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 ℓqLp$\ell ^qL^p$ decoupling for functions and ℓqSp$\ell ^q{\mathcal {S}}^p$ decoupling for operators on bounded sets in R2d${\mathbb {R}}^{2d}$. We also show that the equivalence depends only on the bounded set Ω$\Omega$ and not on the values of p,q$p,q$ nor on the partition of Ω$\Omega$. The proof relies on a quantum version of Wiener's division lemma.