  Let $0<p,q \leq \infty$ and denote by $\mathcal S_p^N$ and $\mathcal S_q^N$
the corresponding finite-dimensional Schatten classes. We prove optimal bounds,
up to constants only depending on $p$ and $q$, for the entropy numbers of
natural embeddings between $\mathcal S_p^N$ and $\mathcal S_q^N$. This
complements the known results in the classical setting of natural embeddings
between finite-dimensional $\ell_p$ spaces due to Sch\"utt, Edmunds-Triebel,
Triebel and Gu\'edon-Litvak/K\"uhn. We present a rather short proof that uses
all the known techniques as well as a constructive proof of the upper bound in
the range $N\leq n\leq N^2$ that allows deeper structural insight and is
therefore interesting in its own right. Our main result can also be used to
provide an alternative proof of recent lower bounds in the area of low-rank
matrix recovery.
