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A central feature of quantum mechanics is the non-commutativity of operatorsused to describe physical observables. In this article, we present a criticalanalysis on the role of non-commutativity in quantum theory, focusing on itsconsequences in the probabilistic description. Typically, a random phenomenonis described using the measure-theoretic formulation of probability theory.Such a description can also be done using algebraic methods, which are capableto deal with non-commutative random variables (like in quantum mechanics). Herewe propose a method to construct a non-commutative probability theory startingfrom an ordinary measure-theoretic description of probability. This will bedone using the entropic uncertainty relations between random variables, inorder to evaluate the presence of non-commutativity in their algebraicdescription.

On non-commutativity in quantum theory (I): from classical to quantum probability