Quantum Speed limit for Information Scrambling

Quantum Speed Limits (QSL) provide fundamental bounds on the rate of evolution in quantum systems. We adapt this framework to study information scrambling in quantum many-body systems, focusing on the dynamics of Out-of-Time-Ordered Correlators (OTOCs). We relate the closed system OTOCs to the second Renyi entropy of a subsystem thereby bounding the rate at which information is scrambled by the norm of the Liouvillian. The derived speed limit constrains the growth of the Renyi entropy, and hence the OTOC, in terms of the decay rates of the Liouvillian dynamics. We then test our bound with the archetypal Bose-Hubbard model, known to exhibit scrambling at strong couplings. We describe the reduced dynamics of the subsystem using a quantum master equation that reproduces the celebrated Redfield equation in the thermodynamic limit of the closed system. We then investigate the validity of the QSL for different values of coupling strengths to assess the validity of Born-Markov approximation. Our numerical results for small systems reveal the importance of our bound despite the undesirable non-Markovian effects and our findings provide deep insight into possible generalized open system models of dissipative quantum chaos.