Inequalities for the Ranks of Quantum States

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