If $\lVert\rho'_n-\rho^{\otimes n}\rVert_1\to0$, then
$$
\liminf_{n\to\infty}\frac1n
\left(D(\rho'_n\|\sigma^{\otimes n})-D(\rho^{\otimes n}\|\sigma^{\otimes n})\right)\geq0.
$$
In finite dimension this follows from <multipartite superadditivity of quantum relative entropy>, <additivity of quantum relative entropy>, contraction of <trace distance> under <partial trace>, and <uniform continuity> on the compact state space.

Lower asymptotic semicontinuity of quantum relative entropy

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

Repeated application of <superadditivity of quantum relative entropy> gives
$$
D(\rho_{1\ldots n}\|\sigma_1\otimes\cdots\otimes\sigma_n)
\geq\sum_{j=1}^nD(\rho_j\|\sigma_j).
$$

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