  We study the central limit theorem for sums of independent tensor powers,
$\frac{1}{\sqrt{d}}\sum\limits_{i=1}^d X_i^{\otimes p}$. We focus on the
high-dimensional regime where $X_i \in \mathbb{R}^n$ and $n$ may scale with
$d$. Our main result is a proposed threshold for convergence. Specifically, we
show that, under some regularity assumption, if $n^{2p-1}\gg d$, then the
normalized sum converges to a Gaussian. The results apply, among others, to
symmetric uniform log-concave measures and to product measures. This
generalizes several results found in the literature.
  Our main technique is a novel application of optimal transport to Stein's
method which accounts for the low dimensional structure which is inherent in
$X_i^{\otimes p}$.
