Precise detection of multipartite entanglement in four-qubit Greenberger--Horne--Zeilinger diagonal states

TL;DR

This paper introduces a method to precisely detect multipartite entanglement in four-qubit GHZ diagonal states using optimized entanglement witnesses, providing necessary and sufficient criteria for tripartite separability.

Contribution

The authors develop a set of tripartite separability criteria for four-qubit GHZ diagonal states, improving detection accuracy and fully characterizing the separable state boundary.

Findings

Criteria are necessary and sufficient for tripartite separability.

The criteria match numerically obtained boundaries.

One criterion applies to generalized Werner states.

Abstract

We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger--Horne--Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.