  We investigate relations between the ranks of marginals of multipartite
quantum states. These are the Schmidt ranks across all possible bipartitions
and constitute a natural quantification of multipartite entanglement
dimensionality. We show that there exist inequalities constraining the possible
distribution of ranks. This is analogous to the case of von Neumann entropy
(\alpha-R\'enyi entropy for \alpha=1), where nontrivial inequalities
constraining the distribution of entropies (such as e.g. strong subadditivity)
are known. It was also recently discovered that all other \alpha-R\'enyi
entropies for $\alpha\in(0,1)\cup(1,\infty)$ satisfy only one trivial linear
inequality (non-negativity) and the distribution of entropies for
$\alpha\in(0,1)$ is completely unconstrained beyond non-negativity. Our result
resolves an important open question by showing that also the case of \alpha=0
(logarithm of the rank) is restricted by nontrivial linear relations and thus
the cases of von Neumann entropy (i.e., \alpha=1) and 0-R\'enyi entropy are
exceptionally interesting measures of entanglement in the multipartite setting.
