Does three-tangle properly quantify the three-party entanglement for Greenberger-Horne-Zeilinger-type states?

TL;DR

This paper investigates whether the three-tangle is an accurate measure of three-party entanglement in GHZ-type states, revealing limitations and properties of the measure through analytical and geometric analysis of mixed states.

Contribution

The paper provides an analytical computation of three-tangle for specific mixed states and introduces the hyper-polyhedron concept to analyze entanglement properties.

Findings

States in the hyper-polyhedron have zero three-tangle.

The three-tangle of certain mixed states is explicitly computed and shown to vanish.

The monogamy inequality is verified for the studied states.

Abstract

Some mixed states composed of only GHZ states can be expressed in terms of only W-states. This fact implies that such states have vanishing three-tangle. One of such rank-3 states, $\Pi_{GHZ}$, is explicitly presented in this paper. These results are used to compute analytically the three-tangle of a rank-4 mixed state $\sigma$ composed of four GHZ states. This analysis with considering Bloch sphere $S^{16}$ of $d=4$ qudit system allows us to derive the hyper-polyhedron. It is shown that the states in this hyper-polyhedron have vanishing three-tangle. Computing the one-tangles for $\Pi_{GHZ}$ and $\sigma$, we prove the monogamy inequality explicitly. Making use of the fact that the three-tangle of $\Pi_{GHZ}$ is zero, we try to explain why the W-class in the whole mixed states is not of measure zero contrary to the case of pure states.