Speed limits and scrambling in Krylov space

We investigate the relationship between Krylov complexity and operator quantum speed limits (OQSLs) of the complexity operator and level repulsion in random/integrable matrices and many-body systems. An enhanced level repulsion corresponds to increased OQSLs in random/integrable matrices. However, in many-body systems, the dynamics is more intricate due to the tensor product structure of the models. Initially, as the integrability-breaking parameter increases, the OQSL also increases, suggesting that breaking integrability allows for faster evolution of the complexity operator. At larger values of integrability breaking, the OQSL decreases, suggesting a slowdown in the operator's evolution speed. Information-theoretic properties, such as scrambling, coherence, and entanglement, of Krylov basis operators in many-body systems are also investigated. The scrambling behavior of these operators exhibits distinct patterns in integrable and chaotic cases. For systems exhibiting chaotic dynamics, the Krylov basis operators remain a reliable measure of these properties of the time-evolved operator at late times. However, in integrable systems, the Krylov operator's ability to capture the entanglement dynamics is less effective, especially during late times.