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We consider a problem in random matrix theory that is inspired by quantuminformation theory: determining the largest eigenvalue of a sum of p randomproduct states in (C^d)^{otimes k}, where k and p/d^k are fixed while d grows.When k=1, the Marcenko-Pastur law determines (up to small corrections) not onlythe largest eigenvalue ((1+sqrt{p/d^k})^2) but the smallest eigenvalue(min(0,1-sqrt{p/d^k})^2) and the spectral density in between. We use the methodof moments to show that for k>1 the largest eigenvalue is still approximately(1+sqrt{p/d^k})^2 and the spectral density approaches that of theMarcenko-Pastur law, generalizing the random matrix theory result to the randomtensor case. Our bound on the largest eigenvalue has implications both forsampling from a particular heavy-tailed distribution and for a recentlyproposed quantum data-hiding and correlation-locking scheme due to Leung andWinter. Since the matrices we consider have neither independent entries norunitary invariance, we need to develop new techniques for their analysis. Themain contribution of this paper is to give three different methods foranalyzing mixtures of random product states: a diagrammatic approach based onGaussian integrals, a combinatorial method that looks at the cycledecompositions of permutations and a recursive method that uses a variant ofthe Schwinger-Dyson equations.

Random tensor theory: extending random matrix theory to random product states