Linear independence of quantum translates

Abstract

If $A$ is in the $p$-Schatten class on ${\mathbb{R}}^n$, $1\le p\le \frac{4n}{2n-1}$, then the quantum translates of $A$ are linearly independent. Moreover, there exists a non-zero operator in the $p$-Schatten class on ${\mathbb{R}}^n$, $p>\frac{4n}{2n-1}$ whose quantum translates are linearly dependent.