Probabilistic bound on extreme fluctuations in isolated quantum systems

TL;DR

This paper establishes probabilistic bounds on how likely an isolated quantum system can contract into a small subspace over time, providing insights into entropy fluctuations and quantum state evolution.

Contribution

It introduces a probabilistic framework to bound the maximum likelihood of quantum state contraction into a subspace over indefinite times.

Findings

Maximal probability approaches 1/2 for real initial states.

Maximal probability approaches π²/16 for complex initial states.

Contraction corresponds to roughly a twofold entropy reduction.

Abstract

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. 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 two, thus bounding large downward fluctuations in entropy from generic initial states.