  Recent theoretical and experimental studies have shown significance of
quantum information scrambling (i.e. a spread of quantum information over a
system degrees of freedom) for problems encountered in high-energy physics,
quantum information, and condensed matter. Due to complexity of quantum
many-body systems it is plausible that new developments in this field will be
achieved by experimental explorations. Since noise effects are inevitably
present in experimental implementations, a better theoretical understanding of
quantum information scrambling in systems affected by noise is needed. To
address this problem we study indicators of quantum scrambling --
out-of-time-ordered correlation functions (OTOCs) in open quantum systems. As
most experimental protocols for measuring OTOCs are based on backward time
evolution we consider two possible scenarios of joint system-environment
dynamics reversal: In the first one the evolution of the environment is
reversed, whereas in the second it is not. We derive general formulas for OTOCs
in those cases as well as study in detail the model of a spin chain coupled to
the environment of harmonic oscillators. In the latter case we derive
expressions for open systems OTOCs in terms of Feynman-Vernon influence
functional. Subsequently, assuming that dephasing dominates over dissipation,
we provide bounds on open system OTOCs and illustrate them for a spectral
density known from the spin-boson problem. In addition to being significant for
quantum information scrambling, our results also advance understating of
decoherence in processes involving backward time evolution.
