A generic quantum Wielandt's inequality

In this paper, we provide a generic version of quantum Wielandt's inequality, which gives an optimal upper bound on the minimal length $k$ such that length-$k$ products of elements in a generating system span $M_n(\mathbb{C})$ with probability one. We show that $k$ generically is of order $\Theta(\log n)$, as opposed to the general case, in which the best bound to the date is $O(n^2 \log n)$. Our result implies a new bound on the primitivity index of a random quantum channel. Furthermore, we shed new light on a long-standing open problem for Projected Entangled Pair State, by concluding that almost any translation-invariant PEPS (in particular, Matrix Product State) with periodic boundary conditions on a grid with side length of order $\Omega( \log n )$ is the unique ground state of a local Hamiltonian. We observe similar characteristics for matrix Lie algebras and provide numerical results for random Lie-generating systems.Comment: 13 pages, 6 figure