Understanding how to characterize quantum chaotic dynamics is a longstanding question.

In this colloquium, I will discuss recent developments that identify Free Probability --  a generalization of probability theory to non-commuting objects --
as the unifying mathematical framework to describe correlations of chaotic many-body systems. 
I will show how the full version of the Eigenstate Thermalization Hypothesis, the established framework for quantum dynamics that encompasses all the correlations, can be rationalized and simplified using the language of Free Probability. This approach uncovers unexpected connections between quantum information theory and quantum chaos.