Quantum KKL conjecture

Let $({\mathbb M}_{2^{n}},\mathrm{tr})$ be the $n$-fold tensor product of $M_{2\times 2}(\mathbb C)$ equipped with normalized trace. For each $j\in[n]$, let $\mathbf d_j$ be the $j$-th partial derivative operator on ${\mathbb M}_{2^n}$. An element $T\in{\mathbb M}_{2^n}$ is a quantum Boolean function when $T$ is self-adjoint and unitary, that is, $T^*=T$ and $T^*T=\mathbf 1$. Write ${\mathrm{var}}(T)$ for its variance.

Quantum KKL conjecture. There exists a universal constant $C>0$ such that for each $n\in\mathbb N$ and quantum Boolean function $T$,

$$\max_{j\in[n]}\|\mathbf d_j(T)\|^2_{L_2({\mathbb M}_{2^n})}\geq \frac{C{\mathrm{var}}(T)\log(n)}{n}.$$

This is a quantum analogue of the classical KKL inequality for Boolean functions. Earlier work established the inequality for concrete quantum Boolean functions satisfying additional coordinatewise norm conditions, while the conjecture asks for the bound for all quantum Boolean functions.