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  • Open Access

Free Mutual Information and Higher-Point Out-of-Time-Ordered Correlators

Shreya Vardhan1,* and Jinzhao Wang2,†

  • *Contact author: svardhan@caltech.edu
  • †Contact author: wang.jinzhao226@gmail.com
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PRX Quantum 7, 043017 – Published 26 October, 2026

DOI: https://doi.org/10.1103/wt7l-6385

Abstract

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 behaviors 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 four-point OTOC.

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In classical chaos, it is well known that the trajectory of a chaotic particle fills the entire space available to it at late times. It is natural to ask whether a similar notion exists in the much more complex world of quantum chaotic dynamics. In this paper, we address this question, using a new measure of the volume of a quantum operator in the enormous space of all available operators, which is doubly exponential in the number of physical constituents of the system. We show that in quantum chaotic systems, the volume universally grows to fill this entire space. This volume growth indicates that chaotic dynamics typically set the final operator “free” from the initial operator in a precise mathematical sense. We find a general relation that allows us to deduce the value of the abstract volume measure from concrete experimental observables, which have recently been measured on cutting-edge quantum platforms.

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PRX Quantum 7, 043017– Published 26 October, 2026
  • Received 15 May 2026
  • Accepted 10 September 2026
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