Bipartite states of low rank are almost surely entangled

TL;DR

This paper proves that most low-rank bipartite quantum states are entangled, especially when their rank is less than or equal to that of their reduced states, highlighting the prevalence of entanglement in such states.

Contribution

It establishes a probabilistic criterion for entanglement in low-rank bipartite states, advancing understanding of quantum entanglement structure.

Findings

Low-rank bipartite states are almost always entangled.

Entanglement prevalence increases when the state's rank is below that of its marginals.

Provides a probabilistic framework for entanglement detection in low-rank states.

Abstract

We show that a bipartite state on a tensor product of two matrix algebras is almost surely entangled if its rank is not greater than that of one of its reduced density matrices.