If S ( ρ ρ 1 ) − − S ( ρ ρ 13 ) ≤ ≤ 0 and S ( ρ ρ 3 ) − − S ( ρ ρ 13 ) ≤ ≤ 0 , as is always the case for the classical Shannon entropy, this inequality has nothing to say. For the quantum entropy, on the other hand, it is quite possible that the conditional entropies satisfy − − S ( ρ ρ 13 | ρ ρ 1 ) = S ( ρ ρ 1 ) − − S ( ρ ρ 13 ) > 0 or − − S ( ρ ρ 13 | ρ ρ 3 ) = S ( ρ ρ 3 ) − − S ( ρ ρ 13 ) > 0 (but never both!). Then, in this "highly quantum" regime, this inequality provides additional information.