  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.
