Characterizing entanglement dimensionality from randomized measurements

We consider the problem of detecting the dimensionality of entanglement with the use of correlations between measurements in randomized directions. First, exploiting the recently derived covariance matrix criterion for the entanglement dimensionality [S. Liu \textit{et al.}, arXiv:2208.04909], we derive an inequality that resembles well-known entanglement criteria, but contains different bounds for the different dimensionalities of entanglement. This criterion is invariant under local unitary operations and can be used to find regions in the space of moments of randomized correlations. After implementing such an algorithm in practical cases, we show that it detects strictly more states than the other entanglement-dimensionality criteria available in literature, thus providing a method that is both very powerful and potentially simpler in practical scenarios. We conclude by discussing the implementation of our method in the multipartite scenario and its potential applications.