Entropy numbers of embeddings of Schatten classes

Abstract: 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.