Dynamical maps and symmetroids

TL;DR

This paper develops a framework for describing dynamical maps in quantum mechanics using symmetroids and groupoid algebras, introducing new constructions and examples within this mathematical setting.

Contribution

It introduces a novel approach to quantum dynamical maps via symmetroids and von Neumann groupoid algebras, expanding the mathematical tools available in quantum theory.

Findings

Construction of the von Neumann groupoid algebra from a symmetroid

Definition of linear maps on the groupoid algebra via the left-regular representation

Examples illustrating the application of the theoretical framework

Abstract

Starting from the canonical symmetroid $\mathcal{S}(G)$ associated with a groupoid $G$, the issue of describing dynamical maps in the groupoidal approach to Quantum Mechanics is addressed. After inducing a Haar measure on the canonical symmetroid $\mathcal{S}(G)$, the associated von-Neumann groupoid algebra is constructed. It is shown that the left-regular representation allows to define linear maps on the groupoid-algebra of the groupoid $G$ and given subsets of functions are associated with completely positive maps. Some simple examples are also presented.