A well-known property of classical distribution is that they satisfy strong subadditivity, meaning that for any tripartite joint probability distribution $p(x,y,z)$, we have the inequality $$H(AB)+H(BC) - H(B) - H(ABC) \ge 0.$$ This can be equivalently stated as the positivity of the conditional mutual information: $I(A:C|B)\ge0$, which in turn immediately tells us that the inequality is saturated iff $A$ and $C$ are conditionally independent from $B$. Equivalently, this amounts to $A\to B\to C$ being a Markov chain. A nontrivial example (meaning an example where $A,C$ are not unconditionally independent) might be the balanced distribution over three bits given by $$p(000)=p(111) = \frac12.$$ Another possibility is the probability distribution $$p(000)=p(001) = \frac14, \quad p(110) = 1/8, \quad p(111) = 3/8.$$ We can still easily see that this must saturate the strong subadditivity because it's a Markov chain: $Y$ is determined by $Y=X$, and the distribution over $Z$ depends exclusively on $Y$.
Going quantum, it is often mentioned that while strong subadditivity also holds for von Neumann entropies of tripartite quantum states, the statement is less obvious than its classical counterpart. It can still be stated as $I(A:C|B)\ge0$, but this is now the conditional quantum mutual information, which has a less obvious interpretation. Preskill states in his notes that the strong subadditivity of the von Neumann entropy is saturated for tripartite states $\rho_{ABC}$ when there is a direct sum decomposition $\mathcal H_B=\bigoplus_j (\mathcal H_{B_j^L} \otimes\mathcal H_{B_j^R})$ and the state has a decomposition $$\rho_{ABC} = \bigoplus_j p_j \rho_{AB_j^L}\otimes \rho_{B_j^R C}.$$ The proof or source of this statement is not given though, and it's not clear to me whether this condition can somehow be considered as a quantum counterpart of the classical Markov chain condition, or whether it's something entirely different.
What are illustrative examples of quantum states saturating the strong subadditivity of the von Neumann entropy, and therefore satisfying the above condition? Ideally, it would be great to also have examples that illustrate the nontriviality of the statement in the quantum case, whatever that means. Given the tight relation between strong subadditivity and monotonicity of the relative entropy, I suppose this is somewhat equivalent to asking for examples saturating the monotonicity of relative entropy in the tripartite case.
