A generic quantum Wielandt's inequality

TL;DR

This paper establishes that, with high probability, the minimal length for products of generators to span the full matrix algebra is logarithmic in size, significantly improving previous bounds and impacting quantum channel and tensor network theories.

Contribution

The paper introduces a generic version of quantum Wielandt's inequality showing the minimal length is typically Θ(log n), refining the known O(n^2 log n) bound and connecting to quantum channels and tensor network ground states.

Findings

Minimal length generically Θ(log n)

Improved bound on primitivity index of quantum channels

Almost all translation-invariant PEPS are unique ground states

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 $M_n(\mathbb{C})$. 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.