Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 4

For Hermitian $X$, $\Vert X{\Vert }_1={\max }_{-I\le H\le I}\mathrm{Tr}(HX)$. The adjoint of a trace-preserving completely positive map is unital and positive, so $-I\le H\le I$ implies $-I\le {\Lambda }^{\ast }(H)\le I$. ThereforeHence trace distance is contractive under quantum channels.

Let $X=\rho -\sigma =X_{+}-X_{-}$ be its positive-negative spectral decomposition and let $P$ project onto the support of $X_{+}$. Since $\mathrm{Tr}X=0$, $\mathrm{Tr}{X}_{+}=\mathrm{Tr}{X}_{-}=\Vert X{\Vert }_1/2$. Thus

Choose a basis $|a⟩$ of $H$ and define Kraus operators from $H$ to ${\mathbb{C}}^m$ byThenandThe displayed map is therefore a completely positive trace-preserving measure-and-prepare channel.

For any POVM, part c followed by trace-distance contraction givesFor the binary POVM $\{P,I-P\}$ from part b, equality holds. Consequently

For outcome probabilities $p_k=\mathrm{Tr}({\Lambda }_k\rho )$ and $q_k=\mathrm{Tr}({\Lambda }_k\sigma )$, the hint givesThe left side is at most $\Vert \rho -\sigma {\Vert }_1/2$ by part d. Choose a POVM attaining the stated minimum characterization of quantum fidelity; then