 Solution

For any chosen input distribution, the chain rule and nonnegativity of conditional mutual information giveand likewise $I(U;V,W)\ge I(U;W)$. Choosing an input distribution attaining either marginal capacity proves $K\ge \max \{K_1,K_2\}$.

For the upper bound, conditional independence implies $H(V,W\mid U)=H(V\mid U)+H(W\mid U)$, while entropy subadditivity gives $H(V,W)\le H(V)+H(W)$. ThereforeMaximizing over the input law givesMore precisely, under the conditional-independence assumption,The outputs may be dependent through the common input, and this final term measures the information they share rather than add independently.

This is information from two conditionally independent observations.