Entanglement of Local Operators and the Butterfly Effect

The scrambling properties of local operators are analyzed by studying the local operator entanglement and related measures of multi-partite entanglement. The amount of information delocalization is measured by the tri-partite operator mutual information. It is shown that chaotic systems like holographic CFTs and Haar random unitary circuits scramble the maximal amount of information possible, which is proportional to the volume of the input Hilbert space, while integrable systems such as the free Fermion and Clifford circuits scramble only an O(1) amount.