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 $n\ge 1$, every even integer $m\ge 4$, and every mapping $f:{\mathbb{Z}}_m^n\to S_1$, the following holds \begin{align*} &\left(\sum_{j=1}^{n}$\mathbb{E}${x}\left|f\left(x+\frac{m}{2}e{j}\right)-f(x)\right|{1}^{2}\right)^{1/2}\ \leq&\left(3\sqrt{70},m+6\sqrt{n}\right)\left($\mathbb{E}${x,\varepsilon}\left|f(x+\varepsilon)-f(x)\right|_{1}^{2}\right)^{1/2}, \end{align*} where $x$ and $\epsilon$ are independently uniform on ${\mathbb{Z}}_m^n$ and $\{-1,0,1{\}}^n$, respectively. The order $\sqrt{n}$ is necessary for any fixed metric cotype constant. This settles the optimal scaling problem for the Schatten trace class posed by Mendel and Naor \cite[Section 8]{MN} (see also \cite[Question 3]{GMN}).