Inequalities for the ranks of multipartite quantum states

We investigate relations between the ranks of marginals of multipartite quantum states. We show that there exist inequalities constraining the possible distribution of ranks. This is, perhaps, surprising since it was recently discovered that the α-Rényi entropies for α(0,1)(1,∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α(0,1) is completely unconstrained beyond non-negativity. Our results resolve an important open question by showing that 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. We close the paper with an intriguing open problem, which has a simple statement, but is seemingly difficult to resolve.