Graph entropy 6: Separability

This is 6th post on a serie, so you should had read the previous posts (and in the correct order) first. If you just landed here, please start your reading here and follow the links to the next post until you arrive here again, this time with the appropiated backgroud.

In the standard Gibbs-Shannon entropy, the 4th Shannon-Khinchin axiom about separability says 2 different things (that we will label here as sub-axioms 4.1 and 4.2 respectively) that, given two independent distributions P and Q, the entropy of the combined distribution PxQ is:

Axiom 4.1) H(PxQ) = H(P) + H(Q)

When P and Q are not independent, this formula becomes an inequality:

Axiom 4.2) H(PxQ) ≤ H(P) + H(Q)

Graph entropy, being applied to graphs instead of distributions, allows for some more forms of combining two distributions, giving not one but at least three intersting inequalities: