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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 thehigh-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, weshow that, under some regularity assumption, if $n^{2p-1}\gg d$, then thenormalized sum converges to a Gaussian. The results apply, among others, tosymmetric uniform log-concave measures and to product measures. Thisgeneralizes several results found in the literature.  Our main technique is a novel application of optimal transport to Stein'smethod which accounts for the low dimensional structure which is inherent in$X_i^{\otimes p}$.

A CLT in Stein's distance for generalized Wishart matrices and higher order tensors