Given basis $M_1,M_2\dotsc,M_{d^2}$ in $\mathbb C^{d\times d}$, we consider $$\sum_i x_i M_i$$ for random variables $x_i$.
What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1=\sum \sigma_k?$$ Here $\sigma_k$s are singular values of $\sum_i x_i M_i$.
What is the distribution of $$\frac{\lVert\sum_i x_i M_i\rVert_1}{\lVert\sum_i x_i M_i\rVert_2}?$$
What is the eigenvalue distribution (the vector of the eigenvalues) of $\sum_i x_i M_i$, if $x_i$ are independent Gaussian $\mathcal{N}(\mu_i,\sigma_i^2)$? If $x_i$ are uniform distribution in $[a_i,b_i]$?
What is the distribution of $$\lVert\sum_i x_i M_i\rVert_1?$$
What is the distribution of $$\frac{\lVert\sum_i x_i M_i\rVert_1}{\lVert\sum_i x_i M_i\rVert_2}?$$
