Studying and measuring scrambling without out-of-time-ordered correlators

Scrambling in many-body systems refers to the spreading of initially localized information to the entire system and has become a key concept in the study of chaos in quantum systems. Most studies so far have focused on characterizing scrambling using out-of-time-ordered correlation functions (OTOCs), particularly through its early-time decay. However, scrambling is a complex process which involves operator spreading and growth, and a full characterization requires accessing more refined information on the operator dynamics at several timescales. Moreover, OTOCs are intrinsically hard to access experimentally, typically requiring time reversal of the dynamics or the use of auxiliary systems. In this work we study operator scrambling by expanding the target operator in a complete basis and studying the structure of the expansion coefficients treated as a probability distribution in the space of operators. We propose a new approach that allows us to obtain information about scrambling without time reversal nor ancillas, while also avoiding the arduous task of reconstructing the expansion coefficients directly. This is done by using the dynamics of random mixed states and only local measurements together with randomized sampling. We demonstrate our findings by studying diverse cases such as the tilted field Ising model undergoing both regular and chaotic dynamics, circuit models with varying non-Cliffordness, and the kicked collective spin models.