Free mutual information and higher-point OTOCs

Understand

We introduce a quantity called the free mutual information (FMI), adapted from concepts in free probability theory, as a new physical measure of quantum chaos.

• This quantity captures the spreading of a time-evolved operator in the space of all possible operators on the Hilbert space, which is doubly exponential in the number of degrees of freedom.
• It thus provides a finer notion of operator spreading than the well-understood phenomenon of operator growth in physical space.
• We derive two central results which apply in any physical system: first, an explicit ``Coulomb gas'' formula for the FMI of two observables $A(t)$ and $B$ in terms of the eigenvalues of the product operator $A(t)B$; and second, a general relation expressing the FMI as a weighted sum of all higher-point out-of-time-ordered correlators (OTOCs).