  We study operator spreading in many-body quantum systems by its potential to
generate an informationally complete measurement record in quantum tomography.
We adopt continuous weak measurement tomography for this purpose. We generate
the measurement record as a series of expectation values of an observable
evolving under the desired dynamics, which can show a transition from
integrability to complete chaos. We find that the amount of operator spreading,
as quantified by the fidelity in quantum tomography, increases with the degree
of chaos in the system. We also observe a remarkable increase in information
gain when the dynamics transitions from integrable to nonintegrable. We find
our approach in quantifying operator spreading is a more consistent indicator
of quantum chaos than Krylov complexity as the latter may
correlate/anti-correlate or show no explicit behavior with the level of chaos
in the dynamics. We support our argument through various metrics of information
gain for two models: the Ising spin chain with a tilted magnetic field and the
Heisenberg XXZ spin chain with an integrability-breaking field. Our paper gives
an operational interpretation for operator spreading in quantum chaos.
