Probabilistic bound on extreme fluctuations in isolated quantum systems

We ask to what extent an isolated quantum system can eventually "contract" to be contained within a given Hilbert subspace. We do this by starting with an initial random state, considering the probability that all the particles will be measured in a fixed subspace, and maximizing this probability over all time. This is relevant, for example, in a cosmological context, which may have access to indefinite timescales. We find that when the subspace is much smaller than the entire space, this maximal probability goes to 1/2 for real initial wave functions, and to π^{2}/16 when the initial wave function has been drawn from a complex ensemble. For example, when starting in a real generic state, the chances of collapsing all particles into a small box will be less than but come arbitrarily close to 50%. This contraction corresponds to an entropy reduction by a factor of approximately 2, thus bounding large downward fluctuations in entropy from generic initial states.