Criteria for partial entanglement of three qubit states arising from distributive rules

TL;DR

This paper establishes criteria and witnesses for partial entanglement of three-qubit states based on distributive rules, focusing on states with specific diagonal and anti-diagonal structures, including GHZ diagonal states.

Contribution

It provides explicit criteria and witnesses for partial entanglement in three-qubit states arising from distributive rules, especially for states with diagonal and anti-diagonal entries.

Findings

Criteria characterized by diagonal and anti-diagonal entries.

Complete characterization for states with only diagonal and anti-diagonal entries.

Includes important states like GHZ diagonal states.

Abstract

It is known that the partial entanglement/separability violates distributive rules with respect to the operations of taking convex hull and intersection. In this note, we give criteria for three qubit partially entangled states arising from distributive rules, together with the corresponding witnesses. The criteria will be given in terms of diagonal and anti-diagonal entries. They actually characterize those partial entanglement completely when all the entries are zero except for diagonal and anti-diagonal entries. Important states like Greenberger-Horne-Zeilinger diagonal states fall down in this class.