Quantum paving: When sphere packings meet Gabor frames

TL;DR

This paper introduces quantum packing, covering, and paving problems rooted in quantum physics and Gabor analysis, exploring their solutions, conjectures, and applications in non-commutative operator algebras.

Contribution

It formulates new quantum packing, covering, and paving problems and connects them to classical sphere packing and covering, providing initial solutions and conjectures.

Findings

Solutions for specific cases of quantum packing and covering

Formulation of quantum paving problem and related conjectures

Discussion of applications in quantum physics and Gabor analysis

Abstract

We introduce the new problems of quantum packing, quantum covering, and quantum paving. These problems arise naturally when considering an algebra of non-commutative operators that is deeply rooted in quantum physics as well as in Gabor analysis. Quantum packing and quantum covering show similarities with energy minimization and the dual problem of polarization. Quantum paving, in turn, aims to simultaneously optimize both quantum packing and quantum covering. Classical sphere packing and covering hint the optimal configurations for our new problems. We present solutions in certain cases, state several conjectures related to quantum paving and discuss some applications.