A generic quantum Wielandt’s inequality

Quantum 8, 1331 (2024).

https://doi.org/10.22331/q-2024-05-02-1331

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 $M_n(mathbb{C})$. It is conjectured that $k$ should be of order $mathcal{O}(n^2)$ 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 $M_n(mathbb{C})$. 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 $Theta(log n)$, as opposed to the general case, in which the best bound to date is $mathcal O(n^2 log n)$. 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.