The <Umegaki relative entropy> is
$$
D(\rho\|\sigma)
=\operatorname{Tr}\rho(\log\rho-\log\sigma)
$$
when the support of $\rho$ is contained in that of $\sigma$, and $+\infty$ otherwise. Its <additivity of quantum relative entropy> is
$$
D(\rho_A\otimes\rho_B\|\sigma_A\otimes\sigma_B)
=D(\rho_A\|\sigma_A)+D(\rho_B\|\sigma_B),
$$
its <superadditivity of quantum relative entropy> is
$$
D(\rho_{AB}\|\sigma_A\otimes\sigma_B)
\geq D(\rho_A\|\sigma_A)+D(\rho_B\|\sigma_B),
$$
and its <data-processing inequality for quantum relative entropy> is $D(T(\rho)\|T(\sigma))\leq D(\rho\|\sigma)$ for every <quantum channel> $T$.

Additivity follows from the logarithm of a <tensor product>,
$$
\log(\rho_A\otimes\rho_B)
=\log\rho_A\otimes I+I\otimes\log\rho_B,
$$
and the analogous identity for $\sigma_A\otimes\sigma_B$. For superadditivity, subtract the two marginal relative entropies from the joint one. The reference-state terms cancel, leaving
$$
S(\rho_A)+S(\rho_B)-S(\rho_{AB})
=I(A:B)_\rho\geq0.
$$
This is the nonnegativity of <quantum mutual information>, equivalently <Subadditivity of Von Neumann entropy>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-323/4/i/solution