Quantum version of Wielandt's Inequality revisited

TL;DR

This paper improves the bound on the stabilization time of a sequence of operator spaces in a quantum setting, reducing it from O(D^4) to O(D^2 log(D)), thus advancing understanding of quantum operator algebra properties.

Contribution

The paper proves a tighter bound of O(D^2 log(D)) for the stabilization of the sequence of operator spaces, improving upon the previous O(D^4) bound.

Findings

Stabilization occurs after O(D^2 log(D)) steps

Improved bound from previous O(D^4)

Advances quantum operator algebra theory

Abstract

Consider a linear space L of complex D-dimensional linear operators, and assume that some power L^k of L is the whole space of DxD matrices. Perez-Garcia, Verstraete, Wolf and Cirac conjectured that the sequence L^1,L^2,... stablilizes after O(D^2) terms; we prove that this happens after O(D^2 log(D)) terms, improving the previously known bound of O(D^4).