%0 Journal Article
%J Linear Algebra and its Applications
%D 2014
%T Inequalities for the ranks of multipartite quantum states
%A Josh Cadney
%A Huber, Marcus
%A Noah Linden
%A Winter, Andreas
%X 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,∞) 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.
%B Linear Algebra and its Applications
%V 452
%P 153 - 171
%8 07/2014
%! Linear Algebra and its Applications
%R 10.1016/j.laa.2014.03.035