Refining the Understanding of Operator Size Dynamics in Open Quantum Systems

Information scrambling refers to the phenomenon in which local quantum information in a many-body system becomes dispersed throughout the entire system under unitary evolution. It has been extensively studied in closed quantum systems, where it is quantified by operator size growth, revealing deep connections between condensed matter physics, high-energy physics, and quantum information. However, when extending the study of operator size dynamics to open quantum systems, two different definitions of operator size distributions emerge. These definitions are based on different treatments of the bath. In this work, we aim to establish a unified picture for operator size dynamics in open quantum systems, using the solvable Brownian SYK models at generic system size. In particular, we provide the conditions under which the signature of scrambling transition, discovered using one particular definition, appears in operator size dynamics under the other definition. Additionally, we extend previous studies by exploring finite-size effects that are not captured by the scramblon theory. Our results provide a refined understanding of operator size dynamics in open quantum systems.