01680nas a2200181 4500008003900000022001300039245006200052210006200114260001200176300001400188490000800202520111700210100001701327700001801344700001701362700002001379856009901399 2014 d a0024379500aInequalities for the ranks of multipartite quantum states0 aInequalities for the ranks of multipartite quantum states c07/2014 a153 - 1710 v4523 aWe 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 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 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.1 aCadney, Josh1 aHuber, Marcus1 aLinden, Noah1 aWinter, Andreas uhttps://grupsderecerca.uab.cat/giq/publications/inequalities-ranks-multipartite-quantum-states