Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 3 / b / Solution

Data processing states $D(\Phi (\rho )\Vert \Phi (\sigma ))\le D(\rho \Vert \sigma )$ for every quantum channel $\Phi$. Joint convexity statesStrong subadditivity states $S(ABC)+S(B)\le S(AB)+S(BC)$.

For strong subadditivity, apply data processing under ${\mathrm{Tr}}_C$ to ${\rho }_{ABC}$ and $I_A/d_A\otimes {\rho }_{BC}$. Expanding both relative entropies cancels $\log d_A$ and gives precisely the stated inequality. For joint convexity, use flagged states $\hat{\rho }={\sum }_i{p}_i|i⟩⟨i|\otimes {\rho }_i$ and $\hat{\sigma }={\sum }_i{p}_i|i⟩⟨i|\otimes {\sigma }_i$. Part a gives $D(\hat{\rho }\Vert \hat{\sigma })={\sum }_i{p}_{i}D({\rho }_i\Vert {\sigma }_i)$; discarding the flag and applying data processing gives joint convexity of quantum relative entropy.