Both states are block diagonal in $X$. On the support of each block,
$$
\log\rho_{XB}=\bigoplus_x[\log p_x+\log\rho_x],
\qquad
\log\sigma_{XB}=\bigoplus_x[\log q_x+\log\sigma_x].
$$
Substitution into <quantum relative entropy> gives
$$
\boxed{D(\rho_{XB}\|\sigma_{XB})
=D(p\|q)+\sum_xp_xD(\rho_x\|\sigma_x)}.
$$
Taking $-\operatorname{Tr}\rho_{XB}\log\rho_{XB}$ similarly yields the entropy of a <classical-quantum state>:
$$
\boxed{S(\rho_{XB})=H(p)+\sum_xp_xS(\rho_x)}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-323/3/a/solution