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

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.