  A central feature of quantum mechanics is the non-commutativity of operators
used to describe physical observables. In this article, we present a critical
analysis on the role of non-commutativity in quantum theory, focusing on its
consequences in the probabilistic description. Typically, a random phenomenon
is described using the measure-theoretic formulation of probability theory.
Such a description can also be done using algebraic methods, which are capable
to deal with non-commutative random variables (like in quantum mechanics). Here
we propose a method to construct a non-commutative probability theory starting
from an ordinary measure-theoretic description of probability. This will be
done using the entropic uncertainty relations between random variables, in
order to evaluate the presence of non-commutativity in their algebraic
description.
