; the recovery is explicitly labeled, and the proofs in Sections 4 and 5 do not invoke [11] as a premise. The recognition of Blecher-Gao-Xu [2] as an independent proof of the p=1 case (Remark 1.12) is independent corroboration, not a load-bearing self-citation. The proof of Theorem 1.8 contains a suspicious bound at (4.5), where sqrt(W_{≈d}(H_d)) is relaxed to (1/4)sqrt(W_{≈d}(T)) in the negative term; this appears unjustified as written, but it is a correctness gap in an intermediate estimate, not a case of defining one quantity in terms of another or fitting a parameter and renaming it a prediction. The omitted proof of Theorem 4.5, with the reader referred to [2], is an explicit deferral rather than a circular premise. The recoveries of the Rouzé-Wirth-Zhang and Jiao-Luo-Zhou results are plainly labeled as recoveries, and the novel results carry independent proofs, so the derivation is self-contained against external benchmarks.

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

The derivations rely on standard noncommutative Lp and semigroup tools. No free parameters are fitted; the constants K, K_p are universal and explicitly tracked. No new physical entities are introduced.

assumptions (4)

domain assumption Optimal hypercontractivity of the quantum Ornstein-Uhlenbeck semigroup, equation (2.5)
Quoted from [18,17] and used throughout, for example in deriving the log-Sobolev inequality (Lemma 2.3) and the moment comparison lemma (Lemma 5.9).
standard math The log-Sobolev inequality for M_{2^n}, Lemma 2.3, derived from (2.5) following Gross
Used to prove the modified log-Sobolev inequality (Lemma 2.4) and hence Lemma 3.9.
standard math Paley-Zygmund inequality, Lemma 2.6
Used in Corollary 5.16.
domain assumption Curvature and dimension estimates for the semigroup, Proposition 2.5
Proved in the paper for (ii), with (i) and (iii) left to the reader; used in the Buser-type inequality (Theorem 5.4).

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Pith. "Pith review of Quantum KKL-type Inequalities Revisited." pith.science (2026). https://pith.science/paper/J55SEAVI

@misc{pith2026241112399,
  author       = {Pith},
  title        = {Pith review of: Quantum KKL-type Inequalities Revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J55SEAVI}},
  note         = {Machine review of arXiv:2411.12399}
}

read the original abstract

In the present paper, we develop the random restriction method in the quantum framework. By applying this method, we establish the quantum Eldan-Gross inequality, the quantum Talagrand isoperimetric inequality, and related quantum KKL-type inequalities. Our results recover some recent results of Rouz\'e et al. \cite{RWZ2024} and Jiao et al. \cite{JLZ2025}, which can be viewed as alternative answers to the quantum KKL conjecture proposed by Motanaro and Osborne in \cite{MO2010}.

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Works this paper leans on

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Pith tools

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

[1] [1]

Beltran, P

D. Beltran, P. Ivanisvili, and J. Madrid, On sharp isoperimetric inequalities on the hypercube , arXiv preprint arXiv:2303.06738 (2023)

arXiv 2023

[2] [2]

Geometric influences on quantum Boolean cubes

D.P Blecher, L. Gao, and B. Xu, Geometric inﬂuences on quantum boolean cubes , arXiv preprint arXiv:2409.00224 (2024)

work page Pith review arXiv 2024

[3] [3]

Bourgain, J

J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson, and N. Lini al, The inﬂuence of variables in product spaces, Israel J. Math. 77 (1992), no. 1-2, 55–64. MR 1194785

work page 1992

[4] [4]

Carlen and H

A. Carlen and H. Lieb, Optimal hypercontractivity for Fermi ﬁelds and related non commu- tative integration inequalities , Comm. Math. Phys. 155 (1993), no. 1, 27–46. MR 1228524

work page 1993

[5] [5]

Cordero-Erausquin and M

D. Cordero-Erausquin and M. Ledoux, Hypercontractive measures, Talagrand’s inequality, and inﬂuences , Geometric aspects of functional analysis, Lecture Notes i n Math., vol. 2050, Springer, Heidelberg, 2012, pp. 169–189

work page 2012

[6] [6]

Eldan and R

R. Eldan and R. Gross, Concentration on the Boolean hypercube via pathwise stocha stic analysis, STOC ’20—Proceedings of the 52nd Annual ACM SIGACT Symposi um on Theory of Computing, ACM, New York, 2020, pp. 208–221

work page 2020

[7] [7]

Eldan and R

R. Eldan and R. Gross, Concentration on the Boolean hypercube via pathwise stocha stic analysis, Invent. Math. 230 (2022), no. 3, 935–994. MR 4506770

work page 2022

[8] [8]

Eldan, G

R. Eldan, G. Kindler, N. Lifshitz, and D. Minzer, Isoperimetric inequalities made simpler , arXiv preprint arXiv:2204.06686 (2022)

arXiv 2022