  We consider the time evolution of the out-of-time-ordered correlator (OTOC)
of two general observables $A$ and $B$ in a mean field chaotic quantum system
described by a random Wigner matrix as its Hamiltonian. We rigorously identify
three time regimes separated by the physically relevant scrambling and
relaxation times. The main feature of our analysis is that we express the error
terms in the optimal Schatten (tracial) norms of the observables, allowing us
to track the exact dependence of the errors on their rank. In particular, for
significantly overlapping observables with low rank the OTOC is shown to
exhibit a significant local maximum at the scrambling time, a feature that may
not have been noticed in the physics literature before. Our main tool is a
novel multi-resolvent local law with Schatten norms that unifies and improves
previous local laws involving either the much cruder operator norm (cf. [G.
Cipolloni, L. Erd\H{o}s, D. Schr\"oder. Elect. J. Prob. 27, 1-38, 2022]) or the
Hilbert-Schmidt norm (cf. [G. Cipolloni, L. Erd\H{o}s, D. Schr\"oder. Forum
Math., Sigma 10, E96, 2022]).
