### Exercise 9.

Let \(\rho_{AB}= \sum_i p_i \rho_{AB}^i\) be the state of a bipartite quantum system \(AB\). Using the joint convexity of the relative entropy, prove that the quantum conditional entropy is concave in the state \(\rho_{AB}\).

We want to show that for each \(t\in (0,1)\) and pair of quantum states \(\rho_{AB}\) and \(\sigma_{AB}\) on some bipartite Hilbert space \(\mathcal{H}_A\otimes \mathcal{H}_B\), show that \[ S(A|B)_{\omega} \geq t S(A|B)_{\rho} + (1-t) S(A|B)_{\sigma}. \](1) where \(\omega_{AB} = t\rho_{AB} + (1-t) \sigma_{AB}\).

We can use that \(S(A|B)_\eta = -D(\eta_{AB} \| I_A\otimes \eta_B)\), for any state \(\eta\). In this language, (1) becomes \[ D( \omega_{AB}\| I_A\otimes\omega_B) \leq t D(\rho_{AB} \| I_A\otimes \rho_B) + (1-t) D(\sigma_{AB} \| I_A\otimes \sigma_B) \] upon multiplying by \(-1\). But since \(\omega_{AB} = t\rho_{AB} + (1-t) \sigma_{AB}\) and \(I_A\otimes \omega_B = t(I_A\otimes \rho_A) + (1-t) (I_A\otimes \sigma_B)\), the previous exercise establishes this inequality immediately.