Apply bipartite <superadditivity of quantum relative entropy> first to system $1$ and systems $2,\ldots,n$, and then repeat on the remaining joint state. <Mathematical induction> gives
$$
\boxed{
D(\rho_{1\ldots n}\|
\sigma_1\otimes\cdots\otimes\sigma_n)
\geq\sum_{j=1}^nD(\rho_j\|\sigma_j)}.
$$
This is <multipartite superadditivity of quantum relative entropy>. Repeated application of bipartite additivity similarly gives
$$
\boxed{
D(\rho_1\otimes\cdots\otimes\rho_n\|
\sigma_1\otimes\cdots\otimes\sigma_n)
=\sum_{j=1}^nD(\rho_j\|\sigma_j)}.
$$

Solution

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

 Solution

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