Inequalities for the Ranks of Quantum States

TL;DR

This paper establishes nontrivial inequalities constraining the distribution of ranks of marginals in multipartite quantum states, revealing that the case of rank-based entanglement measures is uniquely restricted among R'enyi entropies.

Contribution

It proves that the ranks of marginals in multipartite quantum states are subject to nontrivial linear inequalities, resolving an open question about the constraints on rank distributions.

Findings

Ranks are constrained by nontrivial linear inequalities.

The case of rank-based entanglement measures is uniquely restricted.

This contrasts with other R'enyi entropies which are mostly unconstrained.

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 $\alpha\in(0,1)\cup(1,\infty)$ satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for $\alpha\in(0,1)$ 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.