Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states

TL;DR

This paper investigates the minimal Renyi-Ingarden-Urbanik entropy as a measure of entanglement in multipartite quantum states, analyzing its properties, distributions for random states, and connections to entanglement measures.

Contribution

It introduces and studies the minimal Renyi-Ingarden-Urbanik entropy for multipartite states, linking it to known entanglement measures and analyzing its distribution for random states.

Findings

For bipartite systems, S_1 equals standard entanglement entropy.

Distribution of minimal entropy for random three- and four-qubit states analyzed.

Maximum overlap with closest separable state studied asymptotically.

Abstract

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.