Quantum Version of Wielandt’s Inequality Revisited

Consider a linear space L of complex D-dimensional linear operators and assume that some power Lk of L is the whole set End(CD). 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
).