On non-commutative spreadability

TL;DR

This paper explores the structure of spreadable quantum stochastic processes, analyzing their invariance properties under specific monoids, and extends classical theorems to non-commutative settings for various types of processes.

Contribution

It introduces a framework linking spreadability to monoid actions and extends the Ryll-Nardzewski Theorem to non-commutative processes such as Boolean, monotone, and q-deformed.

Findings

Characterization of monoids acting on the integer index set.

Invariance under monoid actions characterizes spreadability.

Extended Ryll-Nardzewski Theorem for non-commutative processes.

Abstract

We review some results on spreadable quantum stochastic processes and present the structure of some monoids acting on the index-set of all integers $\mathbb Z$. These semigroups are strictly related to spreadability, as the latter can be directly stated in terms of invariance with respect to their action. We are mainly focused on spreadable, Boolean, monotone, and $q$-deformed processes. In particular, we give a suitable version of the Ryll-Nardzewski Theorem in the aforementioned cases.