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

TL;DR

This paper introduces a new decoupling concept for operators in Quantum Harmonic Analysis, establishing an equivalence with classical decoupling for functions, independent of parameters, using a quantum Wiener's division lemma.

Contribution

It develops a novel operator decoupling framework and proves an equivalence with classical decoupling, extending harmonic analysis techniques into the quantum setting.

Findings

Decoupling equivalence depends only on the bounded set, not on parameters.

Established a quantum Wiener's division lemma.

Unified classical and quantum decoupling theories.

Abstract

We introduce the notion of decoupling for operators, and prove an equivalence between classical $\ell^qL^p$ 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 $\Omega$, and not the values of $p,q$ nor the partition of $\Omega$. The proof relies on a quantum version of Wiener's division lemma.