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Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states

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 $S_q$ 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.