  We investigate whether the presence or absence of correlations between
subsystems of an N-partite quantum system is solely constrained by the
non-negativity and monotonicity of mutual information. We argue that this
relatively simple question is in fact very deep because it is sensitive to the
structure of the set of N-partite states. It can be informed by inequalities
satisfied by the von Neumann entropy, but has the advantage of being more
tractable. We exemplify this by deriving the explicit solution for N=4, despite
having limited knowledge of the entropic inequalities. Furthermore, we describe
how this question can be tailored to the analysis of more specialized classes
of states such as classical probability distributions, stabilizer states, and
geometric states in the holographic gauge/gravity duality.
