The Sharp Scale Inequality for the Schatten Trace Class

In this paper, we establish the metric cotype $2$ inequality for the Schatten trace class $S_{1}$ at the optimal order of the scaling parameter. More precisely, for every integer $n\geq 1$, every even integer $m\geq 4$, and every mapping $f:\mathbb{Z}_{m}^{n}\to S_{1}$, the following holds \begin{al