Quantum Signatures of Chaos from Free Probability

Understand

A classical dynamical system can be viewed as a probability space equipped with a measure-preserving time evolution map, admitting a purely algebraic formulation in terms of the algebra of bounded functions on the phase space.

• Similarly, a quantum dynamical system can be formulated using an algebra of bounded operators in a non-commutative probability space equipped with a time evolution map.
• Chaos, in either setting, can be characterized by statistical independence between observables at $ t = 0 $ and $ t \to \infty $, leading to the vanishing of cumulants involving these observables.
• In the quantum case, the notion of independence is replaced by free independence, which only emerges in the thermodynamic limit (asymptotic freeness).