= Mutual information bound for separable states
{title2=$I(A:B)\leq\min(S(A),S(B))$}
For a <separable quantum state> $\rho_{AB}=\sum_kp_k\sigma_A^k\otimes\tau_B^k$, <concavity of quantum conditional entropy> gives $S(A|B)_\rho\geq\sum_kp_kS(\sigma_A^k)\geq0$. Consequently $I(A:B)=S(A)-S(A|B)\leq S(A)$; 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.
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