There exist infinitely many kinds of partial separability/entanglement

TL;DR

This paper demonstrates that in three-qubit systems, there are infinitely many distinct types of partial entanglement, classified through convex sets derived from basic biseparability states.

Contribution

It introduces a novel classification framework for partial entanglement in three-qubit systems using convex sets and proves the existence of infinitely many entanglement types.

Findings

The lattice of convex sets generated by basic biseparable states has infinitely many members.

An increasing chain of convex sets can be distinguished by specific GHZ diagonal states.

There are infinitely many kinds of partial entanglement in three-qubit systems.

Abstract

In tri-partite systems, there are three basic biseparability, $A$-$BC$, $B$-$CA$ and $C$-$AB$ biseparability according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members, to see that there are infinitely many kinds of three qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three qubit Greenberger-Horne-Zeilinger diagonal states distinguishing those convex sets in the chain.