  The delocalization or scrambling of quantum information has emerged as a
central ingredient in the understanding of thermalization in isolated quantum
many-body systems. Recently, significant progress has been made analytically by
modeling non-integrable systems as stochastic systems, lacking a Hamiltonian
picture, while honest Hamiltonian dynamics are frequently limited to small
system sizes due to computational constraints. In this paper, we address this
by investigating the role of conservation laws (including energy conservation)
in the thermalization process from an information-theoretic perspective. For
general non-integrable models, we use the equilibrium approximation to show
that the maximal amount of information is scrambled (as measured by the
tripartite mutual information of the time-evolution operator) at late times
even when a system conserves energy. In contrast, we explicate how when a
system has additional symmetries that lead to degeneracies in the spectrum, the
amount of information scrambled must decrease. This general theory is
exemplified in case studies of holographic conformal field theories (CFTs) and
the Sachdev-Ye-Kitaev (SYK) model. Due to the large Virasoro symmetry in 1+1D
CFTs, we argue that, in a sense, these holographic theories are not maximally
chaotic, which is explicitly seen by the non-saturation of the second R\'enyi
tripartite mutual information. The roles of particle-hole and U(1) symmetries
in the SYK model are milder due to the degeneracies being only two-fold, which
we confirm explicitly at both large- and small-$N$. We reinterpret the operator
entanglement in terms the growth of local operators, connecting our results
with the information scrambling described by out-of-time-ordered correlators,
identifying the mechanism for suppressed scrambling from the Heisenberg
perspective.
