= Solution
Use base-two logarithms, so the <Von Neumann entropy> is $S(\rho)=-\operatorname{Tr}(\rho\log_2\rho)$, with $0\log0=0$. The <quantum mutual information> is $I(A:B)=S(A)+S(B)-S(AB)$; this is the quantity denoted by $S(\rho_A:\rho_B)$ in the question, using the given joint state.
<Strong subadditivity of Von Neumann entropy> states that for every tripartite <density operator>,
$$
\boxed{S(AB)+S(BC)\ge S(B)+S(ABC).}
$$
Equivalently the conditional <quantum mutual information> $I(A:C\mid B)$ is nonnegative. We apply this property to the original state or to a dilation of the relevant <quantum channel>.
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