Product additivity of coherent and private information

Let $\tau$ and $\sigma$ be density operators with positive eigenvalues $({\lambda }_i{)}_i$ and $({\mu }_j{)}_j$, respectively. By [F3], the product eigenvectors form an orthonormal eigenbasis of $\tau \otimes \sigma$ with eigenvalues ${\lambda }_i{\mu }_j$; zero eigenvalues contribute zero. Expanding the entropy sum and using [F4], $\ln ({\lambda }_i{\mu }_j)=\ln {\lambda }_i+\ln {\mu }_j$, gives $S(\tau \otimes \sigma )=S(\tau )+S(\sigma )$.

Choose purifications $\mathrm{\mid }{\psi }_i⟩\in A_i\otimes R_i$ of ${\rho }_i$. Their product purifies ${\rho }_1\otimes {\rho }_2$. The joint output-reference states under the tensor-product channels and the complementary outputs under the product dilations are products. Applying step 1.1 to both entropy terms in [F1] proves the coherent-information equality.

The product ensemble has probabilities $p_{x_1}^{(1)}{p}_{x_2}^{(2)}$ and product receiver and environment output states. Its average receiver and environment states are the corresponding products of the two ensemble averages. Apply step 1.1 to each average and each positive-probability conditional state. Substitution into the two Holevo differences in [F2] separates the finite double sums into the two one-ensemble private-information terms, proving the second equality.