Entanglement of Local Operators and the Butterfly Effect

TL;DR

This paper investigates how local operator insertions affect the spread and robustness of quantum and classical information in many-body systems, revealing maximal delocalization in chaotic systems and limited effects in integrable ones.

Contribution

It introduces a membrane theory to compute entanglement measures in local operator states and compares behavior across chaotic, holographic, and integrable systems.

Findings

Local operators delocalize information at the fastest possible rate in chaotic systems.

Holographic systems exhibit maximal information delocalization consistent with causality.

Integrable systems show limited information delocalization, only O(1) in magnitude.

Abstract

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.