Endpoint quantum KKL inequality for Boolean observables

Problem

Does the Montanaro–Osborne \(L^2\) quantum KKL inequality hold for every Boolean quantum observable?

Let \(A=A^\dagger\) act on \(n\geq2\) qubits and satisfy \(A^2=I\). Write \(\tau(A)=2^{-n}\operatorname{Tr}A\), let \(\mathcal E_i\) completely depolarize qubit \(i\), and define

\begin{equation} \mathcal E_i(A)=\frac{I_i}{2}\otimes\operatorname{Tr}_iA, \qquad D_iA=A-\mathcal E_i(A), \qquad \operatorname{Inf}^{(2)}_i(A)=\tau[(D_iA)^\dagger D_iA]. \tag{1} \end{equation}

Does a universal constant \(c>0\) exist such that every such \(A\) satisfies

\begin{equation} \max_i\operatorname{Inf}^{(2)}_i(A) \geq c\bigl(1-\tau(A)^2\bigr)\frac{\log n}{n}? \tag{2} \end{equation}

Equation (2) uses the squared influences defined in Eq. (1); the balanced case has \(\tau(A)=0\).

Source

The question is explicitly posed or retained as open in the cited primary literature [Montanaro08][Jiao24]. The statement is rewritten here to make its hypotheses and success criterion self-contained.

Progress

For diagonal observables \(A=\sum_x f(x)|x\rangle\langle x|\), the influence in Eq. (1) reduces to the classical bit-flip influence, and Eq. (2) becomes the classical KKL bound. The tensor-product quantum extension was proposed by Montanaro and Osborne [Montanaro08].
Rouzé, Wirth, and Zhang proved important quantum Talagrand, KKL-type, and Friedgut results using geometric or \(L^p\) influences with \(p<2\). Those results do not resolve the displayed endpoint. Jiao, Lin, Luo, and Zhou explicitly preserve this distinction and state that the Montanaro–Osborne conjecture remains open. [Rouz24], [Jiao24]
The February 17, 2026 revision by Chang and Li develops further variance-decay and higher-order inequalities. Its relevant influence results remain sub-\(L^2\); their constants do not give the required endpoint by simply taking \(p\to2\). [Chang26]
An August 6, 2026 paper by Slote, Volberg, and Zhang provides another useful comparison. Its Hermitian-dilation family has small influences on the data qubits but an ancilla of influence one. Section 7 explicitly explains why this is not a counterexample to standard quantum KKL, which counts every qubit. [Slote26]

Comment

Retained as unresolved, with 2026 partial-progress and counterexample checks. A counterexample in a canonical anticommutation-relation algebra uses a different coordinate structure and must not be substituted for a counterexample on the tensor-product qubit cube. [Jiao24]

References

[Montanaro08]: A. Montanaro and T. J. Osborne, Quantum boolean functions, arXiv:0810.2435 (2008); Chicago Journal of Theoretical Computer Science (2010). Paper.arXiv
[Jiao24]: Y. Jiao, W. Lin, S. Luo, and D. Zhou, Quantum KKL-type inequalities revisited, arXiv:2411.12399, November 2024, Introduction, including the explicit distinction between the CAR counterexample and the Montanaro–Osborne conjecture. Full text. See Conjecture 1.4 for the displayed variance-weighted formulation.link
[Rouz24]: C. Rouzé, M. Wirth, and H. Zhang, Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, arXiv:2209.07279; Communications in Mathematical Physics (2024). Paper.arXiv
[Chang26]: F. Chang and P. Li, Quantum Talagrand-type Inequalities via Variance Decay, arXiv:2601.01900v2, February 17, 2026, Introduction and Section 4. Full text.link
[Slote26]: J. Slote, A. Volberg, and H. Zhang, Tightness of and counterexamples to several quantum estimates, arXiv:2608.04411v2, August 6, 2026, Section 7, on the influential ancilla. Full text.link

Endpoint quantum KKL inequality for Boolean observables

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