Let $A,B$ be finite dimensional full-rank positive semidefinite Hermitian matrices with unit trace. Let $|A-B|_1 \leq \varepsilon$ where $|X|_1 = \text{Tr}(\sqrt{X^*X})$ and $X^*$ is the transpose conjugate of $X$. $\varepsilon$ can be chosen arbitrarily small. If so, can one bound
$$|A\log A - B\log B|_1$$
in terms of $\varepsilon$?
Possible related question here and also note the Fannes inequality for entropies does pretty much this for $|\text{Tr}(A\log A) - \text{Tr}(B\log B)|$.
