  Does chaos in the dynamics enable information gain in quantum tomography or
impede it? We address this question by considering continuous measurement
tomography in which the measurement record is obtained as a sequence of
expectation values of a Hermitian observable evolving under the repeated
application of the Floquet map of the quantum kicked top. For a given dynamics
and Hermitian observables, we observe completely opposite behavior in the
tomography of well-localized spin coherent states compared to random states. As
the chaos in the dynamics increases, the reconstruction fidelity of spin
coherent states decreases. This contrasts with the previous results connecting
information gain in tomography of random states with the degree of chaos in the
dynamics that drives the system. The rate of information gain and hence the
fidelity obtained in tomography depends not only on the degree of chaos in the
dynamics and to what extent it causes the initial observable to spread in
various directions of the operator space but, more importantly, how well these
directions are aligned with the density matrix to be estimated. Our study also
gives an operational interpretation for operator spreading in terms of fidelity
gain in an actual quantum information tomography protocol.
