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

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