Inequalities for the ranks of multipartite quantum states

Joshua D Cadney, Marcus Huber, Noah Linden, Andreas Winter*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

21 Citations (Scopus)


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 alpha-Renyi entropies for alpha is an element of (0,1) boolean OR (1, infinity) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for alpha is an element of (0,1) is completely unconstrained beyond non-negativity. Our results resolve an important open question by showing that 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-Renyi 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. (C) 2014 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)153-171
Number of pages19
JournalLinear Algebra and its Applications
Publication statusPublished - 1 Jul 2014


  • Quantum states
  • Marginals
  • Matrix rank
  • Entropy inequalities

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