Towards a quantum sampling theory: the case of finite groups

TL;DR

This paper extends classical sampling theory to quantum systems with finite group symmetries, using frames and invariant subspaces to develop a quantum sampling framework.

Contribution

It introduces a quantum sampling theory for finite groups, adapting classical concepts like frames and invariant subspaces to the quantum setting.

Findings

Developed a quantum sampling framework for finite groups.

Analyzed classical sampling concepts in the quantum context.

Provided illustrative examples of the quantum sampling theory.

Abstract

Nyquist-Shannon sampling theorem, instrumental in classical telecommunication technologies, is extended to quantum systems supporting a unitary representation of a finite group $G$. Two main ideas from the classical theory having natural counterparts in the quantum setting: frames and invariant subspaces, provide the mathematical background for the theory. The main ingredients of classical sampling theorems are discussed and their quantum counterparts are thoroughly analyzed in this simple situation. A few examples illustrating the obtained results are discussed.