Entanglement of local operators and the butterfly effect

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 present 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 after taking the large local Hilbert space dimension limit. 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 free fermionic systems and random Clifford circuits.

Automated summary

This study explores the robustness of quantum and classical information to perturbations implemented by local operator insertions, by computing multipartite entanglement measures in the Hilbert space of local operators. The research reveals the butterfly effect in quantum many-body systems and investigates membrane theory in Haar random unitary circuits to study the phenomenon of information delocalization caused by local operator insertions. Identical behavior is found in conformal field theories with holographic duals, while a limited amount of information is delocalized in free fermionic systems and random Clifford circuits.