  Understanding entanglement of potentially high-dimensional multipartite
quantum systems is crucial across different disciplines in quantum sciences. We
take inspiration from covariance matrix based techniques to derive a nonlinear
criterion that can be used to lower bound the dimensionality vector of mixed
quantum states, revealing both the level of multipartiteness and the
dimensionality of the entanglement in the quantum states. The technique is
based on a system of inequalities that has to be satisfied by all quantum
states with a given entanglement dimensionality vector, which can be checked
via linear programming. We test our condition on paradigmatic classes of
high-dimensional multipartite entangled states like imperfect
Greenberger-Horne-Zeilinger (GHZ) states and find that, in comparison with
other available criteria our method provides a significant advantage, which is
enhanced especially in the case that the dimensions of the individual particles
are different from each other.
