Robust quantum state certification and uncertainty principles for total influence

We show that nonadaptive single-qubit Pauli measurements suffice to test whether an unknown $n$-qubit state $\rho$ is $\varepsilon$-close to or $O(\varepsilon)$-far from an ideal target state $|\psi\rangle$, for all but a $2^{-\Omega(n)}$ fraction of target states. The test uses $O(\varepsilon^{-2}\log(1/\delta))$ copies of $\rho$ to achieve confidence $1-\delta$, which is information-theoretically optimal even among protocols with arbitrary joint measurements. The main technical innovation is an uncertainty principle for weighted generalizations of the total influence of Boolean functions. As a simple example, the unweighted variant states that $\mathbf{Inf}[f]+\mathbf{Inf}[\widehat{f}] = \Omega(n)$, which is a natural hypercube analogue of the Heisenberg uncertainty principle (here $\widehat{\,\cdot\,}$ denotes the $2^{-n/2}$-normalized Fourier transform). The weighted case generalizes $\mathbf{Inf}[\,\cdot\,]$ and $\mathbf{Inf}[\,\widehat{\,\cdot\,}\,]$ to Dirichlet energies associated with Glauber dynamics for certain dual measures on the cube.