Metrological Limits to Information Scrambling

Information scrambling, the rapid delocalization of quantum information in many-body systems, is often quantified using out-of-time-ordered correlators (OTOCs) and Lyapunov exponents. In this work, we derive a generalized Cramer–Rao bound that unifies quantum metrology approaches and scrambling by directly relating OTOCs to the quantum Fisher information (QFI), establishing a geometric constraint on the speed of scrambling framework which leads to a Fisher-Lyapunov bound, showing the growth of QFI by the quantum Lyapunov exponent. Applying this framework to many-body systems that undergo quantum phase transitions, we show that the corresponding QFI metric exhibits universal scaling near quantum critical points. This critical enhancement explains the observed peaks in chaotic indicators at phase transitions. Together, these results reveal a unified picture which bridges information scrambling, critical phenomena and quantum metrology using the intrinsic geometry of quantum states.