Decoupling for Schatten class operators in the setting of Quantum Harmonic Analysis

Abstract

We introduce the notion of decoupling for operators, and prove an equivalence between classical ${\ell }^q{L}^p$ decoupling for functions and ${\ell }^q{\mathcal{S}}^p$ decoupling for operators on bounded sets in ${\mathbb{R}}^{2d}$. We also show that the equivalence only depends on the bounded set $\hat{I}\text{c◯}$, and not the values of $p,q$ nor the partition of $\hat{I}\text{c◯}$. The proof relies on a quantum version of Wiener's division lemma.

13 pages. Accepted to Bulletin of the London Mathematical Society