The Quantum KKL Inequality

Abstract:In this paper, we resolve the quantum KKL conjecture of Montanaro and Osborne \cite{MO2010} for quantum Boolean functions on the $n$-qubit hypercube. More precisely, for every self-adjoint unitary $T$, we bound the largest $L_2$-influence from below by a constant multiple of $\mathrm{Var}(T)\log(n)/n$. The proof heavily depends on upper and lower commutator estimates on the Hilbert space.

Quantum Physics (quant-ph); Functional Analysis (math.FA)

arXiv:2609.21900 [quant-ph]

(or arXiv:2609.21900v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.21900 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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