= Lower asymptotic semicontinuity of quantum relative entropy
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.
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