Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$.
We would like to bound the following expectation on the trace norm
$\mathbb{E}|\sum_{1\leq i\leq d^2}\frac{\sum_{j=1}^mx_{i,j}}{m}H_i|_{tr},$
where $x_{i,j}$ are iid sample of $X_i$.
