Signature of quantum chaos in operator entanglement in 2d conformal field theories

We study operator entanglement measures of the unitary evolution operators of (1+1)-dimensional conformal field theories (CFT), aiming to uncover their scrambling and chaotic behaviors. In particular, we compute the bi-partite and tri-partite mutual information for various configurations of input and output subsystems, and as a function of time. We contrast three different CFTs: the free fermion, the compactified free boson at various radii, and CFTs with holographic dual. We found that the bi-partite mutual information exhibits distinct behaviors for these CFTs, reflecting the different information scrambling capabilities of these unitary operators; while a quasi-particle picture can describe well the case the free fermion and free boson CFTs, it completely fails for the case of holographic CFTs. Similarly, the tri-partite mutual information also distinguishes the unitary evolution operators of different CFTs. In particular, its late time behaviors, when the output subsystems are semi-infinite, are quite distinct for these theories. We speculate that for holographic theories the late time value of the tri-partite mutual information saturates the lower bound among quantum field theories.