  Mathematical models are an essential component of quantitative science. They
generate predictions about the future, based on information available in the
present. In the spirit of Occam's razor, simpler is better; should two models
make identical predictions, the one that requires less input is preferred. Yet,
for almost all stochastic processes, even the provably optimal classical models
waste information. The amount of input information they demand exceeds the
amount of predictive information they output. We systematically construct
quantum models that break this classical bound, and show that the system of
minimal entropy that simulates such processes must necessarily feature quantum
dynamics. This indicates that many observed phenomena could be significantly
simpler than classically possible should quantum effects be involved.
