  We study the entanglement of a pure state of a composite quantum system
consisting of several subsystems with $d$ levels each. It can be described by
the R\'enyi-Ingarden-Urbanik entropy $S_q$ of a decomposition of the state in a
product basis, minimized over all local unitary transformations. In the case
$q=0$ this quantity becomes a function of the rank of the tensor representing
the state, while in the limit $q \to \infty$ the entropy becomes related to the
overlap with the closest separable state and the geometric measure of
entanglement. For any bipartite system the entropy $S_1$ coincides with the
standard entanglement entropy. We analyze the distribution of the minimal
entropy for random states of three and four-qubit systems. In the former case
the distributions of $3$-tangle is studied and some of its moments are
evaluated, while in the latter case we analyze the distribution of the
hyperdeterminant. The behavior of the maximum overlap of a three-qudit system
with the closest separable state is also investigated in the asymptotic limit.
