Simple sufficient condition for subspace to be completely or genuinely entangled

TL;DR

This paper presents a simple, geometrically-based sufficient condition to determine whether a subspace in bipartite or multipartite quantum systems is entangled, applicable to both pure and mixed states, with practical scenarios explored.

Contribution

The authors introduce a new, straightforward entanglement criterion based on geometric measures, applicable to various entangled subspaces and states, including mixed states and Dicke states.

Findings

The criterion effectively identifies entangled subspaces in complex quantum systems.

It provides a bound on minimal entanglement using vector entanglement measures.

A formula for the geometric measure of Dicke states is derived.

Abstract

We introduce a simple sufficient criterion, which allows one to tell whether a subspace of a bipartite or multipartite Hilbert space is entangled. The main ingredient of our criterion is a bound on the minimal entanglement of a subspace in terms of entanglement of vectors spanning that subspace expressed for geometrical measures of entanglement. The criterion is applicable to both completely and genuinely entangled subspaces. We explore its usefulness in several important scenarios. Further, an entanglement criterion for mixed states following directly from the condition is stated. As an auxiliary result we provide a formula for the generalized geometric measure of entanglement of the $d$--level Dicke states.