  Observational entropy - a quantity that unifies Boltzmann's entropy, Gibbs'
entropy, von Neumann's macroscopic entropy, and the diagonal entropy - has
recently been argued to play a key role in a modern formulation of statistical
mechanics. Here, relying on algebraic techniques taken from Petz's theory of
statistical sufficiency and on a Levy-type concentration bound, we prove
rigorous theorems showing how the observational entropy of a system undergoing
a unitary evolution chosen at random tends to increase with overwhelming
probability and to reach its maximum very quickly. More precisely, we show that
for any observation that is sufficiently coarse with respect to the size of the
system, regardless of the initial state of the system (be it pure or mixed),
random evolution renders its state practically indistinguishable from the
microcanonical distribution with a probability approaching one as the size of
the system grows. The same conclusion holds not only for random evolutions
sampled according to the unitarily invariant Haar distribution, but also for
approximate 2-designs, which are thought to provide a more physically
reasonable way to model random evolutions.
