Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 323 / 2

The map $\Lambda$ is a completely positive map when $\Lambda \otimes {\mathrm{id}}_m$ maps positive operators to positive operators for every $m$. In a basis $\{|i⟩\}$ define the unnormalized Choi matrixIf $\Lambda$ is completely positive, $J(\Lambda )=(\mathrm{id}\otimes \Lambda )(|\Omega ⟩⟨\Omega |)\ge 0$, where $|\Omega ⟩={\sum }_i|ii⟩$. Conversely, decompose $J={\sum }_k|a_k⟩⟨a_k|$ and reshape each vector into an operator $A_k$. The Choi reconstruction formula gives $\Lambda (X)={\sum }_k{A}_{k}X{A}_k^{†}$, which is completely positive. Thus

For every ancillary system,when $X\ge 0$, proving complete positivity. Cyclicity of trace givesThis equals $\mathrm{Tr}\rho$ for every $\rho$ exactly whenthe trace-preserving condition in the Kraus representation.

With $P_i=|y_i⟩⟨y_i|$, the channel has Kraus form $Y(\rho )={\sum }_i{P}_i\rho P_i$ and ${\sum }_i{P}_i^{†}{P}_i=I$, so it is completely positive and trace preserving. It measures in the $y$ basis and prepares the observed basis state, hence is a measure-and-prepare channel. If $p_i=⟨y_i|\rho |y_i⟩$, then $Y(\rho )={\sum }_i{p}_i{P}_i$ andMoreover $\log Y(\rho )={\sum }_i(\log p_i)P_i$ on its support, so

Write $q_j=⟨z_j|\rho |z_j⟩$, $r_j=⟨z_j|Y(\rho )|z_j⟩$, and $c={\max }_{i,j}|⟨y_i|z_j⟩{|}^2$. Thenand thereforeThe data-processing inequality for quantum relative entropy applied to $Z$ and part c giveCombining them proves