0000000000951056
AUTHOR
Matthew B. Hastings
Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States
We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in $${(\mathbb {C}^d)^{\otimes k}}$$ , where k and p/d k are fixed while d → ∞. When k = 1, the Marcenko-Pastur law determines (up to small corrections) not only the 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 method of 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 the Marcenko-Pastur law, generalizing the random matrix…
Random tensor theory: extending random matrix theory to random product states
We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product 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 only the 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 method of 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 the Marcenko-Pastur law, generalizing the random matrix theory result to the random tensor case.…