A generic quantum Wielandt's inequality

Abstract

Quantum Wielandt's inequality gives an optimal upper bound on the minimal length k such that length-k products of elements in a generating system span Mn(C). It is conjectured that k should be of order O(n2) in general. In this paper, we give an overview of how the question has been studied in the literature so far and its relation to a classical question in linear algebra, namely the length of the algebra Mn(C). We provide a generic version of quantum Wielandt's inequality, which gives the optimal length with probability one. More specifically, we prove based on [KS16] that k generically is of order Θ(log⁡n), as opposed to the general case, in which the best bound to date is O(n2log⁡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 Ω(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.