Growth in Matrix Algebras and a Conjecture of Pérez-García, Verstraete, Wolf and Cirac

Let S be a family of n× n matrices over a field, and let S^ℓ 𝔽 be the linear span of all products of length ℓ of the matrices in S. Assume that, for some ℓ , the set S^ℓ 𝔽 is the full n× n matrix algebra. We show that, in this case, the set S^k 𝔽 is the full n× n matrix algebra, for any integer k⩾ n^2+2n-3 .