Abstract
The notion of operator growth in quantum systems furnishes a bridge between transport and the generation of entanglement between different parts of the system under quantum dynamics. We define a measure of operator growth in terms of subsystem operator R\'{e}nyi entropy, which provides a state-independent measure of operator growth, unlike entanglement entropies, and the usual measures of operator growth like out-of-time-order correlators. We show that the subsystem operator R\'{e}nyi entropy encodes both spatial and temporal information, and thus can directly connect to transport for a local operator related to a conserved quantity. We construct a unified Schwinger-Keldysh (SK) field theory formalism for the time evolution of operator R\'{e}nyi entropy and entanglement entropies of initial pure states. We use the SK field theory to obtain the operator R\'{e}nyi and state entanglement entropies in terms of infinite-temperature and vacuum Keldysh Green's functions, respectively, for non-interacting systems. We apply the method to explore the connection between operator and entanglement growth, and transport in non-interacting systems with quasiperiodic and random disorder, like the one- and two-dimensional Aubry-Andr\'{e} models and the two-dimensional Anderson model. In particular, we show that the growth of subsystem operator R\'{e}nyi entropy and state von Neumann and R\'{e}nyi entanglement entropies can capture both ballistic and sub-ballistic transport behaviors, like diffusive and anomalous diffusive transport, as well as localization in these systems.