  We study the equilibration properties of isolated ergodic quantum systems
initially prepared in a cat state, i.e a macroscopic quantum superposition of
states. Our main result consists in showing that, even though decoherence is at
work in the mean, there exists a remnant of the initial quantum coherences
visible in the strength of the fluctuations of the steady state. We back-up our
analysis with numerical results obtained on the XXX spin chain with a random
field along the z-axis in the ergodic regime and find good qualitative and
quantitative agreement with the theory. We also present and discuss a framework
where equilibrium quantities can be computed from general statistical ensembles
without relying on microscopic details about the initial state, akin to the
eigenstate thermalization hypothesis.
