# On the convex cones arising from classifications of partial entanglement in the three qubit system

## TL;DR

This paper explores the structure of convex cones formed by X-shaped three-qubit states to classify partial entanglement, identifying extreme rays and criteria for separability.

## Contribution

It characterizes convex cones of X-shaped states, finds their extreme rays, and provides criteria for partial separability in three-qubit systems.

## Key findings

- Identified all extreme rays of the convex cones.
- Provided necessary criteria based on diagonal and anti-diagonal entries.
- Applied results to important classes like GHZ diagonal states.

## Abstract

In order to classify partial entanglement of multi-partite states, it is natural to consider the convex hulls, intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems. In this paper, we consider convex cones consisting of X-shaped three qubit states arising in this way. The class of X-shaped states includes important classes like Greenberger-Horne-Zeilinger diagonal states. We find all the extreme rays of those convex cones to exhibit corresponding partially separable states. We also give characterizations for those cones which give rise to necessary criteria in terms of diagonal and anti-diagonal entries for general three qubit states.