  Let $0<p,q\leq \infty$ and denote by $\mathcal S_p^N$ and $\mathcal S_q^N$
the corresponding Schatten classes of real $N\times N$ matrices. We study
approximation quantities of natural identities $\mathcal S_p^N\hookrightarrow
\mathcal S_q^N$ between Schatten classes and prove asymptotically sharp bounds
up to constants only depending on $p$ and $q$, showing how approximation
numbers are intimately related to the Gelfand numbers and their duals, the
Kolmogorov numbers. In particular, we obtain new bounds for those sequences of
$s$-numbers. Our results improve and complement bounds previously obtained by
B. Carl and A. Defant [J. Approx. Theory, 88(2):228--256, 1997], Y. Gordon, H.
K\"onig, and C. Sch\"utt [J. Approx. Theory, 49(3):219--239, 1987], A. Hinrichs
and C. Michels [Rend. Circ. Mat. Palermo (2) Suppl., (76):395--411, 2005], and
A. Hinrichs, J. Prochno, and J. Vyb\'iral [preprint, 2020]. We also treat the
case of quasi-Schatten norms, which is relevant in applications such as
low-rank matrix recovery.
