  We adopt a continuous weak measurement tomography protocol to explore the
signatures of chaos in the quantum system(s). 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 chaos. We
find that the rate of information gain depends on the degree of chaos in the
dynamics, the choice of initial observable, and how well the operator is
aligned along the density matrix. The amount of operator spreading in the
Krylov subspace, as quantified by the fidelity in quantum tomography and
various other metrics of information gain, increases with the degree of chaos
in the system. We study operator spreading in many-body quantum systems by its
potential to generate an informationally complete measurement record. Our
quantifiers for operator spreading are more consistent indicators of quantum
chaos than Krylov complexity. Our study gives an operational interpretation for
operator spreading in terms of fidelity gain in quantum tomography.
  Continuing in our journey of finding the footprints of chaos in the quantum
domain, we explore the growth of errors in noisy tomography. For random states,
when the measurement record is obtained from a random operator, the subsequent
drop in the fidelity obtained is inversely correlated to the degree of chaos in
the dynamics. This gives us an operational interpretation of Loschmidt echo for
operators by connecting it to the performance of quantum tomography. We find a
quantity to capture the scrambling of errors, an out-of-time-ordered correlator
(OTOC) between two operators under perturbed and unperturbed dynamics that
serves as a signature of chaos. Our results demonstrate a fundamental link
between Loschmidt echo and scrambling of errors, as captured by OTOCs, with
operational consequences in quantum information processing.
