Operator size and error propagation: the Loschmidt echo in many-body open quantum systems

The dispersal of quantum information in many-body systems---i.e. quantum information scrambling---is a hallmark feature of many-body quantum dynamics. In recent years, quantum simulators have enabled direct experimental measurements of scrambling, motivating an essential open question: What is the effect of experimental error and decoherence, i.e. open quantum dynamics, on information scrambling? In this work, we provide a universal framework for the effect of local errors on scrambling, in which we predict that error propagation is determined nearly entirely by the scrambling dynamics themselves, not the specific form of error. We apply our framework numerically and analytically to a range of examples, and find that scrambling can be unaffected, slowed, stopped, or even reversed by error, depending on the dimensionality, integrability, and conservation laws of the system under study. Finally, within our framework, we demonstrate that this interplay between scrambling and error is responsible for the time profile of the Loschmidt echo decay in many-body systems, which provides a potential theoretical underpinning to recent nuclear magnetic resonance experiments.