Probabilistic bound on extreme fluctuations in isolated quantum systems

Understand

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 $\pi^2/16$ when the initial wave function has been drawn from a complex ensemble.