@article {567,
title = {Inequalities for the ranks of multipartite quantum states},
journal = {Linear Algebra and its Applications},
volume = {452},
year = {2014},
month = {07/2014},
pages = {153 - 171},
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 (\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,$\infty$) 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.},
issn = {00243795},
doi = {10.1016/j.laa.2014.03.035},
author = {Josh Cadney and Huber, Marcus and Noah Linden and Winter, Andreas}
}