  Developing quantum technologies requires the control and understanding of the
non-equilibrium dynamics of quantum information in many-body systems. Local
information propagates in the system by creating complex correlations known as
information scrambling, as this process prevents extracting the information
from local measurements. In this work, we develop a model adapted from
solid-state NMR methods, to quantify the information scrambling. The scrambling
is measured via time-reversal Loschmidt echoes (LE) and Multiple Quantum
Coherences experiments that intrinsically contain imperfections. Considering
these imperfections, we derive expressions for out-of-time-order correlators
(OTOCs) to quantify the observable information scrambling based on measuring
the number of active spins where the information was spread. Based on the OTOC
expressions, decoherence effects arise naturally by the effects of the
nonreverted terms in the LE experiment. Decoherence induces localization of the
measurable degree of information scrambling. These effects define a
localization cluster size for the observable number of active spins that
determines a dynamical equilibrium. We contrast the model's predictions with
quantum simulations performed with solid-state NMR experiments, that measure
the information scrambling with time-reversal echoes with controlled
imperfections. An excellent quantitative agreement is found with the dynamics
of quantum information scrambling and its localization effects determined from
the experimental data. The presented model and derived OTOCs set tools for
quantifying the quantum information dynamics of large quantum systems (more
than $10^{4}$ spins) consistent with experimental implementations that
intrinsically contain imperfections.

Quantum Physics;Mesoscale and Nanoscale Physics