  The eigenstate thermalization hypothesis (ETH) is the leading conjecture for
the emergence of statistical mechanics in generic isolated quantum systems and
is formulated in terms of the matrix elements of operators. An analog known as
the ergodic bipartition (EB) describes entanglement and locality and is
formulated in terms of the components of eigenstates. In this paper, we
significantly generalize the EB and unify it with the ETH, extending the EB to
study higher correlations and systems out of equilibrium. Our main result is a
diagrammatic formalism that computes arbitrary correlations between eigenstates
and operators based on a recently uncovered connection between the ETH and free
probability theory. We refer to the connected components of our diagrams as
generalized free cumulants. We apply our formalism in several ways. First, we
focus on chaotic eigenstates and establish the so-called subsystem ETH and the
Page curve as consequences of our construction. We also improve known
calculations for thermal reduced density matrices and comment on an inherently
free probabilistic aspect of the replica approach to entanglement entropy
previously noticed in a calculation for the Page curve of an evaporating black
hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a
sufficient mechanism for thermalization, in general. In particular, we show
that reduced density matrices relax to their equilibrium form and that systems
obey the Page curve at late times. We also demonstrate that entanglement
velocities, which govern the spreading of entanglement, are encoded in higher
correlations of the EB. Lastly, we examine the chaotic structure of eigenstates
and operators together and reveal previously overlooked correlations between
them. Crucially, these correlations encode butterfly velocities, a well-known
dynamical property of interacting quantum systems.
