Average randomness verification in sets of quantum states via observables

Abstract: We present a hierarchical test, average randomness, that verifies the compatibility of a set of quantum states $S$ with the $t$-moments of the Haar-random distribution. To check such compatibility, we consider the expectation values of states in $S$ with respect to a chosen observable, with focus on their statistical moments. Our first result is a connection between Haar-randomness and the Dirichlet distribution, providing a closed-form expression for the expectation values, as well as and their statistical moments, including simple bounds for the latter. The average randomness metric compares the measured statistical properties of $S$ with those arising from Dirichlet distribution. When it vanishes, $S$ is compatible with being a $t$-design, as seen through the observable $\Obs$, defined as $\Obs$-shadowed $t$-designs. By permutation- and unitary-equivalent randomization of observable, we are able to extend the analysis of average randomness to statistically verify the compatibility of $S$ with $t$-designs. We envision the use of average randomness verification as a practical test for the randomness sets of states with no prior information available.