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We study the eigenvalue distributions for sums of independent rank-one$k$-fold tensor products of large $n$-dimensional vectors. Previous results inthe literature assume that $k=o(n)$ and show that the eigenvalue distributionsconverge to the celebrated Mar\v{c}enko-Pastur law under appropriate momentconditions on the base vectors. In this paper, motivated by quantum informationtheory, we study the regime where $k$ grows faster, namely $k=O(n)$. We showthat the moment sequences of the eigenvalue distributions have a limit, whichis different from the Mar\v{c}enko-Pastur law. As a byproduct, we show that theMar\v{c}enko-Pastur law limit holds if and only if $k=o(n)$ for this tensormodel. The approach is based on the method of moments.

On spectral distribution of sample covariance matrices from large dimensional and large $k$-fold tensor products