@article {pub6605,
	title = {Hierarchically discriminating Haar-randomness in quantum states from a black-box device},
	author = {Xavier Bonet-Monroig AND Hao Wang AND Adri{\'a}n P{\'e}rez-Salinas},
	year = {2026},
	month = {July},
	abstract = {We present a hierarchical discrimination algorithm for sets of states from a quantum d
the compatibility of a set of quantum states $S$ with the $t$-moments of the Haar-random distribution over states.
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.},
	publisher = {IOP Publishing},
	journal = {New Journal of Physics},
	volume = {28},
	number = {074512}
}