A hierarchy of entanglement criteria for four qubit symmetric Greenberger-Horne-Zeilinger diagonal states

TL;DR

This paper develops a hierarchical set of necessary and sufficient entanglement criteria for four-qubit symmetric GHZ-diagonal states using a two-step optimization method, enhancing entanglement detection accuracy.

Contribution

It introduces a novel hierarchy of four criteria, combining linear and nonlinear conditions, with a nested structure tailored for symmetric GHZ-diagonal states.

Findings

The criteria set is necessary and sufficient for certain state subsets.

The criteria exhibit a nested structure applicable to Werner and GHZ diagonal states.

The method improves entanglement detection precision for symmetric states.

Abstract

With a two step optimization method of entanglement witness, we analytically propose a set of necessary and sufficient entanglement criteria for four qubit symmetric Greenberger-Horne-Zeilinger (GHZ) diagonal states. The criterion set contains four criteria. Two of them are linear with density matrix elements. The other two criteria are nonlinear with density matrix elements. The criterion set has a nest structure. A proper subset of the criteria is necessary and sufficient for the entanglement of a proper subset of the states. We illustrate the nest structure of criterion set with the general Werner state set and its superset the highly symmetric GHZ diagonal state set, they are subsets of the symmetric GHZ diagonal state set.