This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies S ( ρ ρ 12 | ρ ρ 1 ) = S ( ρ ρ 12 ) − − S ( ρ ρ 1 ) and S ( ρ ρ 12 | ρ ρ 2 ) = S ( ρ ρ 12 ) − − S ( ρ ρ 2 ) are both non-negative. In the quantum case, however, both can be negative, e.g. S ( ρ ρ 12 ) can be zero while S ( ρ ρ 1 ) = S ( ρ ρ 2 ) > 0 . Nevertheless, the subadditivity upper bound on S ( ρ ρ 12 ) continues to hold. The closest thing one has to S ( ρ ρ 12 ) − − S ( ρ ρ 1 ) ≥ ≥ 0 is the Araki–Lieb triangle inequality