Mutual information bound for separable states ($I(A:B)\le \min (S(A),S(B))$)

For a separable quantum state ${\rho }_{AB}={\sum }_k{p}_k{\sigma }_A^k\otimes {\tau }_B^k$, concavity of quantum conditional entropy gives $S(A|B{)}_{\rho }\ge {\sum }_k{p}_{k}S({\sigma }_A^k)\ge 0$. Consequently $I(A:B)=S(A)-S(A|B)\le S(A)$; swapping the parties gives the displayed bound. In finite dimension it is at most ${\log }_2\min (d_A,d_B)$. Thus more than one bit of mutual information witnesses entanglement of two qubits. Smaller positive mutual information need not be entanglement, because separable states can be classically correlated.