  We imagine an experiment on an unknown quantum mechanical system in which the
system is prepared in various ways and a range of measurements are performed.
For each measurement M and preparation rho the experimenter can determine,
given enough time, the probability of a given outcome a: p(a|M,rho). How large
does the Hilbert space of the quantum system have to be in order to allow us to
find density matrices and measurement operators that will reproduce the given
probability distribution? In this note, we prove a simple lower bound for the
dimension of the Hilbert space. The main insight is to relate this problem to
the construction of quantum random access codes, for which interesting bounds
on Hilbert space dimension already exist. We discuss several applications of
our result to hidden variable, or ontological models, to Bell inequalities and
to properties of the smooth min-entropy.
