# Exercise 9

## from Example Sheet 3

### 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.