Algebraic structures underlying quantum independences : Theory and Applications

TL;DR

This survey unifies algebraic and physical approaches to quantum probabilities, introducing group and bialgebra techniques to advance understanding of quantum independence and related noncommutative probability concepts.

Contribution

It provides a unified algebraic framework for quantum probabilities, connecting different approaches and applying group and bialgebra methods to noncommutative probability theories.

Findings

Unified algebraic framework for quantum independence

Application of group and bialgebra techniques

Recent results on cumulants and Wick polynomials

Abstract

The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture introducing jointly to several fields of applications, many of which are probably not all familiar (at leat at the same time and in the form we use to present them) to the readers. Lastly, we take the opportunity to present various results obtained recently that use group and bialgebra techniques to handle notions such as cumulants or Wick polynomials in the various noncommutative probability theories.