Abstract

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 (lpha-R’enyi entropy for lpha=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 lpha-R’enyi entropies for $lphain(0,1)p(1,infty)$ satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for $lphain(0,1)$ is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of lpha=0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., lpha=1) and 0-R’enyi entropy are exceptionally interesting measures of entanglement in the multipartite setting.