  We study the robustness of quantum and classical information to perturbations
implemented by local operator insertions. We do this by computing multipartite
entanglement measures in the Hilbert space of local operators in the Heisenberg
picture. The sensitivity to initial conditions that we explore is an
illuminating manifestation of the butterfly effect in quantum many-body
systems. We derive a "membrane theory" in Haar random unitary circuits to
compute the mutual information, logarithmic negativity, and reflected entropy
in the local operator state by mapping to a classical statistical mechanics
problem and find that any local operator insertion delocalizes information as
fast as is allowed by causality. Identical behavior is found for conformal
field theories admitting holographic duals where the bulk geometry is described
by the eternal black hole with a local object situated at the horizon. In
contrast to these maximal scramblers, only an $O(1)$ amount of information is
found to be delocalized by local operators in integrable systems such as free
fermions and Clifford circuits.
