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 Rényi–Ingarden–Urbanik entropy Sq 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 → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 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 distribution of the three-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.