Free mutual information and higher-point OTOCs

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). This second result provides a precise information-theoretic interpretation for the higher-point OTOCs as collectively quantifying operator ergodicity and the approach to freeness. This physical interpretation is particularly useful in light of recent progress in experimentally measuring higher-point OTOCs. We identify universal behaviours of the FMI and higher-point OTOCs across a variety of chaotic systems, including random unitary circuits and chaotic spin chains, which indicate that spreading in the doubly exponential operator space is a generic feature of quantum many-body chaos. At the same time, the non-generic behavior of the FMI in various non-chaotic systems, including certain unitary designs, shows that there are cases where an operator spreads in physical space but remains localized in operator space. The FMI is thus a sharper diagnostic of chaos than the standard 4-point OTOC.