  Quantum chaos imposes universal spectral signatures that govern the
thermofield dynamics of a many-body system in isolation. The fidelity between
the initial and time-evolving thermofield double states exhibits as a function
of time a decay, dip, ramp and plateau. Sources of decoherence give rise to a
nonunitary evolution and result in information loss. Energy dephasing gradually
suppresses quantum noise fluctuations and the dip associated with spectral
correlations. Decoherence further delays the appearance of the dip and shortens
the span of the linear ramp associated with chaotic behavior. The interplay
between signatures of quantum chaos and information loss is determined by the
competition among the decoherence, dip and plateau characteristic times, as
demoonstrated in the stochastic Sachdev-Ye-Kitaev model.
