The Quantum KKL Inequality

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 $\mathrm{V}\mathrm{a}\mathrm{r}(T)\log (n)\mathrm{/}n$. The proof heavily depends on upper and lower commutator estimates on the Hilbert space.