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

Both states are block diagonal in $X$. On the support of each block,Substitution into quantum relative entropy givesTaking $-\mathrm{Tr}{\rho }_{XB}\log {\rho }_{XB}$ similarly yields the entropy of a classical-quantum state:

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.

For an ensemble $\{p_x,{\rho }_x\}$ and measurement outcome $Y$, the Holevo bound iswhere $\bar{\rho }={\sum }_x{p}_x{\rho }_x$. Form the classical-quantum state ${\omega }_{XB}={\sum }_x{p}_x|x⟩⟨x|\otimes {\rho }_x$. Part a givesThe measurement is a quantum channel from $B$ to a classical register $Y$. Data processing for relative entropy, applied to the mutual-information representation, gives $I(X:Y)\le I(X:B)$ and proves the bound.

The quantity ${\chi }^{\ast }(\Lambda )$ is the one-use Holevo information of the channel: by the Holevo--Schumacher--Westmoreland theorem, its regularization gives the classical capacity, and without entangled inputs across uses it is the asymptotically achievable classical rate using collective output measurements.

If an ensemble contains a mixed ${\rho }_x={\sum }_j{q}_{j|x}|{\psi }_{xj}⟩⟨{\psi }_{xj}|$, refine its label to $(x,j)$ with probability $p_x{q}_{j|x}$. The average channel output is unchanged, while Concavity of Von Neumann entropy givesThus refinement cannot decrease the Holevo quantity, so the maximum may be restricted to pure input states.